GROUPS GENERATED BY INVOLUTIONS,
GELFAND-TSETLIN PATTERNS AND
COMBINATORICS OF YOUNG TABLEAUX
ANATOL N. KIRILLOV
RIMS, Kyoto University,
Kyoto 606, Japan
Steklov Mathematical Institute,
Fontanka 27, St.Petersburg, 191011, Russia
and
ARKADIY D. BERENSTEIN
Council on Cybernetics,
Russian Academy of Sciences,
Moscow, Russia
ABSTRACT
We construct families of piecewise linear representations (cpl-representations) of the
symmetric group Sn and the ane Weyl group Sen of type A(1)
n;1 acting on the space
of triangles Xn . We nd a nontrivial family of local cpl-invariants for the action of
the symmetric group Sn on the space Xn and construct one global invariant w.r.t. the
action of the ane Weyl group Sen (so called cocharge). We nd the continuous analogs
for the Kostka-Foulkes polynomials and for the crystal graph. We give an algebraic
version of some combinatorial transformations on the set of standard Young tableaux.
Introduction.
In this paper we dene and study a new class of representations of the symmetric group Sn , namely, the continuous piecewise linear representations (cplrepresentations) of Sn in the space of triangles Xn. By denition, a triangle x 2 Xn
is a triangular array of real numbers x = (xij ) xij 2 R 1 i j n. More
precisely, following GZ1],
GZ2],
BZ1], we consider the Gelfand-Tsetlin cone Kn
consisting of all triangles x 2 Xn such that
xij xi+1j+1 1 i j n ; 1
xij xij;1 1 i j n
xij 0 1 i j n:
nn
, having
This is a nondegenerate convex polyhedral cone in the space Xn =R
n
2 generators (see Remark 2.2). It is well known (e.g. GZ1]) that the integral
( +1)
2
2 Anatol N. Kirillov and Arkadiy D. Berenstein
points set, (Kn)Z , of the cone Kn is in a one-to-one correspondence with the set
STY (n) of standard Young tableaux, having all entries not exceeding n. The cplaction of the symmetric group Sn on the space Xn , given in our paper, is such that
it conserves the Gelfand-Tsetlin cone Kn and that on the set STY (n) it coincides
with the action of the symmetric group on the set of standard Young tableaux given
by A. Lascoux and M.-P. Schutzenberger LS2], LS3] (see Theorem 2.3).
Our main observation is that a great many of combinatorial constructions on
the set of standard Young tableaux, e.g. the Schutzenberger involution Sch1],
EG],
Ki1], the dual Schutzenberger involution Sch1], a promotion transformation Sch1],
EG], the action of the symmetric group LS2],
LS3], the crystal graph structure on
the set STY ( n), Ka1],
Ka2], the construction of cocharge LS1],
Ki1], may be
transfered to the Gelfand-Tsetlin cone Kn , and even to the whole space of triangles
Xn . Our constructions are based on a consideration of \elementary transformations" tj 1 j n ; 1, and T 2 R.
Denition 0.1 Assume that x 2 Xn , then tj (x) = xe T (x) = ex, where
xeik = xik if k 6= j
xeij = min(xij+1 xi;1j;1 ) + max(xij;1 xi+1j+1) ; xij (0:1)
and we presuppose that x0j := +1 xjj;1 := ;1 1 j n ; 1
exik = xik if (i k) 6= (1 1)
ex11 = x11 ; :
We denote by Gn =< t1 tn;1 > the group generated by ti 1 i n ; 1.
The transformations ti 1 i n ; 1, satisfy the following relations (see Corollary
1.1):
(i) t2i = 1 ti tj = tji if jj ; ij 2
(ii) (t1t2 )6 = 1
(ii) (t1qi )4 = 1 if 3 i n ; 1
(0:2)
where qi := t1 |{z}
t2 t1 t|3{z
t2t1} t|i ti;{z
1 t1 :
}
The restriction of the involutions ti to the set of standard Young tableaux
STY ( ) of a given shape and content admits a simple combinatorial interpretation. It is these restrictions that are ordinary used in order to prove that
Schur functions are the symmetric, e.g. BK],
SW],
Sa] and Section 2A. We assume
that the relations (0.2) are the dening relations for the group Gn . By any way,
the group Gn seems to be very interesting. It is easy to see that the order of the
group G3 is equal to 12. But if n 4 then Gn is innite and for any N there exist
an epimorphism of the group G4 on the symmetric group SN (see comments after
Corollary 1.3). The group Gn admits an extension Gen by means of R1 :
Gen :=< t1 tn;1 T 2 R > :
Combinatorics of the Gelfand-Tsetlin patterns
3
We have the following relations between the generators in the group Gen :
(i) T T = T+ (ii) (t1 T )2 = 1 tj T tj = T 3 j n ; 1 2 R
(iii) ti tj = tj ti if jj ; ij 2
(iv) (T t2t1 T t2 t1)3 = 1 for any 2 R
(v) T t2 T;+ t2 T; t2 = t2T t2 T;+ t2T; (vi) (t1T qj t1 T qj )2 = 1 3 j n ; 1
(vii) T qj T qj = qj T qj T 3 j n ; 1 2 R
(0:3)
where transformation qj is dened in (0.2).
The proofs of the relations (i)-(v) are based on direct computations. The main
diculties arise in the proof of the (vi). In order to understand better the relations
(0.2) and (0.3), let us consider the following elements in the group Gen :
si := qi t1 qi;1 1 i n ; 1
si = qi T; qi;1 1 i n ; 1
(0:4)
(0:5)
The main result of Section 1 is Theorem 1.1, which is equivalent to the relations
(i)-(vi), and asserts that
10. The involutions si 1 i n ; 1, satisfy the relations of the symmetric
group Sn , i.e.
a)
b)
c)
s2i = 1 1 i n ; 1
(si si+1)3 = 1 1 i n ; 2
si sj = sj si if ji ; j j 2:
(0:6)
20. The transformations s(i) 1 i n ; 1 2 R, satisfy the colored braid
relations, i.e. for any 2 R we have
a)
b)
c)
s(i) s(i) = s(i+) 1 i n ; 1
+) s() = s() s(+) s() 1 i n ; 2
s(i) si(+1
i
i+1 i
i+1
()
( )
( )
()
si sj = sj si if ji ; j j 2:
(0:7)
The relation (0.3),(vi) is equivalent to the statement that the transformations s(i)
and sj() commute if ji ; j j 2. In oder to prove the last statement, we use another
expression for s(i) 1 i n ; 1, as a product of the Lusztig involutions Lu2]. We
recall the corresponding denitions, because we use not exactly the same involutions
that contained in Lu2], but their analogs for the space of triangles Xn.
4 Anatol N. Kirillov and Arkadiy D. Berenstein
Denition 0.2. For each triple of integers (ijk) 1 i < j < k n, let us
dene a transformations Rijk : Xn ! Xn in the following manner
Rijk (x) = xe x 2 Xn where
xeij = xij + xik ; xik;1 ; min(xjk ; xjk;1 xij ; xij;1 )
xejj = xjj ; xik + xik;1 + min(xjk ; xjk;1 xij ; xij;1 )
xe = x if (
) 6= (i j ) or (j j ):
(0:8)
Let us denote by Ln the group generated by all Rijk with 1 i < j < k n.
We have the following relations between the generators in the group Ln :
(i) (Rijk )2 = 1
(ii) Rijk Ri j k = Ri j k Rijk if j(ijk) \ (i0 j 0 k0)j 6= 2
(iii) (Rijk Rijl Rikl Rjkl )2 = 1
(iv) (Rijl Rikl )3 = 1 if 1 i < j < k < l n:
0 0
0
0 0
0
(0:9)
We assume that the relations (0.9) are the dening ones for the group Ln .
To go further, let us dene the transformations T(i) acting on the space Xn :
T(i) = xe x 2 Xn where
xeii = xii ; xe = x if (
) 6= (i j ):
The statement that s(i) and s(j) commute if ji ; j j 2, follows from the following
expression for s(i) 1 i n ; 1, (Theorem 1.2):
s(i) = Ri;1i Ri;2i R1i T(i) R1i Ri;2iRi;1i (0:10)
where Rjk := Rjkk+1 1 j < k n ; 1. The proof of identity (0.10) is based on
induction and on the recurence formula for s(i) (see (1.29)).
In order to obtain the corresponding results for the involutions si (see (1.28a)),
we use
Crucial Lemma. Assume x 2 Xn (x) = (1 n ), then
si (x) = s(ii;i ) (x)
+1
(0:11)
where (x) is a weight of the triangle x 2 Xn , i.e. i (x) := jx(i) j ; jx(i;1) j
1 i n x(0) := :
The relations (0.3) between generators of the group Gen allow us to construct
many interesting subgroups in Gen . For example
10. Let 2 R be xed, then the elements s1s(1) s2s(2) sn;1 s(n;) 1 are the
standard generators of the symmetric group Sn .
Combinatorics of the Gelfand-Tsetlin patterns
5
20. Let 0 1 n;1 be the real numbers, 0 6= 0, and sei := se(ii) = si s(ii) for
i = 1 n ; 1. Let us put
se0 := se(0 ) = sen;1 sen;2 se1s(1 ) se2 sen;1 :
0
0
Then se0 es1 esn;1 are the standard generators of the ane Weyl group of the
type A(1)
n;1 . It is important that the cocharge cn (x) of a triangle x 2 Xn (Section 3)
is an invariant w.r.t. the action of any involutions se(ii ) 0 i n ; 1.
Let us summarize the content of the Section 1 of our paper. We construct a
family of the cpl-representations of the symmetric group, nd a family of the cplrepresentations of the ane Weyl group of the type A(1)
n;1 (see item after Corollary
1.3) and construct a cpl-representation of the colored braid relations. We developed
a geometric techniques for proving some (non trivial!) identities between piecewise
linear functions (see Theorem 1.3). Our next step is to give a combinatorial interpretation of the transformations under consideration and to study the continuous
piecewise linear invariants w.r.t. the action of the symmetric group generated by
s1 sn;1 , on the space of triangles Xn . In Section 3 we prove (see Theorem 3.2)
that the following cpl-functions on the space of triangles Xn are sj -invariants:
1j (x) = min(x1j+1 ; x1j xjj ; xj+1j+1 )
ij (x) = min(xi;1j ; xi;1j;1 xij+1 ; xij )
(0:12)
where 2 i j n ; 1
1j (x) = (min(xj;1j + xjj ; xj;1j;1 ; xjj+1
x1j+1 + xj+1j+1 ; x1j ; xjj ))+ 2j (x) = (min(x1j + xjj ; x1j+1 ; xj+1j+1
x1j;1 + x2j+1 ; x1j ; x2j ))+ ij (x) = (min(xi;2j + xi;1j ; xi;2j;1 ; xi;1j+1 xi;1j;1 + xij+1 ; xi;1j ; xij ))+ (0:13)
where 3 i j n ; 1, and for any a 2 R (a)+ := max(a 0). We recall that
a cpl-function ' dened on the space Xn is said to be an sj -invariant if '(sj (x)) =
'(x) for all x 2 Xn .
It seems a very interesting task to nd a fundamental system Ij of cpl-invariants
for sj , that is a system such that any cpl-invariant under the action of sj is a
min ; max linear combination of that from Ij . It is not clear whether or not this
set is nite, because we have the trivial examples of the following kind
xik if k =
6 j or min('(x) '(sj (x)))
where '(x) is any cpl-function on Xn . However, the number of fundamental cpl-sj invariants having a complexity bounded by some integer k, is nite. Here we dene
6 Anatol N. Kirillov and Arkadiy D. Berenstein
the complexity of a cpl-function as the least number of min and max needed for
its representation. Remark. In a previous denition of complexity we considered
min and max as functions of two variables, e.g. min : R R ! R. Thus, the
complexity of the function min(x1 x2 xm) is equal to m ; 1]. By any way, we
may construct (see the proof of Theorem 3.2) other nontrivial examples of cpl-sj invariants and it is not clear whether or not they are independent (in cpl-sense)
from the invariants (0.12) and (0.13).
The next interesting problem is to describe the fundamental system of continuous
piecewise polynomial functions (cpp-functions) on the space Xn, which are invariant
w.r.t. the action of the subgroup Sn Gn . Of course, we
have such trivial example
P
as: xin 1 i n or minf '((x)) j 2 Sn g, or 2Sn '((x)), where '(x)
is any cpp-function on Xn . But the question is the following one: does there exist
nontrivial cpl-invariants under the action of the symmetric group Sn on the space
Xn ? In Section 3 we give an armative answer to this question (see Theorem 3.1).
In fact we dene a stable (ibid), cpl-invariant w.r.t. the action of a family of the
ane Weyl groups of type A(1)
n;1 , namely, Sen :=< se0 se1 esn;1 >, see Corollary
1.3, which takes nonnegative values on the Gelfand-Tsetlin cone Kn. Note that
the complexity of the invariant under consideration is equal to (n ; 1)! ; 1, if
n 3. An origin of such invariant is very clear. We have a distinguished Sn invariant function on the set STY (n), namely, the cocharge cn (T ) of a tableau
T 2 STY (n) as dened by A. Lascoux and M.-P. Schutzenberger LS1]. Note that
it is possible (at least for n 5 and hypothetically for all n) to reformulate the
denition of the cocharge of a tableau T 2 STY (n) in terms of the corresponding
Gelfand-Tsetlin pattern x(T ) and obtain some function cn on the set (Kn )Z . We
consider an extension of the function cn to the whole space Xn in a natural manner,
changing its domain of denition from (Kn)Z to Xn . This gives us the desired cplSen -invariant, cn (x) x 2 Xn , which we still call cocharge. Details are contained in
Section 3. We dene another cpl-Sn -invariant (x) (which does't Sen -invariant!) by
setting (x) = cn (qn;1 (x)) x 2 Xn , where the involution qn;1 is dened in (0.2).
As an example we give an expression for cocharge c4:
c4(x) = min(x13 ; x12 x22 ; x33) + x14 ; x13 + x33 ; x44 +
+min(x23 ; x34 x24 ; x23 x13 ; x12 x22 ; x33 4 (x) ; x34 x24 ; 4 (x)):
It is known (e.g. H]) that the volume of the convex polytope K ( ) consisting of
all Gelfand-Tsetlin patterns with highest weight and weight (see Section 1) may
be considered as a continuous analog of the weight multiplicity K := dimV ( ).
We dene a continuous analog of the Kostka-Foulkes polynomial (q-analog of the
weight multiplicity) LS1],
Ma],
Ki1],by means of following integral
K (q) =
Z
K ( )
exp(hcn (x))dx q = exp(h):
(0:14)
We expect to study the properties of the integral (0.14) and the toric variety corresponding to the Gelfand-Tsetlin cone Kn in a separate publication. In Remark
Combinatorics of the Gelfand-Tsetlin patterns
7
3.2 we give a generalization of the Lascoux-Schutzenberger algorithm for computing
the charge of a dominant weight standard Young tableau to the case of a standard
Young tableau with an arbitrary weight.
In Section 2 we study the restrictions of the transformations considered in Section 1 to the integral points set (Kn)Z of the Gelfand-Tsetlin cone Kn . We show
that the involution qi : STY (n) ! STY (n) 1 i n ; 1, (see (0.2)) coincides
with the partial Schutzenberger involution Si : STY (n) ! STY (n). Here for a
given tableau T 2 STY (n), the involution Si acts non trivially only on the part
Ti+1 of the tableau T lling by the numbers 1 i +1, and on this part coincides
with the ordinary Schutzenberger's involution Sch1] or Section 2C. Let us note that
the group generated by the involutions qi 1 i n ; 1, coincides with the group
Gn =< t1 tn;1 >, and thus we may describe the relations between the partial
Schutzenberger involutions (see Remark 1.3). Further we have the following relation
between the partial Schutzenberger involution qi and the action of the symmetric
group Sn on the set STY (n) as dened by A. Lascoux and M.-P. Schutzenberger
Sch2],
Sch3]:
si = qi q1 qi 1 i n ; 1
(0:15)
where si := (i i + 1) 2 Sn is a simple transposition.
The Schutzenberger involution S posesses many interesting properties in connection with the Robinson-Schensted correspondence Sch2], with a rigged congurations Ki1] and so on. In addition we explain in Remark 2.4 that the involution S
allows to give a simple pure combinatorial proof of the following symmetry property
of the Littlewood-Richardson numbers (see Proposition 2.8)
c
= c
:
(0:16)
All other symmetries of the LR-numbers follow from (0.16) and the symmetries of
the Berenstein-Zelevinsky triangles (see BZ3] and Remark 2.4).
Finally, it is interesting to note the following connection between our transformations si(1) 1 i n ; 1, restricted on the integral points set of the convex
polytope K (see Section 1), and the crystal graph corresponding to the irreducible
representation V of the Lie algebra gln with the highest weight , Ka1],
Ka2],
KN].
Namely, let us denote by fi (respectively ei ) the restriction of the map s(i;1)
(1)
(resp. s(+1)
does't
i ) on the GT-polytope K Remark: the transformations si
(;1)
(;1)
conserve GT-cone Kn , so for x 2 Kn we dene fi (x) = si (x), if si (x) 2 Kn
and fi (x) = 0, if s(i;1) (x) 62 Kn ]. Let further (L() B ()) be the crystal base of
an irreducible gln -module V with highest weight and Fei Eei 1 i n ; 1, be
the deformed generators of the Hopf algebra Uq (gln ) as dened by M. Kashiwara
Ka1], Ka2]. Then there exist a bijection
N : B () ! K \ (Xn )Z
8 Anatol N. Kirillov and Arkadiy D. Berenstein
such that
(i) b 2 B () := B () \ V ( ) i N (b) 2 K ( ) \ (Xn)Z (ii) if Fei (b) 2 B (() Fei (b) 6= 0) then N (Fei (b)) = s(i;1) (N (b))
(iii) if Eei (b) 2 B (() Eei (b) 6= 0) then N (Eei(b)) = s(+1)
i (N (b)):
The classical representation theory of the symmetric and general linear groups
is based on dierent combinatorial constructions among which the ones with Young
tableaux play an essential role. The main reason is that Young tableaux in a
natural way parametrize a basis of irreducible representation of the symmetric or
general linear groups Ru],
JK],
St]. In some sense the choice of a concrete realization of an irreducible representation predetermine the corresponding combinatorial
structures. If we consider the realization of irreducible representations of the general linear group in the space of the Gelfand-Tsetlin patterns, GZ1], then we deal
with combinatorics of convex polytopes of a special kind. If we consider the realization of representations of the symmetric (or general linear) group by means of
Specht (or Weyl) module, JK], then we deal with combinatorics of Young tableaux.
While the combinatorics of Young tableaux is extensivly developed (see e.g. Sa]),
the combinatorics of the Gelfand-Tsetlin patterns seems only began to be studied
GZ1],
GZ2],
BZ1].
The main goal of our paper is to show that the majority of combinatorial constructions over Young tableaux admits an \algebraization" and may be dened on
the space of triangles. Of course we did not exhaust the subject and we believe that
other combinatorial constructions, e.g. Robinson-Schensted correspondence (e.g.
Sch2]), also admit natural continuous analogs.
Acknowledgements. We are very grateful to many people who encouraged us
on the dierent step of this work. We are obliged to L.D. Faddeev, I.M. Gelfand,
M.-P. Schutzenberger, A.V. Zelevinsky, I.V. Cherednik, D. Foata, A. Lascoux,
M.Kashiwara, P. Mathieu, T. Miwa and I. Pak for interesting discussions and useful
comments.
One of us (AK) gratitude goes to Dr.A. Schnizer for help in a computation of
the examples of the action t2 t3 s4 s5 , which lead to better understanding of
the transformations under consideration, J.H.H. Perk for help in preparing English
text, and also thank with much gratitude the colleagues at Kyoto University and
RIMS for their invitation, their hospitality which made it possible to nish this
work. I would like to acknowledge my special indebtedness to Dr. N.A. Liskova for
the inestimable help in preparing the manuscript to publication.
Combinatorics of the Gelfand-Tsetlin patterns
9
x1. Groups acting on the space of triangles.
Let n be a positive integer. In this section we dene the group Gn generated
by involutions and Gen its extension by means of R1, which acts on a space of
triangles Xn . By denition the space Xn consists of all sequences
x = (x(n) x(n;1) x(1))
n(n+1)
where x(j) = (x1j xjj ) 2 Rj . As a vector space Xn ' R
. We will call
(n)
n
the vector x 2 R the highest weight of the triangle x 2 Xn and denote it by
(x) := x(n) . Let us dene a weight := (x) of a triangle x 2 Xn as a vector
= (1 n ) 2 Rn such that
2
P
j = jx(j) j ; jx(j;1) j 1 j n
(1:1)
where jx(j) j := ji=1 xij x(0) = . For given vectors 2 Rn and 2 Rn we dene
the following subspaces of the space Xn:
X = fx 2 Xn j (x) = g
X ( ) = fx 2 X j (x) = g:
(1:2)
Now we dene the Gelfand-Tsetlin patterns GZ1],
GZ2],
BZ1],
BZ2]. By definition a triangle x 2 Xn is called a Gelfand-Tsetlin pattern (GT-pattern) i the
following inequalities are satised
xij xi+1j+1 1 i j n ; 1
xij xij;1 1 i < j n
xij 0 1 i j n:
(1:3)
We denote by the set of all GT-patterns K = Kn Xn and those which
lie in the subspaces (1.2) correspondingly by K and K ( ). It is clear that K nn
and K ( ) are convex, compact polytopes in the space R+ . It is well known
(see x2, or GZ2]), that if is a partition, and is a composition, then the number
of integral points in the convex polytope K (respectively in K ( )) is equal to
the dimension of the irreducible representation V of the Lie algebra gln with the
highest weight (respectively, the dimension of the subspace V ( ) V of the
weight ). On the other side, as is also well known, the dimension of the weight
subspace V ( ) V admits a pure combinatorial description as the number of
standard Young tableaux of shape and content :
( +1)
2
jK ( ) \ Z
n(n+1)
2
j = dimV ( ) = jSTY ( )j:
(1:4)
The equality (1.4) is a starting point of our investigations. We will try to
translate the combinatorial information about the set of standard Young tableaux
STY ( ) into the language of GT-patterns and vice versa.
10 Anatol N. Kirillov and Arkadiy D. Berenstein
Now let us begin the construction of the main objects of this note: the group Gn
and its extension Gen . At rst we dene the \elementary" transformations tj of the
space of triangles Xn . For this purpose, let us x a positive integer j 1 j < n,
and introduce the sequences of numbers a1 aj and b1 bj . Given a triangle
x = (x(n) x(1)) 2 Xn , we dene
a1 = x1j+1 ai = min(xij+1 xi;1j;1) 2 i j
bj = xj+1j+1 bi = max(xij;1 xi+1j+1) 1 i j ; 1:
(1:5)
Denition 1.1. The transformation tj : Xn ! Xn is given by the following
formulae
tj (x) = xe where
xeij = ai + bi ; xij 1 i j
xekl = xkl if l 6= j:
(1:6)
Proposition 1.1. We have
a) t2j = 1 1 j n ; 1
b) ti tj = tj ti if ji ; j j 2
c) (tj (x)) = (j j + 1) (x) 1 j n ; 1
(1:7)
(1:8)
(1:9)
where the action of transposition (j j + 1) on the weight space Rn is given by
(j j + 1)(1 n ) = (1 j+1 j n ).
Proof. The assertions a) and b) follow directly from the denition (1.6) of the
transformation tj . As for c), let us remark, that
So
bi + ai+1 = xij;1 + xi+1j+1 1 i j ; 1:
jxe(j ) j =
X
i
(ai + bi ) ; jx(j) j = jx(j+1) j ; jx(j)j + jx(j;1) j:
Consequently,
and
j+1 (xe) = jx(j+1) j ; jxe(j) j = jx(j) j ; jx(j;1) j = j (x)
j (xe) = jxe(j)j ; jx(j;1) j = jx(j+1) j ; jx(j) j = j+1 (x):
Now let us consider the restriction of the action of tj on the Gelfand-Tsetlin
polytopes. Fix vectors and in the space Rn .
11
Combinatorics of the Gelfand-Tsetlin patterns
Proposition 1.2. We have for 1 j n ; 1
a) tj : Kn ! Kn (1:10)
b) tj : K ! K tj : K ( ) ! K ((j j + 1) ):
(1:11)
Proof. It is sucient to show that the involutions tj conserve the inequalities
(1.3). Consider \the elementary neighborhood" of x 2 Kn :
a
xij
c
a
tj
;!
min(a b) + max(c d) ; xij
b
d
b
It is clear that (x := xij ) min(a b) x max(c d). So
a min(a b) min(a b) ; (x ; max(c d))
(min(a b) ; x) + max(c d) c:
So all necessary inequalities are satised.
We denote by Gn the group generated by the involutions t1 tn;1 :
c
d
Gn =< t1 tn;1 > :
(1:12)
This group acts on the space of triangles Xn and for any xed vector 2 Rn+ transforms the convex polytope K into itself and interchanges the polytopes K ( ).
The group Gn may be embedded in a bigger group Gen . In order to construct
such an extension, we dene a transformation T of the space Xn in the following
way: assume 2 R1 and x 2 Xn, then let us put T (x) = xe, where xe(j) = x(j) , if
2 j n, and xe(1) = x11 ; . We denote by Gen the group generated by Gn and
T 2 R1:
Gen =< t1 tn;1 T 2 R1 > :
(1:13)
The next proposition gives the description of some relations between the generators ti 1 i n ; 1, and T 2 R1.
Proposition 1.3. We have
a) t1 T = T; t1 or equivalently (t1 T )2 = 1
b) ti T = T ti 3 i n ; 1
c) (t1t2 )6 = 1
d) ti tj ] = 0 if ji ; j j 2
e) t2 T t2 T;+ t2 T; = T t2 T;+ t2 T; t2:
(1:14)
f ) (T t2 t1 T t2t1 )3 = 1 2 R1 :
Proof. The assertions a), b) and d) are evident.
12 Anatol N. Kirillov and Arkadiy D. Berenstein
c) It is sucient to show that
(t1t2 )3 = (t2t1 )3 : X3 ! X3:
By direct computation we nd t2t1 (x) = xe, where
xe12 = x13 ; x12 + max(x23 2 (x))
xe22 = x33 ; x22 + min(x23 2(x))
2 (xe) = 3 (x) xe11 = 2 (x)
xe13 = x13 xe23 = x23 xe33 = x33:
Consequently, (t2t1 )3 (x) = xe, where xe = x , if (
) 6= (1 2) or (2,2), and
xe12 = x13 ; x12 ; max(x23 3 (x)) + max(x23 2(x)) + max(x23 1(x))
xe22 = x33 ; x22 ; min(x23 3(x)) + min(x23 2(x)) + min(x23 1(x)): (1:15)
Similarly, we nd t1 t2 (x) = x, where
x12 = x13 ; x12 + max(x23 1 (x))
x22 = x32 ; x22 + min(x23 1(x))
x11 = 3 (x) 3 (x) = 2 (x)
x3 = x3 = 1 2 3:
Using these formulae, it is easy to see that for (t1 t2)3 (x) we obtain the same expression (1.15).
e) Let us dene t2 := T t2T; t2 2 R1 . It is sucient to show that
t2t2 = t2 t2 : X3 ! X3 for any 2 R1. Given x 2 X3 , the following relations hold
x (t2 (x)) = x if (
) 6= (1 2) or (2 2)
x12 (t2(x)) = x12 + max(x23 x11 + ) ; max(x23 x11)
x22 (t2(x)) = x22 + max(x23 x11 + ) ; max(x23 x11):
Consequently, x12(t2 t2(x)) =
= x12 + max(x23 x11 + ) + max(x23 x11 + ) ; 2max(x23 x11) = x12(t2 t2(x)):
By the same reasoning
x22(t2 t2(x)) = x22 (t2t2 ):
Combinatorics of the Gelfand-Tsetlin patterns
13
f) It is sucient to prove (1.14,f) for n = 3. Let us consider a correspondence
: X3 ! X3 , given by
:
x13
x12
x23
x11
x22
x33
;!
x13 + + x23 + x33
x12 + + x22 + x11 + + It is clear that is an automorphism of X3. The following identities, which may
be veried by a direct computation, reduce the proof of (1.14 f) to that of point c):
;1 t1 T = t1 ;1 T; t2 t1 T; t2 t1 = (t2 t1 )2 :
Note that c) is a particular case of f ).
One of our main results of this note, namely Theorem 1.1, gives the description
of additional relations between the generators in the group Gen . We don't know
whether or not the set of relations given by (1.24) and (1.25) contains all relations,
but assume that it is complete. In any case the validity of the relations (1.24) allows
us to construct the section of the following exact sequence
*
1 ! Ker ! Gn )
Sn ! 1 (ti) = (i i + 1)
and to nd a subgroup (not normal subgroup !), which is isomorphic to the symmetric group Sn .
Now let us pass to the construction of the involutions si , which will generate the
symmetric group and the transformations si() . The transformations s(i) will satisfy
the colored braid relations ( Theorem 1.1 ).
For this purpose consider the following elements in the groups Gn and Gen :
pi = ti ti;1 t1
vi = ti ti+1 tn;1 qi = p1 p2 pi ui = vn;1 vn;2 vn;i = t1 t2 tn;1 := p;n;1 1 = v1 si = i;1 t1 1;i s(i) = i;1 T 1;i (1:16)
where 2 i n ; 1 and put p1 = q1 = s1 = t1 vn = 1.
Proposition 1.4. We have
a) qi = qi;1 pi = p;i 1 qi;1 ui = ui;1 vn;i = vn;;1 i ui;1 consequently qi2 = 1 u2i = 1 1 i n ; 1:
b) si = qi t1qi s(i) = qi T; qi : 1 i n ; 1:
(1:17)
14 Anatol N. Kirillov and Arkadiy D. Berenstein
c) qn;1 si qn;1 = sn;i qn;1 s(i) qn;1 = s(n;) i 1 i n ; 1:
d) (si;1 si )3 = qi;1 t1 pi t1(t2 t1 )6 t1 p;i 1 t1qi;1 2 i n ; 1:
e) Crucial Lemma: Assume x 2 Xn (x) = (1 n ) then
si (x) = s(ii ;i ) (x):
(1:18)
f ) si s(i) = s(i;) si 1 i n ; 1:
g) Assume that si tj = tj si for all j < i ; 1 then
(1:19)
si = ti;1 ti si;1 ti ti;1 :
h) Formulas for the weights
(si (x)) = (i i + 1) (x)
(s(i) (x)) = (1 i;1 i ; i+1 + i+2 n )
((x)) = (n n ; 1 2 1) (x)
(qn;1 (x)) = (x) := w0 (x)
where w0 2 Sn is the element of the maximal length in the symmetric group Sn .
Proof. a) We must prove that pi+1 qi pi+1 = qi : Let us use induction:
+1
pi+1 qi pi+1 = ti+1 pi qi;1 pi ti+1 pi = ti+1 qi;1 ti+1 pi = qi;1 pi = qi because ti+1 qi;1 ] = 0. Consequently,
qi2 = qi qi = qi;1 p;i 1 pi qi;1 = qi2;1 = = q12 = t21 = 1:
Similarly we may prove that
vn;i ui;1 vn;i = ui;1 and u2i = 1:
b) By induction, it is easy to see that
i;1 = qi ui+1 v1;1 v2;1 t1 2 i n ; 1
n;1 = qn;1 un;1 v1;1 n = qn;1 un;1 :
So we have
si = i;1 t1 ;(i;1) = qi (ui+1 v1;1 v2;1 t1v2 v1 u;i+11 )qi = qi t1 qi because ui v1;1 v2;1 t1] = 0, if i 2.
c) It is sucient to show that
(1:20)
qn;i qn;1 qi t1 ] = 0 qn;i qn;1 qi T ] = 0 1 i n ; 1:
(1:21)
Let us assume that 2 i n ; 2. For i = 1 or i = n ; 1 the equalities (1.21) are
clear. From a) by induction we nd
1
qi = qj pj+1 pi = p;i 1 p;j+1
qj if i j:
(1:22)
Combinatorics of the Gelfand-Tsetlin patterns
15
Consequently,
qn;i qn;1 = pn;i+1 pn;1 :
Now it is easy to see by induction that
1
pn;i+k;1 = vn;i+k vk;+1
pk 2 k i:
So
pn;i+1 pn;1 = (
Thus we have
qn;i qn;1 qi = (
Yi
k=2
Yi
k=2
1
vn;i+k vk;+1
) t1 qi :
1
vn;i+k vk;+1
) t1 :
Q
But the product ik vn;i k vk; does not contain the involutions t and t and
+
=2
1
+1
therefore commutes with t1 .
d) We use the identity (1.22). Let i ; j 1, then
1
2
1
si sj = qi t1 qi qj t1 qj = qj (pj+1 pi t1 p;i 1 p;j+1
t1 )qj :
Now let us take j = i ; 1. Then
1
1
1
t1qj =
(sj+1sj )3 = qj (pj+1 t1p;j+1
t1 )3qj = qj t1 pj+1 t1(t1 p;j+1
t1pj+1 )3 t1 p;j+1
1
= qj t1 pj+1 t1(t2 t1)6 t1p;j+1
t1 qj since
1
t1 p;j+1
t1 pj+1 = t1 t1 tj+1 t1 tj+1 t1 = t2 t1 t2t1 :
e) It is sucient to show (see (1.17) of Proposition 1.4) that if
(x) = (1 n ) then
t1 (qi (x)) = Ti ;i (qi (x)):
+1
We note that
(qi (x)) = (i+1 i 1 i+2 n ) 1 i n ; 1:
Consequently,
x11 (t1(qi (x)) = i x11 (Ti ;i (qi (x))) = i+1 ; (i+1 ; i ) = i :
+1
Crucial Lemma allows us to reduce any statment about si to that about si . For
example, the relation (1.24,b) follows from that (1.25,b) if = i+1 ; i+2 and
= i ; i+1 .
16 Anatol N. Kirillov and Arkadiy D. Berenstein
g) In fact, by (1.17) and part a), we have
si = qi t1 qi = p;i 1 qi;1 t1qi;1 pi = p;i 1 si;1 pi =
= t1 ti;2 (ti;1 ti si;1 ti ti;1 )ti;2 t1 :
However, by assumption we have sei t1] = = sei ti;2 ] = 0, where sei is given by
(1.19). So si = sei = ti;1 ti si;1 ti ti;1 .
f ) Follows from Proposition 1.3, point a).
Note that from (1.17) we obtain the formula for si as a product of (i2 + i ; 1)
simple involutions tj , whereas formula (1.19) gives the expression for si as a products
of 4i ; 3 involutions. For example
s1 = t1 q1 = t1 s2 = t1 t2t1 t2 t1 q2 = t1 t2 t1 q3 = t1 t2 t1t3 t2 t1
s3 = t1 t2t1 t3 t2t1 t2 t3t1 t2 t1 = t2 t3 t1t2 t1 t2t1 t3 t2:
(1:23)
Note that equality (1.23) for s3 is equivalent to the following relation in the
group Gn , n 4 (see Corollary 1.1): (t1q3 )4 = 1.
Now let us give a formulation of our main result of this Section.
Theorem 1.1. 10: The involutions si 1 i n ; 1, satisfy the relations of
the symmetric group Sn , i.e.
a) s2i = 1 i = 1 n ; 1
b) (sisi+1 )3 = 1 i = 1 n ; 2
c) si sj = sj si if 1 i j n ; 1 ji ; j j 2:
(1:24)
20: The transformations s(i) satisfy the colored braid relations, i.e. for any
2 R1 we have
a) s(i) s(i) = s(i+) in particular s(i;) = (s(i));1 +) s() = s() s(+) s() 1 i n ; 2
b) s(i) s(i+1
i
i+1 i
i+1
()
( )
( )
()
c) si sj = sj si if ji ; j j 2:
(1:25)
Corollary 1.1 (of the Theorem 1.1) 1 . We have the following relations between
0
the generators ti in the group Gn
1)
2)
3)
t2i = 1 ti tj = tj ti if ji ; j j 2
(t1 t2)6 = 1
(t1 qi )4 = 1 if 3 i n ; 1
where qi = p1 p2 pi = t1 |{z}
t2 t1 |t3{z
t2t1} t|i ti;1{z t2 t1} :
Combinatorics of the Gelfand-Tsetlin patterns
17
20. Besides the relations 1) - 3) of Corollary 1.1, part 10, we have the following
relations between the generators ti , and T in the group Gen
1)
2)
3)
4)
5)
6)
T T = T+ t1T = T; t1 ti T = T ti if i 3
(T t2t1 T t2 t1)3 = 1 for any 2 R1
T t2 T;+ t2 T; t2 = t2T t2 T;+ t2T; T qj T qj = qj T qj T 3 j n ; 1
(t1T qj t1 T qj )2 = 1 3 j n ; 1:
Proof. 10 The assertions 1) and 2) are proved in Proposition 1.3. On the other hand,
if i ; j 2, then sj si = j;1 (t1si;j+1 );(j;1) and (sj si)2 = 1 i (t1si;j+1 )2 = 1.
But if i 3, then (t1 si )2 = (t1qi )4 :
20. The properties 1) - 4) follow from Proposition 1.3. The identity 5) follows
from the fact that the transformations s(i) and s(j) commute, if ji ; j j 2 (see
Theorem 1.1). The last identity 6) follows from an observation that the elements
si s(i) and sj s(j) commute, if ji ; j j 2, which is a consequence of Corollary 1.4.
Corollary 1.2 (of the Theorem 1.1). Let us dene
s0 := s(0) = sn;1sn;2 s2s1 T s2 sn;1 :
Then s0 s1 sn;1 are the standard generators of the ane Weyl group of the
()
()
type A(1)
n;1 , and (s0 (x)) = s0 ( (x)) = (n + 2 n;1 1 ; ).
Proof. It is sucient to show that (s0s1)3 = 1 and (s0sn;1 )3 = 1. We have the
following chains of the equivalent statements
i) (s0s1)3 = 1 () (s2 s1T s2 s1)3 = 1 () (T s1 s2)3 = 1.
The last relation is equivalent to the following one (T t2t1 t2 t1)3 = 1, which is a
particular case of (1.14,f) when = 0.
ii) (s0sn;1 )3 = 1 () (s2s3 sn;2sn;1 sn;2 s3s2)s1 T ]3 = 1.
The last relation is equivalent to one of the form (s2s1 T )3 = 1, which is also
a particular case of (1.14,f). It is also clear, that s20 = 1 and s0sj = sj s0, if
2 j n ; 2.
Corollary 1.3. The elements si s(i) 1 i n 2 R1 satisfy the following
relations
1) (si s(i) )2 = 1 1 i n ; 1
) )3 = 1 1 i n ; 2 2 R1 2) (si s(i) si+1 s(i+1
3) si s(i) sj s(j) = sj s(j) si s(i) if ji ; j j 2 2 R1 :
Proof. We must prove 2) only when i = 1, i.e. (s1T s2s(2) )3 = 1. But this relation
is exactly (1.14,f). The statement 3) follows from Corollary 1.4.
18 Anatol N. Kirillov and Arkadiy D. Berenstein
Using the same arguments as in the proof of Corollary 1.2, we may prove that if
se0 := s(0) = sn;1s(n;) 1 sn;2 s(n;) 2 s2s(2) s1s(1) T s2 s(2) sn;1 s(n;) 1 then se0 s1 s(1) sn;1sn(;) 1 are the standard generators of the ane Weyl group
of the type A(1)
n;1 and
(s(0) (x)) = (n + n + 2 n;1 1 ; n ; ):
Let us give some comments.
i) Part 10 of Theorem 1.1 follows from part 20 and the equality (1.18).
ii) The assertion b) of part 20 follows from Proposition 1.3, part e). In fact,
formula (1.25, b) is equivalent to the following one
T s(2+) T = s(2) T+ s(2) which in turn coincides with (1.14).
iii) Assertion c) of part 20 is equivalent to the following identity
T s(j;)i+1 = s(j;)i+1 T if j ; i 2:
(1:26)
In fact, using (1.16) we nd that if j i, then
s(i) s(j) = i;1 (T s(j;)i+1 )1;i :
iv) We assume that relations between the generators ti , 1 i n ; 1, (respectively between ti and T ) as described in Corollary1.1, are the dening relations for
the group Gn (respectively for the group Gen ).
v) Let us say a few words about the group Gn for n = 3 and n = 4.
G3 = f t1 t2 j t21 = t22 = (t1 t2)6 = 1 g
G4 = f t1 t2 t3 j t21 = t22 = t23 = (t1t3 )2 = (t1t2 )6 = (t1 t2t1 t3 t2)4 = 1 g
The group G3 is nite of the order 12, contains the normal subgroup
N :=< 1 (t1t2 )2 > of the order 3 with the factor group G=N = Z2 Z2.
The group G4 seems to be very interesting. For example, for arbitrary n the
symmetric group Sn is a factor of the group G4 . Namely, let us dene an epimorphism := n : G4 ! Sn by setting
(t1) = s1 (t2) = s2s4 s6 (t3) = s3s5s7 where si = (i i + 1) 2 Sn 1 i n ; 1, is a simple transpositions.
In fact, one can show that is agree with the relations in the group G4 and
(t1) (t2) and (t3) are really generate the symmetric group Sn . Consequently,
the group Gn is innite, if n 4.
The main diculties in the demonstration of Theorem 1.1 appear to be to prove
that the generators s(i) and s(i) commute, if ji ; j j 2, or equivalently, to verify
identity (1.26). Our strategy is to prove more precise result about the action of
transformations s(i) on the triangles. Before stating the theorems about the structure of the mapping s(i) , let us dene some additional operators acting on the space
of triangles Xn .
Combinatorics of the Gelfand-Tsetlin patterns
19
Denition 1.2. For each triple of integers (ijk) 1 i < j < k n, let us give
a transformation Rijk : Xn ! Xn by the following formulae
Rijk (x) = xe x 2 Xn where
xeij = xij + xik ; xik;1 ; min(xjk ; xjk;1 xij ; xij;1)
xejj = xjj ; xik + xik;1 + min(xjk ; xjk;1 xij ; xij;1 )
(1:27)
xe = x if (
) 6= (i j ) or (j j ):
Proposition 1.5. The operators Rijk satisfy the following relations:
a) (Rijk )2 = 1
b) Rijk Ri j k = Ri j k Rijk if j(ijk) \ (i0 j 0k0 )j 6= 2
c) (Rijk Rijl Rikl Rjkl )2 = 1
d) (Rijl Rikl )3 = 1
e) (Rijk (x)) = (x) x 2 Xn :
Proof. The assertions a) b) and e) are almost evident, and c) d) may be checked
by direct computation.
0 0
Remark 1.1.
0
0 0
0
i) The assertions a) ; c) are essentially due to G. Lusztig Lu2].
ii) It seems plausible that relations a) ; d) are the dening ones for the group
Ln , generated by all Rijk with 1 i < j < k n.
To go further, let us dene the transformations T(i) and 'i acting on the space
Xn by the formulae:
T(i) (x) = xe 'i (x) = xe x 2 Xn where
xeii = xii ; xeii = xii + i+1 (x) ; i (x)
both xe and xe are equal to x i (
) 6= (i i):
It is clear from the denitions that if x 2 Xn = (x) = (1 n ), then
'i (x) = Ti ;i (x), and
(T(i) (x)) = (1 i ; i+1 + n )
('i (x)) = (i i + 1) (x):
Now we are ready to state our result about the structure of the transformations
si and s(i) .
Theorem 1.2. The following equalities are fullled
si = Ri;1iRi;2i R1i 'i R1i Ri;2iRi;1i (1:28a)
s(i) = Ri;1i Ri;2i R1i T(i) R1i Ri;2i Ri;1i
(1:28b)
where 1 i n ; 1 and Rjk := Rjkk+1 1 j < k n ; 1.
The proof of Theorem 1.2 will be given in Appendix.
+1
20 Anatol N. Kirillov and Arkadiy D. Berenstein
Corollary 1.4. Let x 2 Xn x = (x n x ). Then
( )
(1)
10: si (x) = (x(n) x(i+1) xe(i) x(i;1) x(1)) 1 i n ; 1 and a vector
xe(i) 2 Ri depends only on the components of the vectors x(i;1) x(i) and x(i+1) .
(i)
(i)
20: s(i) (x) = (x(n) x(i+1) ex x(i;1) x(1) ) and a vector xe depends
only on and the components of vectors x(i;1) x(i) and x(i+1) .
In particular, from Corollary 1.2, part 10, follows that si tj = tj si if 1 j < i ; 1
and consequently, si sj = sj si , if i ; j 2. This proves the part 10 of Theorem 1.1
and also the recurrence relation (1.19) for si .
Similarly, from Corollary 1.2, part 20, it follows that T s(i) = s(i) T for any
2 R and 3 i n ; 1. This proves part 20 of Theorem 1.1 (see (1.26)).
As another application of Theorem 1.2 we will deduce a recurrence relation
for the transformations s(i) , but before doing so it is necessary to introduce some
additional notations.
Denition 1.3. Let us give a mapping i i + 1] : Xn ! Xn 1 i n ; 1 by
i i + 1](x) = xe x 2 Xn where
xeki = xki+1 + xki;1 ; xki if k < i
xeii = xii+1 + xi+1i+1 ; xii xeik = xi+1k and xei+1k = xik if k > i
and for all remaining elements xe = x .
Lemma 1.1. The mappings i i + 1] 1 i n ; 1, satisfy the relations of the
symmetric group Sn , i.e.
a) i i + 1]2 = 1
b) i i + 1] i + 1 i + 2] i i + 1] = i + 1 i + 2] i i + 1] i + 1 i + 2]
c) i i + 1] j j + 1] = j j + 1] i i + 1] if ji ; j j 2
d) (
i i + 1](x)) = (i i + 1) (x) x 2 Xn:
It is not dicult to show that the group generated by all i i + 1] 1 i n ; 1,
is really isomorphic to the symmetric group Sn .
Proposition 1.6. Let us take i 2. Then
si = Ri;1i i ; 1 i]
i i + 1]si;1
i i + 1]
i ; 1 i]Ri;1i
s(i) = Ri;1i i ; 1 i]
i i + 1]s(i;)1 i i + 1]
i ; 1 i]Ri;1i:
(1:29)
Proof. It is easy to see, that
Rki = i ; 1 i]
i i + 1]Rki;1
i i + 1]
i ; 1 i]
T(i) = i ; 1 i]
i i + 1]T(i) i i + 1]
i ; 1 i]
(1:30)
Combinatorics of the Gelfand-Tsetlin patterns
21
where 1 k < i n ; 1. Identity (1.29) follows by induction from (1.19), (1.30)
and Theorem 1.2, parts 10 and 20 .
The recurrence relations (1.19) and (1.29) are used as an induction base in a rst
proof of Theorem 1.2. However, it is possible to solve these recurrence relations in
an explicit form and consequently to obtain a second proof of Theorem 1.2. Before
describing the solutions of (1.19) and (1.29), let us give the appropriate denitions.
It is convenient to use the notations
(a)+ = max(a 0) (a); = min(a 0) a 2 R:
At rst we dene the sequence of piece-wise linear functions Qkn (a1 an),
1 k n, inductively, in the following way:
Q11 (a1) := ;a1 Q1n (a1 an ) := (Q1n;1 (a1 an)); + (Q1n;1 (a2 an ))+
Qkn (a1 an ) := Q1n (ak an a1 a2 ak;1) 1 k n:
(1:31)
Secondly, we dene the linear functionals
'ij := '(ijn) : Xn ! R 1 i j n ; 1
on the space of triangles Xn by the formulae:
'ij (x) := xi;1j + xij ; xi;1j;1 ; xij+1 if 2 i j n ; 1
(1:32)
'11 (x) := 0 '1j (x) := x1j + xjj ; x1j+1 ; xj+1j+1 if 1 < j n ; 1:
Theorem 1.3. Let us x a positive integer k, 2 k n ; 1, and a triangle
x 2 Xn . Assume that
Then
sk (x) = xe s(k) (x) = xe:
xeik = xik + Qik ('2k (x) 'kk (x) '1k (x))
(1:33)
exik = xik + Qik ('2k (x) 'kk (x) '1k (x) + k+1(x) ; k (x) + ): (1:34)
The proof of Theorem 1.3 will be given elsewere.We shall give the exact formulae
for sk , if k 3, in Section 3.
Remark 1.2.h We know (see (1.20)) that n = qn;1 un;1 . Assume additionally
that (qn;1 un;1 ) = 1 h may be equal to 1. Consider the subgroup n Gn
generated by the elements si = i;1 t11;i 1 i < nh. Then from Corollary 1.1
it follows that
a) s2i = 1 1 i < nh
b) (si si+1)3 = 1 1 i nh ; 2
c) si sj = sj si if 2 ji ; j j n ; 2:
22 Anatol N. Kirillov and Arkadiy D. Berenstein
Proof. From the denition it is easy to see that sisj = ;i (t1sj;i+1 )i , if j i.
So (sisi+1 )3 = 1 i (t1s2)3 = 1: But t1 s2 = (t2t1 )2 and by Proposition 1.3 part c)
we know that (t1t2 )6 = 1. Similarly (si sj )2 = 1 i (t1 sj;i+1 )2 = 1 and under the
assumption 2 ji ; j j n ; 2 the assertion c) follows from Theorem 1.1 part 10).
In particular, for any 1 a (n ; 1)h the involutions sa sa+n;2 generate
a subgroup of Gn which is isomorphic to the symmetric group Sn .
Remark 1.3. The involutions qi 1 i n;1, also give a system of generators
for the group Gn , because we have
t1 = q1 ti = qi;1 qi qi;1 qi;2 if i 2 (q0 := 1):
(1:35)
The proof of (1.35) follows from Proposition 1.4, point a). In fact, we have
qi;1 qi qi;1 qi;2 = qi;1 qi;1 pi p;i;11 qi;2 qi;2 = pi p;i;11 = ti if i 2:
The relations between the generators qi 1 i n ; 1, follow from Corollary
1.1 and have the following form
1)
2)
3)
4)
qi2 = 1 1 i n ; 1
(q1 q2 )6 = 1 (q1 qi )4 = 1 if 3 i n ; 1
qi+1 qi qi;1 qi ] = 0 if 3 i n ; 2
qi qi+1 qi qi;1 qj qj+1 qj qj;1 ] = 0 if ji ; j j 2:
(1:36)
In Section 2 we will show that in the case when 2 Zn+ is a partition, the
restriction of the involution qn;1 on the set KZ ' STY ( n) coincides with
the Schutzenberger involution S (see e.g. Sch1],
Sch2],
EG],
Ki1]). The similarly
combinatorial interpretation admits the involutions qi 1 i n ; 1: So our
construction gives the extension of the Schutzenberger involutions qi on the space
of triangles Xn and describes the group generated by q1 qn;1 i.e. the relations
between q1 qn;1 : The next steps are Theorems 1.1 and 2.3 (see Section 2) which
give us the connection between the natural action of the symmetric group Sn =<
s1 sn;1 > on the set STY ( n),which was introduced and studied by A.
Lascoux and M.-P. Schutzenberger LS2],
LS3], and the Schutzenberger involutions
si = qi q1 qi :
(1:37)
The equality (1.37), restricted on the set STY ( n), is a pure combinatorial
assertion and may be deduced directly from the properties of the plactic monoid (e.g.
LS2]) and the Robinson-Schensted correspondence. Details will appear elsewhere.
Combinatorics of the Gelfand-Tsetlin patterns
23
Remark 1.4. In a particular case that the a weight 2 Rn has the form
+
= (an ) i.e. 1 = 2 = = n = a, the group Gn conserves the convex polytope
K ( ) for any highest weight 2 Rn+ = (1 2 n ). It is easy to
see that a restriction of t1 to the polytope K ( ) becomes the identical map and
consequently for all 1 i n ; 1 we have
si jK () = IdK () :
The following example
7 6 3 0
6 5 1
5 3
4
shows that (t3t2 )6 x 6= x (in fact, for this example (t3t2 )24 x = x) and it is not clear
whether or not it is possible to nd the subgroup in Gn which is isomorphic to the
symmetric group SN (for some N ) after the restriction of the action of Gn on the
polytope K ( ) .
We nish this section by introducing some additional transformations of the
space Xn and by considering of the stable behavior of the involutions si . First,
consider the map I: Xn ! Xn which is dened in the following way: let x 2 Xn ,
then I (x) = xe, where
xeij = x1n ; xj;i+1j 1 i j n:
(1:38)
Proposition 1.7. We have
1) i (I (x)) = x1n ; i (x):
2) The map I commutes with the actions of si and qi , i.e.
Isi = si I and qi I = Iqi 1 i n ; 1
3) I T = T; I I si() = s(i;) I 1 i n ; 1:
Proof. Using Denition 1.1 for the involution tj we nd that for x 2 Xn ,
xij (tj (Ix)) = x1n ; xj;i+1j (tj (x)):
Consequently, Itj = tj I , i.e. the actions of tj and I commute.
Secondly, for a partition we consider the restrictions of the maps s(i1) to the
Gelfand-Tsetlin polytope K :
(;1)
ei := pr s(1)
i fi := pr si 1 i n ; 1
where pr : Xn ! K is the projection map.
(1:39)
24 Anatol N. Kirillov and Arkadiy D. Berenstein
Proposition 1.8. We have
si fi si = ei 1 i n:
Proof. The assertion follows from denitions (1.16), Proposition 1.4 point f )
and (1.39).
Now we consider the stable behavior of the involutions si and qi . Given the space
of triangles Xn;1 , we dene the following embeddings ' : Xn;1 ,! Xn = 1 2:
given x = (x(n;1) x(1) ) 2 Xn;1 , then
'1 (x) = (x(n) x(n;1) x(1))
(1:40a)
where x(n) = (x1n;1 xn;1n;1 0) and
'2 (x) = (xe(n) xe(1) )
(1:40b)
where xe(1) = 0 xe(i+1) = (x1i xii 0) 1 i n ; 1:
Proposition 1.9. We have
1) si ('1 (x)) = si (x) if 1 i n ; 2
sn;1 ('1 (x)) = (x1n;1 xn;2n;2 0):
2) ti ('2(x)) = '2 (ti;1 (x))
qi ('2 (x)) = '2 (qi;1 (x))
si ('2 (x)) = '2 (si;1(x)) where 1 i n ; 1:
(1:41)
(1:42)
Remark 1.5. All our main results (including those dealing with a cocharge c
(see Section 3)) after small modication are still valid for the space of truncated
triangles Xnm 0 m n BZ1]. By denition the space Xnm consists of all
sequences
x = (x(n) x(m))
where x(j) = (x1j xjj ) 2 Rj . In what follows, we always consider the space
Xnm as the subspace in Xn under the constraint
xij = xij+1 1 i j m ; 1:
(1:43)
Given a truncated triangle x 2 Xnm, it is convenient to use notations
(x) := x(n) , (x) := x(m) and to dene a weight := (x) of a truncated triangle
x 2 Xnm as a vector = (m+1 n) 2 Rm such that j = jx(j) j ; jx(j;1) j
m < j n. For given vectors 2 Rn 2 Rm and 2 Rn;m we consider the
following subspaces of the space Xnm
= fx 2 Xnm j (x) = g
Xnm
n = fx 2 X j (x) = g
Xnm
nm
n
n j (x) = g:
Xnm( ) = fx 2 Xnm
(1:44)
Combinatorics of the Gelfand-Tsetlin patterns
25
By denition, a truncated triangle x 2 Xnm is called a truncated Gelfand-Tsetlin
pattern i x belongs to the cone Kn . We denote by Knm the cone of all truncated GT-patterns and its intersections with subspaces (1.44) correspondently by
K n and K n ( ). It is clear that K n and K n ( ) are the convex, comKnm
pact polytopes in the space Rc+nm , where cnm = 21 (n ; m)(n + m + 1). It is wellknown (see Section 2 or BZ1 ]), that if and be the partitions, j n j = p
and is a composition of the same integer p, then the number of an integral points
in the convex polytope K n (respectively in K n ( )) is equal to the dimension
of the representation Vn of the Lie algebra gln;m (see Section 2)(respectively the
dimension of the subspace V ( n ) Vn of the weight ). On the other hand,
as is also well known, the dimension of the weight subspace V ( n ) Vn
admits a pure combinatorial description as the number of ( skew) standard Young
tableaux of the shape and content jKZn ( )j := jK n ( ) \ Zcnm j = dimV ( n ) = jSTY ( n )j: (1:45)
Using the constraint (1.43), it is possible to dene the action of the symmetric group
Sn;m on the space of truncated triangles Xnm by setting
i = sm+i 1 i n ; m ; 1:
(1:46)
The fact that the involutions i really generate the symmetric group Sn;m follows
from Theorem 1.1 and Corollary 1.2.
Exercise 1.1. Fix real(ijnumber
and integers i j s.t. 1 i j n. Let us
)
dene a transformation T : Xn ! Xn in the following way
T(ij) (x) := xe x 2 Xn
where xeij = xij ; and xekl = xkl , if (k l) 6= (i j ). Show that
(i) (tj T(ij) )2 = 1
(ii) tk T(ij) = T(ij) tk if jk ; j j 2
(iii) tk T(ij) tk T(;ij) tk T;(ij ) = T(ij) tk T;(ij+) tk T;(ij ) tk if jk ; j j = 1 2 R
which is a generalization of (1.14e).
Exercise 1.2. Fix real number q and integer j 1 j n. Let us dene a
transformation etj : Xn ! Xn :
etj := tj q](x) = xe where
xeik = xik if k 6= j
xeij = min(xij+1 qxi;1j;1) + max(xi+1j+1 qxij;1) ; qxij :
Here we presuppose that x0j := +1 xjj;1 := ;1 1 j n ; 1. Show that
(i) et2j = (1 ; q)etj + q IdXn etietj = etjeii if ji ; j j 2
(ii) (etj (x)) = (1 j;1 j+1 + (1 ; q)j qj j+1 n ):
26 Anatol N. Kirillov and Arkadiy D. Berenstein
where (x) := (1 n) is the weight of triangle x. Thus we obtain a representation of the Hecke algebra Hn (q) on the space of weights Rn .
Problems. (i) To nd the dening relations between the transformations etj .
(ii) Is it possible to extend this representation of the Hecke algebra Hn (q) to
the whole space of triangles Xn ?
(iii) Is it possible to construct a cpl-representation of the braid group Bn on the
space of triangles Xn ?
x2 Combinatorial description of the basic transformations.
In this section we give a combinatorial description of the restrictions of the
transformations constructed in the previous section to the set of standard Young
tableaux of a given shape and content . We will denote this last set by STY ( ).
Also we exploit the notations STY ( n ) for the set of skew standard Young
tableaux of a skew shape n and content , and STY ( n n) for the set of
all skew standard Young tableaux of the shape n with all entries not exceeding
n. We will assume in the sequel that l() n l( ) m l( ) n for some xed
positive integers m n.
At rst we remind the well-known bijection ( e.g.
GZ1],
GZ2],
BZ1]) between the
set STY ( n ) and the set of integral points in the convex polytope K n ( ).
So, take T 2 STY ( n ). As it is well known ( e.g. Ma] ), one may consider the
tableaux T as a sequence of Young diagrams
= (m) m+1 (n) = (2:1)
such that all skew diagrams (i) n (i;1) m < i n are a horizontal strip. Let
us dene the triangle x = x(T ) = (x(n) x(m)), where x(i) is the shape of the
diagram (i) . Then we have
T 2 STY ( n ) i x(T ) 2 KZn ( ):
Let us construct the inverse map KZn ( ) ! STY ( n ). Given a point
x 2 KZn ( ), we are lling the shape n by the numbers 1 n according to the
following rule: in the i-th row ( n )i of the skew diagram n we write exactly
xki+m ; xki+m+1 numbers equal to k, starting from k = 1.
Let us consider an explanatory example. Assume
T :=
1 4
2 2 3 5 = (5 5 4) = (3 1) = (2 2 2 3 1):
1 3 4 4
Then we have the following sequence of shapes for (2.1):
= (3 1) (4 1 1) (4 3 1) (4 4 2) (5 4 4) (5 5 4) = :
Combinatorics of the Gelfand-Tsetlin patterns
27
Consequently,
5 5 4 0 0 0 0
5 4 4 0 0 0
X72 3 x(T ) := 4 4 2 0 0
2 KZn ():
4 3 1 0
4 1 1
3 1
Let us remind brie!y the group-theoretical interpretation of the numbers
Kn := jSTY ( n )j and d(nn) := jSTY ( n n)j. Let gn = gln . The
lattice of weights Pn of the Lie algebra gn is identied in the standard way with Zn the set Pn+ of highest weights of nite-dimensional, polynomial gn -modules is identied with f = (1 n ) 2 Pn j 1 2 n 0g. For each 2 Pn+
let V be an irreducible gn -module with highest weight . For all m 0 m n,
we imbed the Lie algebra gm gn;m gn so that the subalgebra gm is generated
by the elements eij 1 i j m, where feij j 1 i j ng is a standard basis of
gn and gn;m is generated by the elements feij j m + 1 i j ng. Accordingly
we will write the weights 2 Pn;m in the form = (m+1 n ). For all
2 Pn+ and 2 Pm+ we dene (see BZ1]) the skew gn;m -module Vn by putting
Vn = Homg m (V Vjgm ):
(2:2)
When m = 0 it is convenient to assume that Pm+ consists of one element = and
that Vn is the irreducible gn -module V .
Proposition 2.1 (see BZ1]). The multiplicity Kn of the weight in the
skew gn;m -module Vn is equal to the number of all truncated Gelfand-Tsetlin
patterns of the highest weight n and of weight .
As a corollary, we obtain:
d(nn) := jKZn j = jSTY ( n n)j = dimVn :
Note that the numbers Kn admit a completely elementary description in
terms of symmetric functions (e.g. Ma] ): they are the coecients in the expression of the skew Schur function Sn (x) as a linear combination of monomials
x = x1 xnn .
Now we are ready to give a combinatorial interpretation for the restrictions on
the set STY ( n ) of the transformations constructed in Section 1. We will keep
the notations of Remark 1.5 and the beginning of Section 2.
A) The action of ti+m on the set STY ( n ).
Given a skew tableau T 2 STY ( n ). Consider the part Ti of the tableau T
which is lled by the letters i and i + 1. It is clear that Ti is a disjoint union of the
following fragments
1
28 Anatol N. Kirillov and Arkadiy D. Berenstein
z
|
i
{z
a
i i i
i i+1 i+1 i+1 i+1
}|
{z
b
}|
{z
e
}|c
{z
}|d
{
{z
}|c
{
i i+1 i+1
}
We replace every such fragment by the following one
z
|
i
{z
b
i i i
i i+1 i+1 i+1 i+1
}|
{z
a
}|
{z
e
}|d
i i+1 i+1
}
The part T n Ti of the tableau T remains unchanged. As a result we obtain
the new tableau Te 2 STY ( n (i i + 1) ):
Proposition 2.2. We have
ti+m (T ) = Te:
The proof is clear from the construction of the bijection
STY ( n ) $ KZn ( ):
We remark that the action of the transformations ti on the set STY ( n ) as
described above is well-known, see for example BK],
SW],
Sa].
Our main observation is that the action ti on the set of standard Young tableaux
of a given shape and content admits a continuous piecewise linear prolongation on
the set of triangles Xn and other well-known combinatorial operations on the set
of Young tableaux may be presented as a superposition of the transformations ti
and consequently also admit a continuation on the space Xn . Anyhow, using the
involutions ti we may construct for any element of the symmetric group Sn a
piecewise linear one-to-one mapping, dened over Z, between the GT-polytopes
K n ( ) and K n ( ). Note nally that for the denition of the involution ti it is
essential to have the whole tableau T , but not only the corresponding word w(T ),
so we can not dene the action of ti on the words.
Remark 2.1. The action of the involutions ti on the set of standard Young
tableaux STY (), i.e. in the case when weight = (1n ) jj = n, is studied in
many papers. We mention only the work of A.Garsia and T.McLarnan GM], which
contains, among other, the interesting results concerning a connection between the
Robinson-Schensted correspondence and the transformations ti . Now let us say
some words about the group Gn generated by ti 1 i n ; 1,in the case under
Combinatorics of the Gelfand-Tsetlin patterns
29
consideration. It is not dicult to verify that t1 = IdSTY () , and (ti ti+1 )6 =
1 1 i n ; 2. Because of t1 = IdSTY () , we see that si = Id on the set
STY () for all 1 i n ; 1. We note that if the diagram is a hook, then
(ti ti+1 )3 = 1 1 i n ; 2, and the group Gn considered as a subgroup of
Aut(STY ()) is isomorphic to the symmetric group Sn;1 .
Remark 2.2. We give another combinatorial description of the action of the
trasnformation ti on the Gelfand-Tsetlin cone Kn . For this let us describe, at rst,
some special triangulation of the cone Kn . We denote by
Bn = Kn \ f 0 1 g n
n
2+
2
the set of all (0,1) - patterns.
Lemma 2.1. There exist a bijection
Bn ! f 0 1 gn in particular, jBn j = 2n .
Proof. The bijection under consideration is given by a correspondence
x 2 Bn ! (x) 2 f 0 1 gn :
It is easy to see that the GT-pattern consisting of only from zeros and units is
uniquely determined by its weight.
Now let us introduce on the set of triangles Xn a partial order, assuming x xe
i x ; xe 2 (Xn)R = (R+ ) n n .
Proposition 2.3. The set Bn possesses the following properties:
i) Bn is a set of all generators of the cone (Kn )Z = Kn \ (Xn )Z (and, consequently, also for the cone Kn = R+ (Kn )Z ).
ii) Bn is a lattice with respect to the partial order \".
iii) The length of the maximal chain in the lattice Bn is equal to n(n2+1) .
iv) Assume x 2 Kn , then there exist unique decomposition
x = 1x1 + + m xm
(2:3)
where x1 xm 2 Bn 1 m 2 R 1 > 0 m > 0, and x1 x2 xm 0 (m < n 2+n ).
Proposition 2.4 (The connection between the involutions ti and the GelfandTsetlin cone Kn).
i) If x 2 Bn , then ti (x) 2 Bn 1 i n ; 1.
ii) Let us assume that x is written uniquely in the form (2.3). Then
ti (x) = 1 ti (x1) + + mti (xm ):
2+
2
+
2
Note that if the elements x1 xm lie in the same chain of the lattice Bn , then
their images ti (x1) ti (xm) may belong to the dierent chains.
30 Anatol N. Kirillov and Arkadiy D. Berenstein
Conjecture 2.1. Assume that 2 Zn , then all vertices of the Gelfand-Tsetlin
polytope K ( ) belong to the set Kn \ (Xn)Z .
B) The action of transformations pi and p;i 1 .
Let be a partition and be a composition, = (1 n ). Note at rst
that it is sucient to describe the action of the transformations = p;n;1 1 and
;1 = pn;1 on the set
STY ( ) ' KZ ( ) ,! Xn :
In fact, we have a natural embedding Xi ,! Xn x ! xe, where for a triangle
x = (x(i) x(1)) 2 Xi we dene
xe = (x| (i) {z x(i}) x(i) x(1)) 2 Xn :
n;i
On the level of Young tableaux this embedding corresponds to the consideration of
the part Ti of the tableau T ,which is lled by the numbers 1 i. It is easy to
verify the commutativity of the following diagram
Xi ,! Xn
pi;1
??
y
??
y pi;
1
Xi ,! Xn
This means that pi;1 2 Gn acts non-trivially only on the tableau Ti . Remark
further that
(;1 (x)) = (2 3 n 1 ) = (1 2 n) (x)
((x)) = (n 1 n;1 ) = (n n ; 1 1) (x):
Given a tableau T 2 STY ( n ). We may assume that 1 > 0. We delete the
most upperleft box lled with number 1. After this we apply a \jeu de taquin" (e.g.
Sch1],
Sch2]) in oder to obtain a standard tableau T 0 which has the same left border
strip and one box less than T . Repeating this procedure we eliminate all ones from
the tableau T . As a result we obtain a new tableau T 00 of the content (0 2 n )
and a horizontal strip n jnj = 1 . We ll the strip n by the number n. Now
let us subtract one from all numbers situated in the boxes of T 00 . We obtain a
tableau Te of the content e = (e1 en;1 ), where ei = i+1 1 i n ; 1.
The tableau Te and the horizontal strip n dene a tableau T 2 STY ( ), where
= (2 n 1).
Proposition 2.5. We have (in the case = )
;1 (T ) = pn;1 (T ) = T:
Similarly we may describe the action of the transformation = p;n;1 1 . However in
this case it is more convenient to consider the action of on the set of skew Young
Combinatorics of the Gelfand-Tsetlin patterns
31
tableaux. So, let be a partition, . Given a tableau T 2 STY ( n ), we
may assume that n > 0. Let us delete the most right and bottom box in T which
occupied by the number n. After this we apply a \jeu de taquin" (e.g. Sch2],
Sa])
in order to obtain a standard tableau T 0 2 STY ( n e e) with the same right border
strip, which has one box less than T (in fact e = (1 n;1 n ;1) jej = j j+1).
Repeating this procedure we eliminate all numbers equal to n from the tableau T .
As a result we obtain a new tableau T 00 of the content = ( 1 n ), where
1 = 0 i = i;1 2 i n. Further, let us ll up the strip 1 by the number
1. The tableau T and the horizontal strip 1 dene a tableau Tb 2 STY ( n b)
where b = (n 1 2 n;1 ).
Proposition 2.6. We have (in the case = )
(T ) = p;n;1 1 (T ) = Tb:
The proof of Propositions 2.5 and 2.6 is based on the work of M.-P. Schutzenberger
Sch2] and Proposition 2.2 . Details will appear elsewhere.
C) The action of the involutions qi 2 Gn 1 i n.
Let a tableau T 2 STY ( ) be given. Note at the beginning, that involution
qi;1 acts nontrivially only on the tableau Ti (see Section B ), so it is sucient to
describe the action of the transformation qn;1 on the set
STY ( ) ' KZ ( ) Xn :
Remark 2.3. We use a term involution for the restriction of qi to the set
STY ( ) in spite of the fact that in general the transformation qi does't conserve
this set, but, of course, qi2 = IdSTY () .
Theorem 2.1. The restriction of the involution qn;1 to the set STY ( )
coinsides with the Schutzenberger involution S.
Let us recall the corresponding denitions. Let and n be as above. Let
be given a tableau T 2 STY ( n ). We may dene four natural operations on
the set STY ( n ). In this section we give a denition of the rst two operations
and a denition of the remaining one in the Section D). Note, that it is sucient
to consider the case when all parts of the composition are not equal to zero. We
already gave a combinatorial description of the transformations pi and p;i 1 . It is a
\jeu de taquin" that is the crucial step for the description of these transformations.
Let us remind (see Section B ) that pn;1 (T ) = T , where T 2 STY ( n ) =
(2 3 n 1), and in fact, the tableau T is a disjoint union of the tableau Te
of content e = (e1 en;1 ), where ei = i+1 1 i n ; 1, and the lled
horizontal strips n jn j = 1 , with the number n in all boxes. After this we may
act in two ways. In the rst one, let us apply the same algorithm to the tableau Te
and so on. As a result we obtain a sequence of horizontal strips 1 2 n such
that ji j = n;i+1 1 i n, which denes (e.g. Ma] ) a standard Young tableau
32 Anatol N. Kirillov and Arkadiy D. Berenstein
T0 2 STY ( n ;) where ; = (n n;1 1). Therefore there are maps
S : STY ( n ) ! STY ( n ;) T ! T S : KZn ! KZn :
0
(2:4)
The mapping S is called the Schutzenberger involution. If the weight has equal
parts, then we obtain the involution S : STY ( ) ! STY ( ).
The second way is to apply the transformation pn;1 to the tableau T n times,
i.e. consider a tableau pnn;1 (T ). It is interesting to remark that transformation
U := pnn;1 S is also an involution, which will be described in Section D . From our
description of the Schutzenberger involution S it is obvious that qn;1 jSTY () = S.
This ends the proof of Theorem 2.1.
Remark 2.4 (The symmetries of the Littlewood-Richardson numbers). Let
V , V
, V be three irreducible nite-dimensional sln -modules with highest weights
, , . The multiplicity c
of V in the tensor product V V
is called the
Littlewood-Richardson (LR) number and plays an important role in the representation theory of the symmetric and general linear groups. The combinatorial rule for computing the LR-numbers is given by Littlewood and Richardson
(LR-rule), Li] (see also T], Ma]). There are many various interpretations of the
LR-rule:
GZ1],
GZ2], T],
LS2],
KR],
W],
Z]. We will use one in terms of the number
of standard Young tableaux of certain kind GZ2],
KR], and another one in terms
of BZ-triangles BZ3]. Let us remind the corresponding denitions. From the beginning (see e.g. Ma]) we observe that c
6= 0 i both and are contained in and j j jj + jj (mod n). Further, let us consider a tableau T 2 STY ( n )
and its descent set D(T ). We recall that if T 2 STY ( ) then D(T ) is the maximal among of those subsets fx1 xlg T , which satisfy the following condition:
there exists a sequence of entries y1 yl of the tableau T , all lying in the dierent
boxes, such that yi = xi + 1 and n(xi) < n(yi) 1 i l. Here n(x) is an index
of the row which contains x. So, let D(T ) = fx1 xl g be the descent set of
T . We dene a set of exponents for the tableau T as a collection of the integers
fd1 (T ) d2(T ) dn;1 (T )g, where
di (T ) := i+1 ; jfxj 2 D(T ) j xj = igj 1 i n ; 1:
Proposition 2.7 GZ2],
KR]. The LR-number c
is equal to the number of
standard Young tableaux T of the shape and content n , such that
di (T ) i ; i+1 1 i n ; 1:
Let us denote the set of Young tableaux mentioned in Proposition 2.7 by
STY ( n k ).
Proposition 2.8. The involution qn;1 denes a bijection
qn;1 : STY ( n k ) ! STY ( n k )
(2:5)
Combinatorics of the Gelfand-Tsetlin patterns
33
where for a partition = (1 n;1) we put := (1 1 ; n;1 1 ; 2 ).
Now we want to extend the bijection (2.5) to one between some convex polytopes
in the GT-patterns cone Kn, but before let us give the necessary denitions. Given
a triangle x 2 Xn , let us put
d(ji) (x) =
X
k<j
(xki+1 ; 2xki + xki;1 ) + xji+1 ; xji 1
where 1 j i n ; 1. Following GZ1], we will call the numbers d(ji) as the
exponents of the triangle x 2 Xn . Note KR], that if is a partition, is a
composition and x 2 KZ ( ), then
maxf d(ji) (x) j 1 j i g = di (x(T )) 1 i n ; 1:
At last, for 2 Rn let us dene (see GZ1]) a convex polytope
K ( ) = fx 2 K ( ) j d(ji) (x) i ; i+1 1 j i ng:
(2:6)
Proposition 2.9. Let V , V
, V be three irreducible nite-dimensional
sln -modules with the highest weights . Then
i) (Gelfand-Zelevinsky's theorem)
c
= jK ( n ) \ (Xn)Z j
ii) The involution qn;1 denes a bijection
qn;1 : K ( n ) ! K ( n ):
(2:7)
The bijection (2.5) gives a combinatorial explanation of the well known equality
c
= c
. In oder to understand better the other symmetries of the LR-numbers,
it is convenient to use an interpretation of the LR-rule in terms of BerensteinZelevinsky's triangles (BZ-triangles) BZ3]. Let us review brie!y the corresponding
construction from BZ3],
C]. Denote by Tn n 3, the set of vertices of a regular
triangular lattice lling the regular triangle with vertices (2n;3 0 0), (0 2n;3 0)
and (0 0 2n ; 3) this triangle is decomposed into the union of elementary triangles
having all three vertices in a set Qn , and of elementary hexagons centered at points
Hn , where
Hn := f(i j k) 2 Tn j all i j k are oddg
Qn = Tn n Hn :
Let us consider the vector space L consisting of families (z( )) 2 Qn of real
numbers such that for any elementary hexagon the sums of summits of opposite
sides are equal let also K = Kn := L \ RQ+n and KZ := L \ ZQ+n . We dene a linear
projection pr : L ! R3n;3 by the formulas
pr(z) = (l1 ln;1 m1 mn;1 k1 kn;1)
34 Anatol N. Kirillov and Arkadiy D. Berenstein
where
lp = z(2(n ; p) ; 1 2p ; 2 0) + z(2(n ; 1 ; p) 2p ; 1 0)
kp = z(0 2(n ; p) ; 1 2p ; 2) + z(0 2(n ; 1 ; p) 2p ; 1)
mp = z(2p ; 2 0 2(n ; p) ; 1) + z(2p ; 1 0 2(n ; 1 ; p)):
(2:8)
Now let V , V
, V be three irreducible nite dimensional sln-modules with the
highest weights , , and . We recall that the triple multiplicity c
is dened
as dim(V V
V )g . Evidently we have c
= c
, where is the highest
weight of the module V dual to V (see also Proposition 2.8). Let us denote
by BZ ( ) a set of families z( ) 2 Kn such that pr(z( )) = (l m k) where
li = i ; i+1 mi = i ; i+1 and i = ki ; ki+1 for all i 1 i n ; 1 (we assume
that ln = mn = kn = 0). We dene a mapping
: K ( n ) ! BZ ( )
(2:9)
by the following way: given a triangle x 2 K ( n ), let us put (x) = z( ),
where
z(2i ; 2 2(j ; i) ; 1 2(n ; j )) = xij ; xij;1 1 i j n
and observe that the value of z( ) in all other points 2 Qn are uniquely determined
from the condition z( ) 2 BZ ( ) (see BZ3]).
Proposition 2.10. Assume V , V
, V as before.
i) (Berenstein-Zelevinsky BZ3]). The mapping (2.9) is a bijection. In particular,
c
= jBZ ( )j:
(2:10)
;1 qn;1 denes a bijection
ii) A transformation qen;1 := BZ ( )
??
y
K ( n )
eq;!
n
;1
qn
;!1
;
BZ ( )
??
y
(2:11)
K ( n )
Now let as summarize our discussion of the symmetries of the LittlewoodRichardson numbers, or, that is the same, the symmetries of the triple multiplicities c
. Clearly, coecients c
must be invariant under all 6 permutations of
( ), and also under the replacement of ( ) by ( ). These transformations generate a group of 12 symmetries. Obviously, the set Qn and Hn are
invariant under all permutations of indices (i j k). Therefore we have a natural action of S3 on RQn , and it is evident that L, K and KZ are invariant under this action.
Formulas (2.8) imply at once that if pr(z) = ( ) then pr(s1(z)) = ( )
and pr(s2(z)) = ( ), where z := z( ) 2 BZ ( ), and s1 = (1 2) and
Combinatorics of the Gelfand-Tsetlin patterns
35
s2 = (2 3) be two standard generators of S3 . We see that the expression (2.10) of
triple multiplicities (the Berenstein-Zelevinsky theorem BZ3]) makes evident the
following symmetries :
c
= c = c
= c
= c
= c :
(2:12)
The remaining symmetries can be derived from Proposition 2.10, ii), namely, the
transformation qen;1 (see (2.11)) denes the bijection BZ ( ) ! BZ ( ),
which gives a combinatorial proof of the equality c
= c
. All other symmetries
follow from (2.12) for the triple ( ).
Finally, let us consider an explanatory example. Assume n = 4 = (6 2 1),
= (5 2 1), = (6 5 2). It is easy to see that c
= 4. Let us consider the
following tableaux
112224
122334
T : 23
2 STY ( n k) T 0 : 3 3
2 STY ( n k ):
3
4
It is easy to varify that T 0 = S(T ), where S is the Schutzenberger involution (see
Remark 2.3). Futher, let us construct the corresponding GT-patterns
6 2 1 0
6 2 1 0
x(T ) : 5 2 1 2 K ( n ) x(T 0 ) : 5 2 0 2 K ( n ):
5 1
3 0
2
1
It is easy to see that x(T 0 ) = q3 (x(T )) q3 = t1 t2t1 t3 t2t1 . Finally, we construct the
BZ-triangles corresponding to the GT-patterns x(T ) and x(T 0). For lucidity let us
give a construction in the several steps:
1
0
1
6 2 1 0
0
1
0
0
1
0
5 2 1 ! 0 1 !
!
0
0
1
3
5 1
0
1
3
3
3
1
2
1 0 3 1 0 0 1 3 0
K ( n )
6 2 1 0
5 2 0
3 0
1
BZ ( )
0
1
3
1
1 0 1
1
0
!
! 2 2 !
0
2
2
1
0
2
2
3
1
1
1 0 2 1 2 0
1 2 2 K ( n )
BZ ( )
The second triangles are composed from the dierences fxij ;xij;1 1 i < j ng
for the corresponding triangle x 2 Xn .
36 Anatol N. Kirillov and Arkadiy D. Berenstein
D) The
action
of
the
ui
involutions
2
Gn 1 i n ; 1.
Let a tableau T 2 STY ( n ) be given. Note that the involution un;i 1 i n ; 1 acts nontrivially only on the skew tableau Ti , where Ti is the part of
tableau T which is lled by the numbers i i + 1 n. So it is sucient to describe
the action of the involution un;1 on the set
STY ( n ) KZn ( ) Xn:
Theorem 2.2. Assume = . The involution un; on the set STY ( )
coincides with the dual Schutzenberger involution U , where U := pnn; S.
1
1
Let us continue the construction of natural operations on the set STY ( n )
(see Section C). We remind (see Section B) that p;n;1 1 (T ) = Tb, where
Tb 2 STY ( n ), b = (n 1 n;1 ). In fact, the tableau Tb is a disjoint
union of the tableau T of the content = ( 1 n ), where 1 = 0 i = i;1 2 i n, and the lled horizontal strips 1 j1j = n , with the number 1 in all
boxes. As well as in Section C we may go further in two ways. In the rst, let
us apply the same algorithm to the tableau T and so on. As a result we obtain
a sequence of horizontal strips 1 2 n such that i = n;i+1 1 i n,
;
which denes a standard Young tableau T 2 STY ( n ). Consequently there
are maps
U : STY ( n ) ! STY ( n ;) T ! T U : KZn ! KZn :
(2:13)
The mapping U is called the dual Schutzenberger involution. If the weight has
equal parts, we obtain the involution U : STY ( ) ! STY ( ).
Another way is to apply the transformation p;n;1 1 to the tableau T n times, i.e.
consider a tableaux p;n;n1 (T ). It is possible to show that, in fact, the two denitions
of the involution U coincide. So, p;n;n1 = SU. From the denition of the dual
Schutzenberger involution U it is clear that un;1 jSTY () = U.T his ends the
proof of Theorem 2.2.
Remark 2.5. In Sections C and D we gave combinatorial denitions for transformations S U n = SU and ;n = US .
Our notations dier from one in Sch1], Sch2] and EG] for the case = (1n ).
We use words \a jeu de taquin" instead of \a promotion transformation T ! T @ "
(e.g. Sch1], Sch2] ). In fact, using the notations from EG], we have (on the set
STY ()) :
S(T ) = T @ + 1 U(T ) = T S SU = p:
As an illustration of the operations S U n , and ;n , consider the standard Young
tableau
Combinatorics of the Gelfand-Tsetlin patterns
T
1 2 3 8 10
7 13
= 45 69 12
11
0
1
2
4 7
1
0
1 1
1
0 0
S(T )
1 2 3 8 9
5 7 10
= 46 12
13
11
0
1
3
5 7
1
1
0 1
1
0 0
1 4 6 10 11
2 5 8 12
3 7 13
9
0 1
0
0
0 3
2
3
3
0
0 0
0
0
1 2
1 4 5 7 11
SU(T ) = 23 68 139 12
10
1 1
0
0
1 3
3
3
3
0
0 0
0
0
1 2
1 2 4 5 11
US(T ) = 37 126 138 9
10
1
0
2
2
1
2 5
4
U(T ) =
1
0 2
1
37
0 0
Note that the tableaux T and S(T ) have the same conguration and the complementary quantum numbers, i.e.
(k )
Jn
(T ) + Jm(kn)( k );+1n (S(T )) = Pn(k) ( )
( )
see Remark 2.8. It is a general property of the Schutzenberger involution, Ki1].
Remark 2.6. The involutions S and U in general do not commute. It is an
interesting combinatorial task to nd the order of the transformation SU on the
set STY ( ). In some special cases the answer is known (e.g. Sch1], EG]). Using
this answer and formula (1.20), we nd
38 Anatol N. Kirillov and Arkadiy D. Berenstein
Proposition 2.11.
a) If is a rectangular diagram, l() n then
n jKZ = SU = IdKZ :
(2:14)
b) If = (n ; 1 1 0) be a staircase diagram, then
2njSTY () = (SU)2 = IdSTY () :
(2:15)
In fact, if T 2 STY (), then n(T ) is the conjugating of T .
Assertion a) was proved by M.-P. Schutzenberger Sch1] and later rediscovered
by T. Miwa Mi] in a context of the theory of crystal base (e.g. Ka1]). The
assertion b) was proved by P. Edelman and C. Greene EG] in the context of the
theory of balanced tableaux. Both the proof in Sch1] and that in EG] depend on
the consideration of the \geometrical" properties of the involutions S and U. It is
desirable to nd the algebraic proofs of (2.14) and (2.15). Note that we have also
another distinguished involution W0 2 Gn , where w0 is the element of maximal
length in the symmetric group Sn
W = s s|{z}
s s| s{zs} s|n; sn;{z s s}
W : K ! K (W (x)) = w (x) = ;(x):
It seems plausible that the involutions S and W are commuted on the set KZ (but
0
0
1
2 1
3 2 1
1
2
0
2 1
0
0
not on all Xn !). In any case it is interesting to understand the structure of the
group generated by S U and W0 .
E) The action of the symmetric group Sn on the set STY ( ) (compare with
LS2],
LS3]).
Given a tableau T 2 STY ( ) and involution si 2 Gn . Consider a word w(T )
which corresponds to the tableau T . Recall that by denition the word w(T ) may
be obtained from the tableau T in the following way : let us read the elements of the
tableau T consecutively from the right to the left, starting from the upper row. We
obtain the word w(T ) under consideration (see e.g. Ma]). For convenience we will
denote i by a and i +1 by b . First, extract from we word w(T ) a subword w0 which
contains only letters a and b . Secondly, in the word w0 we subtract consecutively
all pairs ab . As a result we obtain a subword of type bm an . Replace it by the word
bn am and after this recover all subtracted pairs and all letters which dier from a
and b . As a result we obtain a new word we . To this word corresponds some Young
tableau Te. Let us denote this tableau by i (T ) := Te.
Theorem 2.3. We have
si (T ) = i (T ) 1 i n ; 1:
Remark 2.7. The fact that the involutions i 1 i n ; 1, generate the
symmetric group was proved by A. Lascoux and M.-P. Schutzenberger LS2] using a
Combinatorics of the Gelfand-Tsetlin patterns
39
delicate analysis of the properties of the plactic monoid and the Robinson-Schensted
correspondence. The same action was rediscovered by M. Kashiwara Ka2] in the
context of the theory of crystal base, as an action of the Weyl group W (in our case
Sn ' W ) on the crystal base, corresponding to an irreducible representation (irrep)
V with the highest weight of the Lie algebra gln . Our denition of the involution
si also appears as an attempt to understand the action of the Weyl group W but
on the Gelfand-Tsetlin basis (e.g. GZ3] ) of the irrep V . In fact, it is possible to
show that if si 2 W ' Sn 1 i n ; 1, then for x 2 KZ we have
si jx >= jsi (x) > +
X
y<x
xy jy >
with respect to some ordering on the set KZ . Here for x 2 KZ the symbol jx >
means a base vector in the space V which corresponds to GT-pattern x.
Remark 2.8 Let us give another description for the action of the symmetric
group on the set of standard Young tableaux, based on the concept of rigged congurations, Ki1],
Ki2]. From the beginning, we remind some denitions from Ki1].
Given a partition and composition l( ) N , a conguration f g of the type
( ) is, by denition, a collection of partitions (1) (2) such that
(i) j (k) j =
X
j j k+1
k
(ii) Pn ( ) := Qn ( (k;1) ) ; 2Qn ( (k) ) + Qn ( (k+1) ) 0 for all
( )
P
k n 1
where (0) := , and for any composition we put Qn ( ) := k min(n k)k .
According to the ultimate traditions of the Bethe ansatz KR], the numbers
(k )
Pn ( ) are called by the vacancy numbers. We dene a cocharge c( ) of the
conguration f g in the following manner:
X ( )0n + 1 X ( k )0n ; ( k )0n c( ) =
+
:
(1)
n1
( +1)
2
kn1
( )
2
(2:16)
Now we want to dene a set of rigged congurations QM ( ) of the type ( ).
By denition, a rigged comguration of the type ( ) is a conguration of the type
(k )
( ) together with a set of integers Jn
k n 1 1 s s := mn ( (k) ),
which satisfy the following inequalities
k) J (k) J (k) P (k) ( )
0 J1(n
sn
n
2n
where mn ( (k) ) = ( (k) )0n ; ( (k) )0n+1 .
The main property of rigged congurations is that they give another way to
describe the set of Young tableaux, namely, there exist a bijection KR],
Ki1]:
" : STY ( ) ;! QM ( ):
40 Anatol N. Kirillov and Arkadiy D. Berenstein
This bijection has many interesting properties KR],
Ki1],
Ki2], but just now let
us remark that the set QM ( ) depends only on the partition, corresponding to
the composition . Thus we may dene an action of the symmetric group SN on
the set of standard Young tableaux STY ( N ) using the commutative diagram
( 2 SN ):
STY ( ) ;!
QM ( )
?
" ?y
(2:17)
;1
STY ( ) ;
QM ( )
Proposition 2.12. Let be a partition, l() N , and T 2 STY ( N ).
Then
si (T ) = "(ii+1) (T )
where the map "(ii+1) corresponds to the vertical arrow in (2.17) when = (i i +1)
is a simple transposition.
From Proposition 2.12 follows that all Young tableaux lying in the same orbit
w.r.t. the action of the symmetric group generated by the involutions si (see (1.16)),
1 i N ; 1, have the same rigged conguration. In particular, the cocharge of
Young tableau (see (2.16)) is an invariant of the action of the symmetric group.
F) Combinatorial interpretation of the transformations fi and ei and the crystal
graph (e.g. Ka1],
Ka2]).
Let us recall at rst the construction of the action of the symmetric group Sn
on the set KZ (see Section E or LS2]). Given tableau T 2 STY ( ), consider the
word w(T ) and the corresponding reduced subword bm an . But now replace this
subword by bm+1 an;1 , if n 1, or 0, if n = 0 (respectively by bm;1 an+1 if m 1,
or 0, if m = 0) and after this recover all subtracted pairs ab and all letters which
dier from a and b. We obtain a new word we (corr. a word w). To this word there
corresponds some Young tableau Te (resp. T ).
Theorem 2.4. We have for T 2 STY ( n):
fi T = Te ei T = T 1 i n ; 1:
Now let us consider a colored graph ;Z (). The vertices of this graph correspond
to all standard Young tableaux T of the shape with all entries n. Two vertices
i T i T = f T (or equivalent,
T1 and T2 are connected by an edge of color i T1 ;!
2
2
i 1
T1 = ei T2 ).
Theorem 2.5. The graph ;Z () coincides with the crystal graph corresponding
to the irreducible representation V of the Lie algebra gln with the highest weight
.
Combinatorics of the Gelfand-Tsetlin patterns
41
Remark 2.9. The description of the edges of the crystal graph B(V ) was
obtained at rst by M. Kashiwara and T. Nakashima KN],
N] in pure combinatorial
terms. We gave in fact the same description. The dierence is that M. Kashiwara
and T. Nakashima in KN] give a description of the action of the generators fei and
ei on the set STY ( n). Our result gives an opportunity to extend this actiono
to the Gelfand-Tsetlin polytope K and obtain some subdivision of the convex
polytope K on coloring parts.
Remark 2.10. Let U = Uq (sl(n)) is the q-analog of the universal enveloping
algebra of the Lie algebra sl(n). The algebra U is the algebra over Q(q) generated
by Ei Fi and Ki1 1 i n ; 1 (see e.g. J],
Ka1]). We denote by U; a
multiplicative monoid in U generated by F1 Fn;1 . Let G be a multiplicative
(1)
monoid in Aut(Xn)) generated by all transformations s(1)
1 sn;1 (see (1.16)).
Conjecture 2.2. The correspondence Fi ;! s(1)
i 1 i n ; 1 denes an
isomorphism U; ' G of the multiplicative monoids.
Let us remark, that for any k l 2 Z+ we have the following relation (e.g. Lu2]):
Fik Fik+1+l Fil = Fil+1 Fil+k Fik+1 1 i n ; 2:
x3 Cocharge and Gelfand-Tsetlin patterns.
First we remind the denition of the charge and cocharge of a tableau, according
to A. Lascoux and M.-P. Schutzenberger LS1],
Ma]. Let and be partitions,
T 2 STY ( ). Consider the word w(T ), which corresponds to tableau T (e.g.
Ma]). We dene the charge c(T ) of tableau T as the charge of corresponding word
w(T ). Now let us dene the charge of a word w. Remind that the weight of a
word w is a sequence = (1 2 ), where i is the number of i appearing in the
word w. We assume that the weight of the word w is dominant, i.e. 1 2 .
(i) We rst assume that w is a standard word, i.e. its weight is = (1N ). Let
us index all elements of w as follows: the index of 1 is equal to 0, and if the index of
k is i, the index k + 1 is either i or i + 1 according to the location of k + 1 either to
the right or to the left of k. The charge c(w) of w is then the sum of all its indices.
(ii) Assume now that w is a word of weight and is a partition. We extract
a standard subword out of w in the following way. Reading w from left to right
we choose the rst entry of 1, then the rst entry of 2 to the right of the 1 chosen
and so on. If at some step there is no s + 1 to the right of the s chosen before, we
come back to the beginning of the word. This operation extracts from w a standard
subword w1 out of w. Let us delete the word w1 from w and repeat the operation,
thus obtaining w2 , etc..
The charge of w is dened toPbe the sum of the charges of the standard subwords
obtained in this way : c(w) = c(wi ). We note that the charge of w is zero i w
is a lattice word.
42 Anatol N. Kirillov and Arkadiy D. Berenstein
Now let and be partitions and T 2 STY ( ). We dene the cocharge of
the tableau T as
c(T ) = n() ; n() ; c(T ):
(3:1)
We give another method for computing of the cocharge of a tableau T 2 STY ( ).
First assume that T 2 STY (). For any x 2 T let us denote by n(x) an index of
the row which contains x. For any x 2 T we dene c(x) by induction : c(1) = 0,
n(x) ; n(x + 1)
c(x + 1) = c(x) + n(x) ; n(x + 1) + 1 ifif nn((xx))<nn((xx++1)1)
(3:2)
P
and put c(T ) = x2T c(x).
Proposition 3.1. Assume that T 2 STY (). Then
(i) c(x) 0 for all x 2 T .
(ii) c(T ) is equal to the cocharge of the tableau T , as it was dened by (3.1).
(iii) Let us dene the descent set D(T ) of a tableau T as
D(T ) = fx 2 T j n(x) < n(x + 1)g
and put
des(T ) =
Then
X
x2D(T )
x p =
X
x2D(T )
(3:3)
1:
c(T ) = pN
N;des(T ) ; n()
c(T ) = 2 ; pN + des(T ) where N = jj:
It is convenient to consider a tableau C (T ), which obtained from the tableau T
by replacement each entries x 2 T by c(x). It is possible to show that C (T ) is, in
fact a reverse plane partition.
Let us consider a clarifying example.
T :=
1
3
6
9
13
2 4 10
5 8 12
7 14
11
C (T ) :=
0
0
1
1
2
0 1 4
1 2 4
1 4
2
In our case we have D(T ) = f2 4 5 8 10 12g p(T ) = 6 des(T ) = 41 n() = 20
and
X
c(T ) :=
c = 23 = 6 14 ; 41 ; 20:
c2C (T )
Secondly, if T 2 STY ( ) be a partition, we act as in step (ii) of
Lascoux-Schutzenberger's algorithm : consider the most left and upper 1 in the
tableau T and compute c(1) = n(1) ; 1, after this consider the most left 2 such that
Combinatorics of the Gelfand-Tsetlin patterns
43
n(2) > n(1) and compute c(2) according to (3.2) and so on. If at some step there
is no x + 1 with condition n(x + 1) > n(x), then consider the most left and upper
x + 1 2 T , and compute c(x + 1)(using (3.2)). This operation extracts from the
tableau T a subset w1 . Let us delete this subset w1 from the tableau T and repeat
the previous operation, thus obtaining w2 etc.. The cocharge of the tableau T is
dene to be the sum
XX
c(T ) =
c(x):
i x2wi
We note that some of the numbers c(x) x 2 T , in general, may be negative.
Thus we dene the charge and cocharge of a tableau T 2 STY ( ), when is a partition. We know (see Section 1, or GZ1],
BZ1]) that STY ( ) is a set
of integral points in the convex polytope K () K , so it is natural to ask is
it possible to continue the cocharge (or charge) from the set STY ( ), is a
partition, on all convex polytope K in a natural manner? It is easy to see that the
charge does't possess a good properties with respect to the action of the symmetric
group Sn on the Gelfand-Tseitlin polytope K .
The main result of this Section, Theorem 3.1, asserts that such continuation
really exists and has the nice properties (at least for n 5). Thus we want to
construct a function cn : Xn ! R, which coincides with cocharge on the integral
points set of the cone Kn (at least for n 5 and hypothetically for all n). To start,
let us dene an imbedding of the symmetric group Sn into the set (Kn)Z :
pn : Sn ,! (Kn)Z :
Denition 3.1. Let us denote by Pn a set of all Gelfand-Tsetlin patterns
p := (pij ) 2 (Kn)Z , which satisfy the following conditions
i) pn;1n;1 = pnn = 0
ii) pij ; pi+1j+1 1 1 i j n ; 1
iii) pin ; 1 pii pi+1n 1 i n ; 1:
(3:4)
Proposition 3.2. There exists a bijection
: Pn ' 0 n ; 1] 0 n ; 2] 0 1] 0 0]:
In particular, jPn j = n!.
Proof. Let p = (p(n) p(1) ) 2 Pn . We will construct a vector k = (k1 kn )
by the following rule. For the beginning, let us observe that for given i 1 i n
from inequality 0 pin ; pii 1 follows an existence of integer j 0 j n ; i,
such that
pii = pii+1 = = pii+j < pii+j+1 ; = pin :
In this case we put ki = j . The correspondence p ! k is desired bijection.
44 Anatol N. Kirillov and Arkadiy D. Berenstein
As an illustrating example, let us take n = 4 and k = (3 1 0 0). At rst we
observe that p34 = 1. Secondly, from inequalities (3.4) we nd that p34 p23 =
p33 p34. Consequently, p23 = p33 = p34 = 1 and p24 = 2. Further, using the
same trick, we obtain that p24 p14 = p13 = p12 = p11 p24. Thus
2 2 1 0
p= 2 1 0
2 1
2
Let us continue and consider the following linear functionals on the space of
triangles Xn+1 :
i) '0n : Xn+1 ! R, given by a formula
'0n(x) =
nX
;1
j =1
(n ; j )(xjn+1 ; xjn):
(3:5)
ii) For every p 2 Pn 'p : Xn+1 ! R, given by a formula
'p (x) =
X
ij n
1
pTij 'n;j+1n;i+1 (x)
(3:6)
where if p = (pij ) 2 Pn , then pT := (pTij ) pTij = pn;j+1n;i+1 , and the linear
maps 'ij : Xn+1 ! R are dened by (1.32).
At last, we dene a function cn+1 : Xn+1 ! R by the following inductive formula
i) c1 (x) = 0 if x 2 X1
ii) cn+1 (x) = cn (x) + '0n (x) + minf'p (x)jp 2 Pn g
(3:7)
where for any triangle x 2 Xn+1 x = (x(n+1) xe) xe 2 Xn , we set by denition,
that cn (x) = cn (xe).
Now we are ready to formulate our main result of this Section.
Theorem 3.1. The function cn : Xn ! R is a continuous, piecewise linear and
satises the following properties
1) invariance:
(i)
(ii)
(iii)
cn (x) = cn (x) for any 2 Sn cn (Ix) = cn (x) where the map I is dened in (1:38)
cn (T1 x) = cn (x) if T+1 x 6= 0 (corr: T;1 x 6= 0):
2) stability: if ' : Xn;1 ! Xn = 1 2, are the embeddings given by (1.40),
then
cn (x) = cn;1 (xe)
.
Combinatorics of the Gelfand-Tsetlin patterns
3) positiveness:
45
cn (x) 0 for all x 2 Kn :
We assume that the function cn (x) given by (3.7) possesses an additional property:
4) Main Conjecture: if and be the partitions l() n l() n,
T 2 STY ( ) and x(T ) be its image (see Section 2) in the set KZ , then
cn (x(T )) = cn (T ):
Corollary 3.1. (of the Main Conjecture) Let and be the dominant weights
of the Lie algebra sln , and K (q) be the Kazhdan-Lusztig polynomial for the
ane Hecke algebra (e.g. Lu1],
Ma]) corresponding to and . Then
K (q) =
X
x2K (
)\(Xn )Z
qcn (x) :
(3:8)
Corollary 3.2. (of Theorem 3.1) Let be a dominant weight and be any
(integral) weight of the Lie algebra sln. Let us dene a polynomial
K (q) :=
X
x2K ( )\(Xn )Z
qcn (x) :
(3:9)
Then, K (q) = K w()(q) any w 2 Sn .
Problem 3.1. Assume 2 Rn . To compute the following integral
cont (q ) :=
K
Z
K ( )
exp(h cn (x))d(x) q = exp(h)
n(n+1)
(3:10)
where (x) is the Lebesque measure on K ( ) induced from R
.
cont
cont (q ) as
It is clear that K (1) = V ol(K ( )). We consider the functions K
a continuons analog of the Kostka-Foulkes polynomials.
We postpone the proof of Theorem 3.1 to the end of Section 3 (see Remark 3.1),
while right now let us consider in more details the particular cases n = 3 and n = 4.
Our goal is to prove for these cases the Main Conjecture.
Let us start with the case n = 3. Given a triangle x 2 X3, we recall that
2
'22 (x) := x12 + x22 ; x11 ; x23
'12 (x) := x12 + x22 ; x13 ; x33:
It is easy to see that
x12 (s2(x)) = x12 ; ('12); ; ('22)+ x22 (s2(x)) = x22 ; ('12)+ ; ('22); '12 (s2(x)) = ;'22 (x) '12(s2(x)) = ;'12 (x)
'22 (s1(x)) = x11 ; x23 '22(s1(x)) = '22 (x):
46 Anatol N. Kirillov and Arkadiy D. Berenstein
Lemma 3.1. In the case n = 3 we have
c3 (x) = min(x13 ; x12 x22 ; x33 )
c3 (q2 (x)) = min(x23 ; x22 x12 ; x11) + (x11 ; x23); (3:11)
(3:12)
where q2 = t1t2 t1 2 G3 , x 2 X3 and (a)+ := max(x 0) (a); := min(x 0).
Proof. According to Denition 3.1, we have
P2 = 1 0 0 0 (P2) = f (0 0) (1 0) g
0
0
T
' 1 0(x) = p12 '12 (x) = '12 (x)
0
' 0 0(x) = 0 '02(x) = x13 ; x12:
0
Consequently,
c3 (x) = x13 ; x12 + ('12); = min(x13 ; x12 x22 ; x33):
The formula (3.12) follows from the description of Schutzenberger involution q2 :
x12 (q2 (x)) = x13 ; min(x11 ; x22 x12 ; x23)
x22 (q2 (x)) = x33 + min(x12 ; x11 x23 ; x22)
x11 (q2 (x)) = 3 (x)
The function (3.11) satises all properties 1) - 4) of Theorem 3.1 (see Proposition 3.3). The function (3.12) satises the properties 1) - 3), except 1(iii) and,
in particular, is invariant with respect to the action of the symmetric group S3 .
Further, for the dual Schutzenberger's involution u2 we have
x12(u2 (x)) = x12 ; ('12); ; (x11 ; x23)+ x22(u2 (x)) = x22 ; ('12)+ ; (x11 ; x23); x11(u2 (x)) = 3 (x):
It is easy to see that c3(u2 (x)) = c3 (x), and
q2 u2 = u2 q2 :
(3:13)
The equality (3.13) is a corollary of the identity (t1 t2)6 = 1, since q2 = t1 t2t1 and
u2 = t2 t1t2 . In general cn (un;1 (x)) 6= cn (x), and SU 6= US.
Now consider the case n = 4. Given a triangle x 2 X4, we recall that
'13 (x) := x13 + x33 ; x14 ; x44
'23 (x) := x13 + x23 ; x12 ; x24
'33 (x) := x23 + x33 ; x22 ; x24:
Combinatorics of the Gelfand-Tsetlin patterns
47
It is possible to check (see Theorem 1.3), that
x13 (s3(x)) = x13 ; ('23 + ('33); )+ ; ('13 + ('33)+ ); x23 (s3(x)) = x23 ; ('33 + ('13); )+ ; ('23 + ('13)+ ); x33 (s3(x)) = x33 ; ('13 + ('23); )+ ; ('33 + ('23)+ ); :
(3:14)
Proposition 3.3. Assume x 2 X , then the function
4
c4 (x) = min(x13 ; x12 x22 ; x33 ) + x14 ; x13 + x33 ; x44+
(3:15)
+ min(x23 ; x34 x24 ; x23 x13 ; x12 x22 ; x33 4 (x) ; x34 x24 ; 4 (x))
satises the all properties 1) - 4) of Theorem 3.1.
Proof. First of all, let us check that our previous denition (3.7) of the cocharge
cn coincides with (3.15). According to Denition 3.1, we have
8
9
< 1 1 0 1 0 0 1 0 0 2 1 0 1 1 0 0 0 0=
P = : 1 0 1 0 0 0 1 0 1 0 0 0 0
0
0
1
1
0
3
(P3) = f (000) (010) (110) (100) (200) (210) g:
Using the denition (3.6), we nd
' 1 1 0 (x) = '12 + '13 + '23
1 0
0
' 1 0 0 (x) = '13 + '23 1 0
0
' 2 1 0 (x) = '12 + 2'13 + '23 + '33 ' 1 0 0 (x) = '13 1 0
1
' 1 1 0 (x) = '12 + '13 + '23 + '33
1 0
1
0 0
0
' 0 0 0 (x) = 0:
0 0
0
Furthermore, it follows from (3.5) that
'03 (x) = 2(x14 ; x13) + (x24 ; x23 ) =
= x14 ; x13 + x33 ; x44 + x24 ; x23 ; '13:
After some not so complicated calculations we can obtain from (3.7) the formula
(3.15).
Secondly, let us check that the function c4 (x) is an invariant with respect to the
action of the symmetric group S4 . Note that the invariance under the action of
generators s1 and s2 is clear from (3.5). In order to prove the s3-invariance of the
c4(x) we make use of a few lemmas.
48 Anatol N. Kirillov and Arkadiy D. Berenstein
Lemma 3.2. Assume x 2 X , then
4
('3 (s3(x))); = x3(s3(x)) ; x3 + ('3(x)); = 1 2 3
(3:16)
where for any a 2 R (a); := min(a 0) (a)+ := max(a 0).
Proof. Using the formulas (3.14) for the action of s3 on the space of triangles
X4 , we nd
'e13 = ;'23 ; '33 + ('23 + ('13)+ ); + ('33 + ('13); )+ where for any function ' on the space of triangles we exploit a notation 'e :=
'(s3 (x)). In order to calculate ('e13); , we make use of the following identity, which
may be varied by direct computation
;a ; b + (a + (c); )+ + (b + (c)+); ]; = (c); ; (b + (a);)+ ; (c + (a)+); :
Consequantly,
('e13 ); = ('13 ); ; ('23 + ('33); )+ ; ('13 + ('33)+ ); = xe13 ; x13 + ('13); :
Similarly we may prove the corresponding formulas for = 2 or 3.
Corollary 3.3 (of Lemma 3.2). The following functions are s -invariance
3
min(x14 ; x13 x33 ; x44 )
min(x13 ; x12 x24 ; x23 )
(3:17)
min(x23 ; x22 x34 ; x33 ):
Lemma 3.3. The expression for c4(x), given below, is equivalent to (3.15):
c4 (x) = x14 ; x13 ; x44 + c3(x) + c3(s3x) + xe33 + ('33 + ('23); ); :
The proof is based on a direct computation.
Thus, we reduce the proof of s3-invariance for the cocharge c4 (x) to that for the
function
('33 + ('13 ); ); ; x13 ; x33:
The s3 -invariance of the last function may be easily deduced from Lemma 3.2.
Finally, we must prove that if T 2 STY ( ) then c4 (x(T )) coinsides with the
cocharge of tableau T .
For this purpose we use a description of Young tableaux by means of rigged
congurations Ki1],
Ki2]. More precisely, let T be a standard Young tableau
T :=
d11
d12
d13
d22 d23 d24
d33 d34
d44
d14
Combinatorics of the Gelfand-Tsetlin patterns
49
where dij is the number of j 's lying in the i-th row of T . We shall construct the
corresponding rigged conguration step by step:
0 d22 0 ;! 0 d22 0 0 d33 0 ;! 0 d22 d23 P 0 d33 0 ;!
0 d33 0
0 d33 0
c3 ( ) = 0
c3 ( ) = 0
c3 ( ) = 0
;! J d22 + d23 P + J
0 d33 0
c3 ( ) = 0
0 d33 0 ,
where P = min(d11 ; d22 d23 ) + (d12 ; d23); J = min(d13 d22 ; d33).
Consequently, c3 (T ) = c( ) + J = min(x13 ; x12 x22 ; x33 ) = c3(x(T )). Let's
go further and start to join successively the parts d44 and d34 .
J d22 + d23 P + J
0 d33 d34 Q
0 d44 0
0 d33 d34 0
0 d44 0
0 d44 0
c3 ( ) = 0
where Q = min(d11 ; d33 d34) + min(d12 + d22 d33 + d34) ; 3d34.
Finally, let us add the parts d24 and d14 . As a result we obtain the
following rigged conguration:
Je1 d22 + d23 d24 ; M Pe
Je2 d33 + d34 M Qe
0 d44 0
d33 + d34 0
d44 0
d44 0
c4 ( ) = 2M
where
M = min(d22 + d23 ; d33 ; d34 min(d13 d22 ; d33 ))
Je1 = min(d14 d22 + d23 ; d34 ; d44 ; M ) + min(d13 d22 ; d33 ) ; d24 ] Je2 = min(d33 ; d44 d14 + d24 ; M ) + min(d13 d22 ; d33) ; M ;
(3:18)
; min(d13 d22 ; d33 ) ; d24 ] 50 Anatol N. Kirillov and Arkadiy D. Berenstein
and := (T ) is equal to 1, if d24 min(d22 + d23 ; d33 ; d34 d22 ; d33 d13) and
equals to 0 otherwise.
The exact values of the vacancies numbers Pe and Qe are not essential for a
computation of the cocharge. Consequently, c4 (T ) =
= c( ) + Je1 + Je2 = c3 (x) + d14 + d33 ; d34 + min(M d14 + d24 ; d33 + d44)+
+ min(M d22 + d23 ; d14 ; d34 ; d44) ; M:
It is easy to check that this expression for c4 (T ) coincides with c4 (x(T )). The proof
of Proposition 3.3 is nished.
Let us write also a formula for the action of Schutzenberger involution
q3 = t1t2 t1 t3t2 t1 on the space of triangles X4: if x 2 X4 and xe = q3 (x), then
xe13 = x14 + x24 ; x13 + ('23 + ('22 )+ )+
xe23 = x24 + x34 ; x23 + ('23 + ('22 )+ ); + ('33 + ('22 ); )+ xe33 = x34 + x44 ; x33 + ('33 + ('22 ); ); xe12 = x14 + x24 + x34 ; x13 ; x23 + ('33 + ('23)+ )+ xe22 = x24 + x34 ; +x44 ; x23 ; x33 + ('23 + ('33); ); xe11 = 4 (x) xe4 = x4 if = 1 2 3 4:
From the description of the action of the symmetric group Sn on the set
STY ( n) one can deduce that all Young tableaux belonging to the same Sn orbit have the same rigged conguration. Thus, we see that all parameters of a
rigged conguration (for example, its quantum numbers or vacancies numbers) are
Sn -invariant. The computations contained in Proposition 3.3 seems to be interesting, because they shed some light on the deep invariants of Young tableau, namely,
its quantum numbers. At this moment we don't know whether or not the quantum
numbers (3.18) considered as the functions on the space X4 are S4 -invariants. By
any way, it seems a very interesting task to nd a \complete list" of the continuous, piecewise linear invariants (cpl-invariants) for the xed involution sj . In this
direction we are going to prove the following result.
Theorem 3.2. Let j be an integer, 2 j n ; 1. Then the following functions
on the space Xn are sj -invariants:
1j (x) = min(x1j+1 ; x1j xjj ; xj+1j+1)
(3:19)
ij (x) = min(xi;1j ; xi;1j;1 xij+1 ; xij ) 2 i j n ; 1
1j (x) = (min('jj (x) ;'1j (x)))+
(3:20)
ij (x) = (min('i;1j (x) ;'ij (x)))+ 2 i j n ; 1:
Conjecture 3.2. The group of all continuous, piecewise linear automorphisms
of the space Xn , having the same set of invariants (3.19) and (3.20), coincides with
the cyclic group of the second order < 1 sj >.
Combinatorics of the Gelfand-Tsetlin patterns
51
It seems plausible that the set of sj -invariants (3.19) and (3.20) is a fundamental
system of continuous, piecewise linear sj -invariants, i.e. any cpl-invariant with
respect to the action of sj on the space Xn is a min ; max linear combination of
that (3.19) and (3.20).
Proof (of the Theorem 3.2) We have to prove that for 2 j n ; 1 all functions
ij and ij , 1 i j , are the sj -invariants. We shall prove it using the inductive
formula (1.29) for sj :
sj = Rj;1j j ; 1 j ]
j j + 1]sj;1
j j + 1]
j ; 1 j ]Rj;1j :
(3:21)
The case j = 1 is almost evident, namely, it is necessary to check the s1 -invariance
only one function 11(x) = min(x12 ; x11 x11 ; x22 ), but this is clear from the
denition of s1. So, we may assume that ij;1 and ij;1 (i = 1 j ; 1) are
the invariants with respect to the action of sj;1. Now, using the expression (3.21),
we are going to reduce proof of the sj -invariance of the functions (3.19) and (3.20)
to that for the more simple ones. At rst, let us dene a map J : Xn ! Xn by the
following manner (compare with (1.38)):
(J (x))ij := ;xj;i+1j 1 i j n:
Lemma 3.4. We have
(i) J 2 = 1 tj J = Jtj sj J = Jsj 1 j n ; 1
(ii) 1j (J (x)) = 1j (x) 1j (J (x)) = 2j (x)
ij (J (x)) = j;i+2j (x) 2 i j
ij (J (x)) = j;i+3j (x) 3 i j:
The proof of this Lemma based on a direct computation.
Lemma 3.5. If : Xn ! R is an sj -invariant function (i.e. (sj (x)) = (x)
for all x 2 Xn ), then e(x) = (J (x)) is also the sj -invariant one.
Secondly, let us introduce the notations
ij0 (x) := ij (Rj;1j (x))
0ij (x) := ij (Rj;1j (x))
ij00 (x) := 0ij (
j j + 1]
j ; 1 j ](x))
00ij (x) := 0 ij (
j ; 1 j ]
j j + 1](x))
Accoding to the formulas
(3.21) and (3.22) below, we have only to prove that the
00
00
functions ij and ij are the sj;1-invariants.
52 Anatol N. Kirillov and Arkadiy D. Berenstein
Lemma 3.6 (Reduction formulas). We have
a) ij0 = ij (x) and
ij00 = ij;1 (x) if 2 i j ; 2:
b) 0ij (x) = ij (x) and
00ij (x) = ij;1 (x) if 3 i j ; 2:
The proof based on a direct calculation, using the following formulas for the action
of the involution Rj;1j (see Denition 1.2): if xe := Rj;1j (x), then
xej;1j = xj;1j+1 ; min(xjj+1 ; xjj xj;1j ; xj;1j;1 ) = xj;1j+1 ; jj (x)
xejj = xjj ; xj;1j+1 + xj;1j + min(xjj+1 ; xjj xj;1j ; xj;1j;1 )
xe = x if (
) 6= (j j ) or (j ; 1 j ):
Further, using the induction assumptions and Lemma 3.6, we see, that it remains
to prove the sj -invariance only for the following functions:
1j 1j 2j j;1j jj j;1j jj :
But according to Lemmas 3.4 and 3.5, we may reduce the proof of the sj -invariance
of the last functions to that of the following ones:
1j 1j 22 33 34 33 44 45 :
In order to nish the inductive step and thus to prove our Theorem, we are going to
give for the functions under consideration the expressions, which are sj -invariance
either by the inductive assumption or by the evident reasons. The following Lemmas
contain all necessary formulas.
Lemma 3.7 (Formula (3.22)). We have
a) 10 j (x) = min(x1j+1 ; x1j xjj+1 ; xj+1j+1 + xj;1j ; xj;1j+1 2xj;1j ; xj;1j+1 ; xj;1j;1 + xjj ; xj+1j+1)
00
1j (x) = min(1j;1 (x) xj;1j+1 ; xjj ):
b) jj0 (x) = xj;1j+1 ; xj;1j jj00 (x) = xjj ; xjj+1 :
0 = min(x24 ; x23 x34 ; x33 x45 ; x44 )
c) 34
00 (x) = min(33 (x) x34 ; x35 ):
34
d) 0jj (x) = (min(xj;2j ; xj;2j;1 ; xj;1j + xj;1j;1
xjj+1 ; xjj ; xj;1j + xj;1j;1 ))+
00
jj = (j;1j;1 (x) + xj;1j+1 ; xj;1j )+ :
Combinatorics of the Gelfand-Tsetlin patterns
e) 45(x) = (min(x14 ; x13 ; x25 + x24 x34 ; x33 ; x24 + x23
x45 ; x44 ; x24 + x33))+ 00
45(x) = min(44 (x) (x45 ; x46 + x33 ; x34)+ ):
f ) 01j (x) = (min(x1j+1 ; x1j + xj;1j+1 ; xjj+1
xj;1j ; xj;1j;1 ; xjj+1 + xjj ))+ j 3
00
1j (x) = (1j;1(x) + xjj ; xj;1j+1 )+ j 3:
g) 012(x) = 12(x) 0012(x) = (11 + x22 ; x13)+ :
53
(3:22)
Lemma 3.8. a) The function 00 (x) is a s -invariant i 00 (R (
34]
45])(x))
45
is a s3-invariant.
b) We have the following equality
4
45
34
0045 (R34(
34]
45])(x)) = (min(33(x) + x25 ; x24 x45 ; x44 ; x36 + x35))+ :
The proofs of Lemma 3.7 and Lemma 3.8 00based on a direct calculation. We need
Lemma 3.8, because the s4 -invariance of 45 does't evident from Lemma 3.7, e).
As noted above, these Lemmas complete the inductive step and so the proof of
the Theorem 3.2 is over.
Corollary 3.4 (of Theorem 3.2). Let Q1n (a1 an), be the piecewise linear
function dened by means of the recurrence relation (1.31). Then
(a1 + Q1n (a1 an ) + Q1n (a2 an a1)); = (a1); + Q1n (a2 an a1)
(3:24)
Proof. The identity (3.24) is equivalent to the sn-invariance of the function 1n (x)
(see (3.19)) on the space Xn+1. When n = 3, we obtain the following identity (see
Lemma 3.2):
;a ; b + (a + (c);)+ + (b + (c)+ ); ]; = (c); (b + (a);)+ ; (c + (a)+); where a1 = c a2 = a a3 = b. Note that as a corollary of Theorem 3.2 we obtain a
\geometrical" proof of the (3.24). It is an interesting task to nd an algebraic proof
for (3.24).
Remark 3.1. The proof of Theorem 3.1, i.e. the sn-invariance of the cocharge
cn+1 (x), based on the similar, but more rened, technics. The crucial step is the
analogs of Lemmas 3.6 and 3.7. The details will be appear elsewhere.
Remark 3.2. Note that Theorem 3.1 gives opportunity to dene the charge of
a tableau T 2 STY ( ) in the case, when weight is a composition. In fact we
may dene the charge c(T ) of tableau T 2 STY ( ) as
c(T ) = n( ) ; n() ; c(T )
54 Anatol N. Kirillov and Arkadiy D. Berenstein
P
where n( ) = i (i;1)i and c(T ) is the cocharge of tableau T 2 STY ( ) ,! KZ .
Now we give a direct method for calculation of the charge, which is a generalization of the Lascoux-Schutzenberger algorithm (e.g.
Ma]). Given a standard tableau
T 2 STY ( ), consider the corresponding word w(T ) and a subdivision
w(T ) = w1
a a a
w2
wk
of the word w(T ) on a standard subwords wi according to the Lascoux-Schutzenberger algorithm, but now the standardness of a word means that its weight satises
the condition i 1 for all i, i.e. the word w = a1 aN ai 2 1 n] is called
standard, if all elements ai are dierent. We denote by a(w) the minimal element
in a word w and by a; := maxf b 2 w j = b < a g. Now let us dene the index
indwk (a) of a letter a 2 wk for all k 1. At rst, we want to attribute the indeces to
the minimal elements a(wl ) of the standard subwords wl . We shall use an induction.
(i) Let us put indw a(w) = a(w) ; 1
(ii) Assume that we already dened the indeces of all minimal elements < a(wk )
(for some k 1). Let k0 be the maximal number such that a(wk ) < a(wk ), and let
us put a := a(wk ) a0 := a(wk ). In this case we dene
1
0
0
ind (a0) + a ; a0 ; 1 if a0 lies at the right from a in w
wk
indw (a) =
0
0
0
k
0
indwk (a ) + a ; a 0
if a lies at the left from a in w:
Secondly, using the known values of the indeces for a(wk ), we shall nd all others
by induction.
(iii) Let us assume that a 2 wk and a > a(wk ), then there exist a; 2 wk . We
dene in this case
ind (a;) + (a ; a; ; 1) if a lies at the right from a; in w indwk (a) = indwk (a; ) + (a ; a; ) if a lies at the left from a; in w :k
wk
k
Finally, we dene
c(w) =
XX
k a2wk
indwk (a):
Proposition 3.4. If t 2 STY ( ) then
c(T ) + c(T ) = n( ) ; n():
Let us give an explanatory example. Take = (11 9 6 2) = (3 4 4 4 4 4 4 4)
and consider the the following tableau:
1 1 1 2 2 2 4 4 6 7 8
2 3 3 3 4 5 7 8 8
2 STY ( )
T:= 3 4 5 6 7 8
5 5 7
6 6
Combinatorics of the Gelfand-Tsetlin patterns
55
Our algorithm gives for the charge of the tableau T the following answer:
w1 (T ) = 8 4 1 5 2 6 3 7 c(w1 ) = 6
2 1 0 1 0 1 0 1
w2 (T ) = 7 2 1 8 3 4 5 6 c(w2 ) = 9
2 1 0 2 1 1 1 1
w3 (T ) = 4 2 1 7 3 8 5 6 c(w3 ) = 14
2 1 0 3 1 3 2 2
w4 (T ) = 6 2 8 4 3 7 5 c(w4 ) = 16
3 1 4 2 1 3 2
Consequently, c(T ) = 45, and c(T ) := n( ) ; n() ; c(T ) = 112 ; 38 ; 45 = 29.
We may use another method (see Ki1]) for calculating the cocharge based on an
application of rigged congurations. First of all, we nd the rigged conguration,
which corresponds to the tableau T :
1
1
T ! 4
3
1
1
4
7
4
2
03
B2
=B
B@ 1
1
3
2
1
1
1 1
1
1
0
2
4
2
1
1
0
1 0
1
0
0
0 0
0
4 ;2 ;1 1
1 1 1C
2 1 0C
CA :
0 0 0
0 0 0
P
The charge of the conguration is equal to c( ) := ij 2ij = 26, and
the charge of unique tableau Te with the dominant weight, which lies in the S -orbit
corresponding to the tableau T , is equal to
8
c(Te) := c( ) + f sum of the quantum numbers g = 26 + 12 = 38:
Consequently,
c(Te) = n( ) ; n() ; c(Te) = 105 ; 38 ; 38 = 29
where the dominant weight = (47 3). Of couse, our two methods give the same
result.
56 Anatol N. Kirillov and Arkadiy D. Berenstein
Appendix. Proof of Theorem 1.2.
We give here only a proof of the statement (1.28a). The statement (1.28b) may
be proved by a similar method. Let n be a xed positive integer, consider the spaces
of triangles Xn (see Section 1) and partitions Dn . By denition, a partition d 2 Dn
is a triangular array of real numbers d = (dij ) 1 i < j n. As a vector space
Dn ' R n n . We dene a weight of partition d as a vector = (1 n) with
components
X X
j = ; dij + djk 1 j n:
( ;1)
2
i<j
k>j
Let us dene a mapping @ = @n : Xn ! Dn by putting @ (x) = d, where dij =
xij ; xij;1 1 i < j n.
Lemma A.1. (i) A map (@ ) : x ! ((x) @ (x)) x 2 Xn gives a bijection of
vector spaces Xn ! Rn Dn .
(ii) (@ (x)) = (x) ; (x):
The bijection (@ ) allows us to transfer an action of any transformation of the space
Xn to the space Dn and vice versa. We will denote the operators F : Xn ! Xn
and (@ ) F (@ );1 : Rn Dn ! Rn Dn by the same letter F if this does not
lead to a confusion.
Lemma A.2. Let i i + 1] be the involution as given in Denition 1.3, then
i i + 1]( d) = ( de) where
(i) deii+1 = ;dii+1 deij = di+1j if j > i + 1
deki+1 = dki if k < i
and for all remaining elements xe = x .
(ii) (de) = (i i + 1) (d):
It follows from Lemma 1.1 that the involutions i i + 1] generate a group which is
isomorphic to the symmetric group Sn .
Let us x a positive integer j 1 j n, and dene an operator @ (j) : Xn ! Dn
in the following way: @ (j)(x) = d, where (see Fig.1)
dil = xil ; xil;1 if l > j and l > i
dil = xi;1l;1 ; xi;1l;2 if 2 < i < l j
d1l = xl;1j;1 ; xlj if 1 < l j:
Combinatorics of the Gelfand-Tsetlin patterns
& d16
57
& d26
& d36
& d46 & d56
d12 .
d13 .
d14 . d15 .
& d25
& d35
& d45
& d24
& d34
& d23
Fig:1
@ (5) : X6 ! D6 :
It is clear that @ (1) = @ . Further, let us dene a mapping (@ (j)) : Xn ! Rn Dn
by means of a rule
x ! ((x) @ (j) (x))
Lemma A.3. The map (@ (j) ) is a vector space isomorphism.
Now we rewrite the involutions Rijk and 'i (see Denition 1.2) of the space Xn
in terms of partitions.
Lemma A.4. We have (i) Rijk ( d) = ( de), where
deij = dik + max(dij ; djk 0)
dejk = dik + max(djk ; dij 0)
deik = min(dij djk )
and for all remaining elements de = d .
(ii) 'i ( d) = ( de) where
deii+1 = dii+1 + i ; i+1 ; i + i+1 :
For all remaining elements de = d . Here (1 n) = (d) is a weight of d.
Note that the involutions Rijk 1 i < j < k n, satisfy the relations of
Proposition 1.5.
At last, we dene s_i = (@ )si(@ );1 : Rn Dn ! Rn Dn 1 i n ; 1,
where the involutions si are given by (1.16).
Theorem A.1. We have the following equality
s_i = Ri;1'i Ri 1 i n ; 1
where Ri := R1ii+1 R2ii+1 Ri;1ii+1, if 2 i n ; 1, and R1 = Id.
It is clear that the identity (1.28a) follows from Theorem A.1.
58 Anatol N. Kirillov and Arkadiy D. Berenstein
Proof of Theorem A.1. We use the formula (1.19) as an inductive step. Note
that Theorem A.1 is evident if i = 1. So it is sucient to prove that
(@ (i;1) )si(@ (i;1) );1 = Ri;1'i Ri :
Now we use an inductive hypothesis and the formula (1.19). So we have
(@ (i;1) )si((i;1) );1 = eti;1eti sei;1 (eti );1 (eti;1 );1 where
sei;1 = (@ (i+1) )si;1(@ (i+1) );1 eti = (@ (i))ti(@ (i+1));1
eti;1 = (@ (i;1))ti;1(@ (i));1:
(A:1)
Proposition A.1. (i) If Theorem A.1 is valid for the involution s_i; , then
1
sei;1 = (Rei );1 'i Rei (ii) eti = R(i+1) eti;1 = R(i) where
R(j) = R12j R13j R1j;1j Rei = R2ii+1 Ri;1ii+1 = R1ii+1 Ri :
(A:2)
The statement (ii) may be proved by direct computation. Let us prove the statement (i). We have
sei;1 = (@ (i+1) )(@ );1((@ )si;1(@ );1)(@ )(@ (i+1));1 :
According to an inductive step we have an identity
(@ )si;1(@ );1 = Ri;;11'i;1 Ri;1:
The following Lemma completes the proof of the statement (i):
Lemma A.5. The following identities are valid
(@ (i+1) )(@ );1Ri;1 (@ )(@ (i+1));1 = Rei (@ (i+1) )(@ );1'i;1 (@ )(@ (i+1));1 = 'ei :
This Lemma follows immediately from the following one
Lemma A.6. Let ( d) 2 Rn Dn . Then (@ (i+1) )(@ );1(@ ) = ( de), where
dekj = dkj if j > i + 1
dekj = dk;1j;1 if j i + 1 k > 1
de1j = j;1 ; dj;1n ; dj;1n;1 ; ; dj;1j ;
; j + djn + djn;1 + + djj +1 if j i + 1
We nish the prove of Theorem A.1 and thus Theorem 1.2 by means of the following
relation between transformations (A.2).
Combinatorics of the Gelfand-Tsetlin patterns
59
Proposition A.2.
R(i)R(i+1) (Rei );1 = (Rei );1R1ii+1R(i+1) R1ii+1R(i) :
(A:3)
In fact, it follows from (A.1) - (A.2) that
(@ (i;1) )si (@ (i;1));1 = R(i) R(i+1)(Rei );1 'i (R(i) R(i+1)Rei );1
Now using (A.3) we obtain (@ (i;1) )si (@ (i;1));1 =
= Ri;1(R(i+1) R1ii+1R(i)'i (R(i)R1ii+1R(i+1) );1 )Ri = Ri;1'i Ri because both R(i) and R(i+1) R1ii+1 commute wuth 'i .
The proof of Proposition A.2 consists of successive application of the relations c)
and b) of Proposition 1.5 to the quadruples (1 k i i +1) with k = i ; 1 i ; 2 2.
References
BK] Bender A., Knuth D. Enumeration of plane partitions, J.Comb. Th. A, 1972,
v.13, 40-54.
BZ1] Berenstein A.D., Zelevinsky A.V. Involutions on Gelfand-Tsetlin patterns and
multiplicities in skew gln -modules, Doklady AN USSR, 1988, v.300, 1291-1294.
BZ2] Berenstein A.D., Zelevinsky A.V. Tensor product multiplicities and convex polytopes in partition space, Journ.Geom. and Phys., 1988, v,5, 453-472.
BZ3] Berenstein A.D., Zelevinsky A.V. Triple multiplicities of sl(r + 1) and the spectrum of the exterior algebra of the adjoint representation of sl(r + 1), Tech. rep.
90-60, MSI, Cornell, 1990.
C] Carre C. Littlewood-Richardson's rule in a Berenstein-Zelevinsky's construction,
Prepr., 1991.
EG] Edelman P., Greene C. Balanced Tableaux, Adv. Math., 1987, v.63, 42-99.
GM] Garsia A.M., McLarnan T.J. Relations between Young's natural and the Kazhdan-Lusztig representation of Sn , Adv. Math., 1988, v.69, 32-92.
GZ1] Gelfand I.M., Zelevinsky A.V. Polytopes in the pattern space and canonical basis
in irreducible representations of gln , Funct. Anal. i ego pril., 1985, v.19, 72-75
(in Russian).
GZ2] Gelfand I.M., Zelevinsky A.V. Multiplicities and regular bases for gln , in \Group
theoretical methods in physics. Proc. of the third seminar Yurmala, May 22-24,
1985", Nauka, Moscow, 1986, v.2, 22-31 (in Russian).
GZ3] Gelfand I.M., Zelevinsky A.V. Canonical basis in irreducible representations of
gl3 , Ibid, 31-45.
H] Heckman G.J. Projections of the orbit and asymptotic behavior of multiplicities
for compact connected Lie groups, Invent. Math., 1982, v.67, 333-356.
J] Jimbo M. A q-dierence analog of U (g) and the Yang-Baxter equation, Lett.
Math. Phys., 1986, v.11, 247-252.
60 Anatol N. Kirillov and Arkadiy D. Berenstein
JK] James G.D., Kerber A. The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, v.16, Addison-Wesley, MA, 1981.
Ka1] Kashiwara M. On the crystal bases of the q-analog of universal enveloping algebras, Duke Math. Jour., 1991, v.63, 465-516.
Ka2] Kashiwara M. Global crystal bases of quantum groups, Prepr. RIMS-756, 1991.
Ki1] Kirillov A.N. On the Kostka-Green-Foulkes polynomials and Clebsch-Gordan
numbers, Journ. Geom. and Phys., 1988, v.5, 365-389.
Ki2] Kirillov A.N. Decomposition of symmetric and exterior powers of the adjoint
representation of gln , Prepr. RIMS-840, 1991.
KN] Kashiwara M., Nakashima T. Crystal graphs for representations of the q-analogue of classical Lie algebras, Prepr. RIMS-786, 1991.
KR] Kirillov A.N., Reshetikhin N.Yu. The Bether ansatz and the combinatorics of
Young tableaux, J.Soviet Math., 1988, v.41, 925-955.
Li] Littlewood D.E. The theory of group characters, 2-nd edn. Oxford University
Press, 1950.
LS1] Lascoux A., Schutzenberger M.-P. Sur une conjecture de H.O. Foulkes, C.R.
Acad Sci. Paris, 1978, v.286A, 323-324.
LS2] Lascoux A., Schutzenberger M.-P. Le monoide plaxique, Quaderni della Ricerca,
1981, v.109, 129-156.
LS3] Lascoux A., Schutzenberger M.-P. Keys & standard bases, in \Invariant theory
and tableaux", IMA vol. in Maths. and Appl., 19, Springer, 1990, 125-144.
Lu1] Lusztig G. Green polynomials and singularities of unipotent classes, Adv. Math.,
1981, v.42, 169-178.
Lu2] Lusztig G. Canonical bases arising from quantized enveloping algebras, J. Amer.
Math. Soc., 1990, v.3, 447-489 II, Progress of Theor. Phys., supplement, 1990,
No.102, 175-201.
Ma] Macdonald I. Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.
Mi] Miwa T. Private communication.
N] Nakashima T. Crystal base and a generalization of the Littlewood-Richardson
rule for the classical Lie algebras, Prepr. RIMS-783, 1991.
Ru] Rutherford D.E. Substitutional analysis, Hafner Publishing Company, New York,
1947.
Sa] Sagan B. The symmetric group, Wadsworth & Brooks/Cole, Mathematics Series,
California, 1991.
Sch1] Schutzenberger M.-P. Promotion des morphisms d'ensemble ordonnes, Discrete
Math., 1972, v.2, 73-94.
Sch2] Schutzenberger M.-P. Le correspondance de Robinson, Lecture Note in Math.,
1977, v.579.
St] Stanley R. GL(n C) for combinatorialists, in \Surveys in Combinatorics" (ed.
E.K.Lloyd), London Math. Soc., Lecture Note Series, No.82, Cambridge University Press, Cambridge, 1983, 187-199.
Combinatorics of the Gelfand-Tsetlin patterns
61
SV] Shabat G., Voevodsky V. Drawing corves over number elds, in \The Grothendieck festschrift", 1990, v.III, 199-227.
SW] Stanton D., White D. Constructive combinatorics, Srringer-Verlag, New York,
1986.
T] Thomas G. Schensted's construction and the multiplication of Schur functions,
Adv. Math., 1978, v.30, 8-32.
W] Weyman J. Pieri's formulas for classical group, Contemporary Mathematics,
1989, v.88, 177-184.
Z] Zelevinsky A.V. A generalization of the Littlewood-Richardson rule and the
Robinson-Schensted-Knuth correspondence, J. Algebra, 1981, v.69, 82-94.