JGP - Vol. 5, n. 3, 1989
Tensor product multiplicities
and convex polytopes in partition space
A. D. BERENSTEIN, AN. ZELEVINSKY
Scientific Council for Cybernetics
Academy of Sciences
Moscow, USSR
Dedicated to I.M. Ge/land
on his 75th birthday
Abstract. We study multiplicities in the decomposition of tensorproductof two lireducible finite dimensional modules over a semisimple complex Lie algebra. A conjecturalexpression for such multiplicity is given as the number of integral points of a
certain convexpolytope. We discuss some special cases, corollaries andconfirmations
of the conjecture.
0. INTRODUCTION
In this paper we shall deal with irreducible finite-dimensional modules over a
semisimple complex Lie algebra g. The fundamental role in theory of these modules
and their numerous physical applications is played by two kinds of multiplicities. The
first is the weight multiplicity K~ of weight ~ in the irreducible g-module VA with
highest weight )~.. The second one is the tensor product multiplicity c~ of V~ in
VA 0 T/~. In fact, the weight multiplicity can be obtained as limit case of the tensor
product multiplicity, so we shall mostly study multiplicities ct,.
Our main <<result>> is the conjectural expression for c~ as the number of integral
points in certain convex polytope. For g s~ (or g~) such representation was given
by I.M. Gelfand and one of the authors in [1], [2], [3]. Convex polytopes constructed
there lie in the space of Gelfand-Tsetlin patterns. Here we present new approach replacing Gelfand-Tsetlin patterns by partitions into sum of positive roots. This language
Kcy-Words: SemisimplecomplexLie algebras,partition space, multiplicity, con vexpolyt opes.
I98OMSC: 17B20,52A25.
454
A.D. BERENSTEIN, A.V.ZELEVINSKY
makes sense for any semisimple Lie algebra g.
Our approach comes back to the idea of F.A. Berezin and I.M. Gelfand [4], who
emphasized the fundamental role of the hierarchy
P(i)
—4
.-
where P(’y) is the Kostant partition function [5]. The functions P(’y), KA/3 and c1~,
depend respectively on 1,2 and 3 weights, and we have the following expressions due to
B. Kostant:
(0.1)
KA$= ~detw.P(w~+p)—~—p),
WEW
(0.2)
c~, ~
det w
WEW
where W is the Weyl group of g, and p is the half-sum of positive roots. L let
R be the root system of g, and R~ the set of positive roots. By definition,
= card {(m
0)~ER, : m0 E Z~.,>mn a = ‘y}, i.e., P(-y) is the number
‘
of partitions of
into the sum of positive roots. This number is naturally interpreted
as the number of integral points of the convex polytope i~(‘y) in the <<partition>> space
RR
with coordinates (Zo)aER* . Namely, i~(‘y) is the intersection of the affine plane
{~Z.~ a = ~ of codimension rkg with the <<positive octant>> {x~~ 0 for all a}.
‘~
-
Integral points of L~(~)will be called g-partitions of weight ‘y.
lt is well-known that
(0.3)
c~~ KA,~_~ P(~+ v
0
—
(see § 1 below). This suggests to look for expressions for multiplicities KAP and c’~,
also in terms of g-partitions. Our conjecture is that there are convex polytopes z~ c
C i~( )~+ i.’ ~) such that ~ is that number of integral points of i~, (and
similarly for KA,~_U
Note that P( )~+ v ~) is the <<leading term>> corresponding to w = e in alternating
sum (0.1) for KA,~_~
; similarly, KA~_~
is the leading term in (0.2) for c~. It follows
that once the polytopes i~ and
are found, one can in principle prove the conjecture using formulas (0.1) and (0.2) by cancelling alternating terms. This method was
used in [6] for g = ~ In this case the expression for c~,in terms of g-partitions (see
§2 below) is equivalent to the one given in [1-3] and is also equivalent to the classical
Littlewood-Richardson rule (see [7], [8]).
Now we give a more precise version of our conjecture. Let fl C R~be the set of
simple roots of g, and w0 (a E fl) be the fundamental weight corresponding to a.
—
).
—
CONJECTURE 0.1.
There are linear forms ~s), ~t) ~ RR~ (where a E H and
for each a s and t run over certain finite sets) such that for every highest weights
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TENSOR PRODUCT MuLTIPLICrFIE5 AND CONVEX POLYTOPES IN PARTITION SPACE
p, A
=
~
~
and
ii
=
~eH
~ n~w~
the multxplicity c~,is equal to the number
nEIl
of g-partitions m of the weight A
+
u
—
~ satisfying the inequalities max 4~)
( m) <
L,~,maxn~(m)<na forallaEll.
Explicit (conjectural) expression of forms 4s), n(~ for all classical Lie algebras will
be given in §2 (the case C2 will be considered in another publication).
Conjecture 0.1 implies the following two expressions for weight multiplicities.
COROLLARY. Jf 4s), n~ are linear forms from Conjecture 0.1 then for any highest
weight A = ~ £0w,~and any weight /3 the weight multiplicity KAP is equal to the
nED
numberofg-partitions m of weight A—~3suchthat max 48) ( m) <en forall a and is
,
also equalto the numberofg-partitions m ofweight A —~3such that max n~( m) <£~
foraila.
.
Conjecture 0.1 is closely connected with the problem of constructing <<good>> bases
in the modules
VA weight
(see [1-3]).
To explain
thisweight
relationship
weany
need
someweight
definitions.
3) be the
subspace
of VA of
/3. For
highest
u =
Let VA(/
>InEfl n~w~
let VA(/3,zI) = {v E VA(13) : e~1v = 0 for a E fl},where en is the
root vector of weight a. It is well-known (see e.g. [9]) that
(0.4)
c~, dim VA(~ z.’,v).
—
Following [1-3] we say that a basis B in VA is good if each subspace VA(/3, v) is
spanned byapartof B.
CONJECTURE 0.2. For any g and A there exists a good basis in VA.
.
This conjecture was independently given by I.M. Gelfand and A.V. Zelevinsky (see
[10]) and K. Baclawski [11]. In fact in [11] the conjecture was stated as theorem but,
unfortuantely, the proof is incorrect. In [2] the conjecture is proven for g = a~ using
the deep results of [12]. In [13] it is proven for g = sp
4
It is easy to see that conjectures 0.1 and 0.2 together imply the following
CONJECTURE 0.3. Let 4~and n~ be linear formsfrom Conjecture 0.1. Then foreach
highest weight A there is a good basis B = {Vm } in VA indexed by g-partitions m as
follows:
(1) For any weight ~3the subset B fl VA (/3) is indexed by g-pa.rtitions m of weight
A
—
/3
satisfying inequalities max
(2)ForanyaEll,vmEB
48)
( m)
<en for all a.
min{n:e~vm=0}=maxn~)(m).
456
A.D. BERENSTEIN, A.v. ZELEVINSKY
It was explained in [2] and [13]that the notion of good basis is closely connected with
the problem of canonical definition of Clebsch-Gordan coefficients (or equivalently of
canonical definition of tensor operators, see e.g. [14]and references there). We hope that
the parametrization ofbasis vectors by g-partitions as in Conjecture 0.3 will be helpful
for physical applications.
There are two other general methods of computation of multiplicities. The first one
comes back to D.E. Littlewood and is developed by B.G. Wyboume, R.C. King and
others (see [15-18]). This is so called <<S-function method>> based on the theory of
symmetric polynomials. This method enables one to express multiplicities KA$ and
in terms of Young tableaux. Unfortunately, the answer in general is not completely
combinatorial i.e., includes some negative terms (produced by so called modification
rules). Thus, it is difficult to use this method for parametrization of basis vectors.
Another approach to multiplicities and special bases is so called <<Standard monomial
theory>> (see [19] and references there). This method gives an unified combinatorial expression for weight multiplicities and also the special basis in modules VA <<of classical
type>> ([19]). Note that this basis is not <<good>> in the sense of Conjecture 0.2.
Recently P. Littelmann [20] obtained the unified combinatorial expression for c~,,
for all modules of classical type. His result is based on the standard monomial theory.
It would be very interesting to compare these two approaches to multiplicities with
our approach.
The paper is organized as follows. In §1 we collect some general results on multiplicities. In particular, we show here that multiplicities in reduction of g-modules to a
Levi subalgebra g’ as well as tensor product multiplicities for semisimple part of B’
are special cases of tensor product multiplicities for g.
In §2 we give the explicit form of Conjecture 0.1 for all classical Lie algebras (Conjecture 2.2). There are given also some special cases of Conjecture 2.2 which can be
verified independently.
In §3 Conjecture 2.2 is proven for classical Lie algebras of small rank. For g =
sp4 we deduce it from the results of [131, and for 04, 05 and 06 use the exceptional
isomorphisms D2 = A1 x A1, B2 = C2, D3 = A3.
Finally, in §4 some reduction multiplicities including weight multiplicities are studied. For any classical Lie algebra B we introduce the notion of a B-pattern generalizing
famous Gelfand-Tsetlin patterns, and establish the correspondence between g-patterns
and B-partitions from Conjecture2.2.
notion
02r+ The
r and
o of a ~P2 -pattern pattern is essentially
due to D.P. Zhelobenko (see [9]);
2~-pattemsseem to be new although they
are closely connected with generalized Young tableaux considered in [15].
1. GENERAL PROPERTIES OF MULTIPLICITIES
Let B be a complex semisimple Lie algebra, /1 a Cartan subalgebra of B~R C 11” the
457
TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE
root system. We fix the set of simple roots U
=
{ai,...,ar}. Let q= ~
~ B(a)
nCR
be the root decomposition of g. Choose standard generators e±n.E ~g(±a1),hn
[en, e_n.] E Fl (i
=
1
,. . . ,
=
r). Let w1,... , w~E Fl~be the fundamental weights of g
defined by Wi(hn,) =
Recall that A ~ h~is a highest weight for g (i.e. the highest weight of an irreducible
finite-dimensional g-module VA ) if and only if A = ~ Lw1 with nonnegative integers
I <i<r
L~.Integral linear combinations of fundamental weights are called integral weighLs~,it is
well-known that each weight of a finite-dimensional B-module is integral.
3, ii) defined in
Now
consider
a
weight
subspace
VA(/3)
C
VA
and
its
subspace
VA(1
§0.
PROPOSITION 1.1. a) Wehave 0 <KAfl <P(A—/3) (see~0).Jnparticular, VA(/3)
0 unless A /3 is a nonnegative integral linearcornbination ofsimple roots.
b) Wehave VA(/3,z1) = cr’. Inparticular, 0 <ce’ < K~.
=
—
This proposition is well-known (see e.g. [9]). Having it in mind it is natural to introducethefollowingnotation: C~~(y)= dim VA(A—’y, 71) = ~
If A = ~
ii =
~
‘y
=
~g
1a~ then we shall sometimes write ~
(gi,... ,gr) for
3, ii) (and so that of CA~(’y))to the case
cA~(~y)
. We shall extend the definition of VA(/
when some
of the coefficients n
1 in decomposition u = ~ n1w1 may be equal to co;
this simply means that the corresponding condition en~ v = 0 is omitted. Since all en
act on VA as nilpotent operators the infinite value of n~can be replaced by sufficiently
large positive value. Since VA ® V~~ V~® VA it follows that cA~(’1) ç~(~y)
For each fl’ C H we define the reductive Lie subalgebra g’ = ~(U’) C B to be
B’~h~~(a),
where the sum is over all roots a of the form ~
c1a~.The subalgebras ~q(11’) will
cz1fl’
be called Levi subalgebras of B- In particular, ~(0) = 11 is a Levi subalgebra. We
denote by V~the irreducible B’-module with highest weight /3.
PROPOSITION 1.2. The multiplicity of VA’~in therestriction VA g’ is equal to CA~(~)‘
where the coefficient n~in the decomposition ii =
n1w1 is co for a1 ~ U’ , and 0
for a E H’. In particular, the weight multiplicity KA,A_,, is equal to cA~(’1),where
all coefficients n~of v are oc.
This follows at once from definitions since the subspace of highest vectors of weight
A—1 in VAI,. isjust VA(A—~,v)={vEV~(A—’y):env=0 for aEH’}.
•
458
AD. BERENSTEIN, A.V.ZELEVINSKY
With a subset U’ C H we also associate the semisimple Lie subalgebra Bo =
Bo (11’) generated by e±n,hn for a E fl’. The Cartan subalgebra !i~of Bo is spanned
by hn for a E [I’, so the natural projection p : Fr
fi~sends w1(a~~
fl’) to 0
—~
Without loss of generality we can assume that U’
{a1,... a~}for some k < r.
=
,
PROPOSITION 1.3. Let A,v be highest weights for B~and ‘y the weight of the form
~ g1a1. Let A’ = p(A), ii’
p(v), 1’ = p( 1) be the cormsponding Bo -weights.
1<i’(k
Then
CA~(1) =
dent on
Lk+1,
..
cA,U,(’y’)
-
.
In other words, ~
~k+1~-~
,
~
n~and equals
,0) isindepen(gi,...
~
‘9k)
-
Proof Let VA~ denote the irreducible Bo -module with highest weight A’, let Bo =
~ 1L~~
where n~j(resp. n~) is generated by en,
,e,~,,(resp. C~n,~••~ —n,,)’
and n! = ~ a), the sum over positive roots a involving some of the roots
,..
,
.
a,.. Our proposition follows readily from the next statement:
The Bo -module (VA) ~eof j?-invariants in VA is isomorphic to VA and this isomorphism identifies the weight subspace VA (A’ ‘y’) with [VA(A
The isomorphism of (VA)” with VA, is clear from the fact that (VA)” as a Bo -module
has the unique highest vector. The inclusion [VA(A 1)]” C VA (A’ 1’) is evident,
and converse inclusion follows from theequality VA, (A’ —1’) = U~VA (A’) where U
is the weight subspace U(n~)(—’y’) ofthe universal enveloping algebra U(n~)
ak+ 1’~
,
—
—
—
—
,
.
Proposition 1.3 shows that tensor product multiplicities for <<standard>> semisimple
Lie subalgebras of B are special cases of these multiplicities for g.
Next proposition will be used as a test to our conjectures on multiplicities.
PROPOSITION 1.4. Let g be simple, and 8 be themaximalrootofg ([21]). Then c~,,=
ce~(8) is equal to the number ofnonzero coefficients n~, in the expansion u = ~
Proof By definition, V0
g is the adjoint B-module, and V0(0)
Fl. Therefore,
c0~(O) = dim V0(0,v) = dim{h E 2(h)
Fl : (ad
en)”~(h)
=
0,
i
1,...,r}.
But
= 0 , which implies our statement.
ad e~.( h) = at( h)
and (ad Ca)
2. MAIN CONJECTURE
In this section we give a precise form of Conjecture 0.1 for all classical simple Lie
algebras. We use the standard numeration of simple roots a
1,
a,. and corresponding
fundamental weights w1,... ,w,. (sec [21]). Thus, the integral weights for each type
B,., G,., D,. will be written as linear combinations of standard basis vectors e~,
. ..
,
. . . ,
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TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE
in )R (with integer or half-integer coefficients); for Ar_l the lattice of integral weights
is identified with Z~/Z(s1+
+
...
Let m13 = rnE, rn~,= mE+E, m~= mE ,soa o2~~1-partition
of weight ‘yE Z’~
isavector rn (m11(l <1<3< r), m~,(I <1<1< r), m1(1 <1< r)) such
that all its components are nonnegative integers, and
>
m1~(e1— e,)
+
~m~,(c~+
e,)
+
>rn1c~ = ‘l-
8P2r~ and o
It will be convenient for our purposes to realize ,sL,.-,
2,.-partitions in terms
of o2,.~1-partitions Namely, we identify an sL,.-partition with an o2,~1-partition
(m~,,m~,m1) suchthat rn1 = m~,= 0 for all
Similarly, an o2,.-partition isan
o2,.~1-partition (rn22, m~,m1) such that rn = 0 for all i. Finally, a sp2~-partitionis
identified with a o2,.÷1-partition~
m~,m~) such that all m1 are even. It is clear
that such identification preserves the weight of a partition.
Now we give an expression for
.
i,j.
= CAP(’y)
C~,,,
=
forg= .sL,.. Forany ,sLr-partition rn= (m1~) and anypair
(2.1)
=
A13(m)
=
(see §1)
(9l’~• . , g,.~)
~
(1,5)
put
m~1 m~+1~~1
—
(with the convention that m21 = 0 unless I < I < j < r, so for example
= rn1,.). We define the linear forms ~
= ~(8)()
~~t) =
(1 <i,j<r—1) by
(2.2)
L~
=
—
~
L\,~ (0 <s<
~
~t)
OP8
~
i.\~,
i~
=
(i < t ~ r).
tpi
The following theorem is the first motivation of Conjecture 0.1.
THEOREM 2.1. The multi~plicity~
.,,,
...,~,
of sL,.-partitionsrn of weight ~
max n~t)(rn)<n~for all 1,5.
(g~,. g,...~)is equal to 1(m)
the number
<La.
. . ,
suchthatmax4~
The proof will be given in §4.
REMARKS. 1. Let A,p,v E Z~/Z.(c~+
e,.) bethreehighestweights for aL,..
Choose representatives A, i7 E Z ~ of these weights so that [~I = I + I Dj, where
...+
~,
I/1iEi
+
..
+
p,.e,.~=
+
...
+
p,.. Denote by LR~ the coefficient given by the
460
A.D. BERENSTEIN, A.V. ZELEVINSKY
Littlewood-Richardson rule (see [8]), i.e. the tensor product multiplicity for gL,.. Then
it is clear that c’~,, LR~V
2. Theorem 2.1 makes the statement of Proposition 1.3 for g = .sL,.~1‘Bo =
evident: since g,. = rn1 ,~+ + m2
+
+ rn,.,..~i~ and all m13 are nonnegative, it
follows that g,. = 0 implies m1,.~1 =
= m,.,.1 = 0, and we are left with an
sL,.-partition.
3. Consider the linear involution /3
on integral sL,.-weights such that ~ =
—c,.+1_~,and extend it to sL,.-partitions by I = rn,.~1_1,.÷i_j.
By definitions, n~
(m) = L(r:t)(,~) for all I, t. Therefore, Theorem 2.1 implies cA~(’y)= c~(~)(this
equality follows at once from the tensor product interpretation of multiplicities and the
fact that V~,= (VA)*, the dual module).
Now consider other classical Lie algebras of types B,., C,. and D,.. Let I = I,. =
{I,2,...,r,O,1,...,i}
belinearlyorderedby O <I < ~ <2 <~ <
r< i~.
For each a, t E I define the linear form A3,~on 02,.+ 1-partitions m = (m11, m,~,m~)
by
,..~.
...
...
—
...
=
(2.3)
A
—
U
A
U
—
m~3— rn~,
<
L~çy=
—
J m~1÷~
m~~1~÷1
forj°< r
—
1~m~
—
—
m1~1
forj
=
r
(with the convention that rn11 = m,~•= 0 unless 1 < 1 <j < r, arid rn1 = 0 unless
1 <1 < r).
For
= I 2,.
r
1, at E I and r = 0, 1 define the linear forms L~8), (t,E)
by formulas
1,1
,
—
. -
A
(2.4)
I
I
—
O~<8
—
(tO) =
(t I)
=
A~,
A~,
+ ~÷i<~<~
1~<P<,.
(to)
+ ~
where in each sum p runs over the indicated interval in I, and empty sums are understood as 0.
We shall also need the involution m
on the set of all o
2,.-partitions defined
by
—~
(2.5)
=
{
We define the forms
fory<r
forj = r,
rn
~
(a E I) and
e~8)
~
~(rI)
{
m’
forj<r
m~j forj = r.
separately in each of the cases B,.
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TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES INPARTITION SPACE
C,. , D,.
D,. :
4~~(m)
= 4_l(m),
B,. :
~(a)
=
—
C,.:
~
=
—~-
~r~I)(m)
~IAp,.,
~r,1)
( rn)
=
=
r4~~(fn)= ~
m,.
(2.6)
>A~,.,
n~.~’~(m)
= m,./2.
O<p<8
(here ~
means that the summands A1,., A2,.,
...
are taken with coefficient 2).
MAIN CONJECTURE 2.2. Let g be one of the classical simple Lie algebras of type B,.,
C,., or D,. (i.e. g = 02,.+i, ap2,. or °2,.)~ Then ~
(g1,. .g,.) is equal
to the number of g-paititions m of weight ~ ga1 such that max
( m) <
.
4~
max ~
( m)
< n,, for all i, 5
=
1,.
.
. ,
r, where maximums are over all possible
e with the only exception that in case D,. the forms n~’~(I
to n~
,.,0) are not taken into account.
i ~ r — 1) equal
REMARKS. 1. Actually, the linear forms contributing to Conjecture 2.2 are the following:
B,.,c,. :L~ (1
<5<
r, 0 <a
<5),
and<t<rforr=
,~t~E)(li~r—l,itrfore=0,
1)
4rI)
D,.:L~8)(lj~r_l,0~s<5),
L~(O<s<r—1),
~t~E)(l<i<r_2,i<t<r_IforE=0,and~<~<rfors=1),
(ri)
(ri)
n~_1, n,.’
-
2. Conjecture 2.2 is compatible with Proposition 1.3 in the same way as Theorem
2.1 (cf. Remark 2 to the Theorem 2.1). Namely, putting g,. = 0 we obtain the multiplicity for aL, predicted by Proposition 1.3. Similarly, putting 9i 0 we obtain the
multiplicity for the subalgebra g~({a2, . .. a,.)) i.e. the previous member of the same
series as g.
3. It is easy to see that the expression for c~~(’y)given by Conjecture 2.2 in case
C,. can be reformulated as follows:
,
(8P~r)
~
i
~
462
AD. BERENSTEIN, A.V. ZELEVINSKY
is equal to the number of
~
~2,.+
1-partitions m = (rn13 m~,m1) contributing to
2g,.)
(°2 +,)
(gi,~-~g,._i,
and such that all m~are even.
4. In case D,. let A —f A be the linear involution on weights such that f~,=
for I < r I
= —s,. (clearly, it is compatible with the involution m—* th on
—
~,.
,
o
2,.-partitions defined by (2.5)). Then Conjecture 2.2 easily implies ct,, = c~ (which
is of course evident since the involution A
A is an automorphism of the root system
D,.).
—~
Applications of Theorem 2.1 and Conjecture 2.2 to Lie algebras of small rank will be
given in §3. Now we give some confirmations of Conjecture 2.2. First we show that the
Theorem 2.1 and Conjecture 2.2 imply Proposition 1.4. It suffices to prove
PROPOSITION 2.3. Let B be one of classical simple Lie algebras of rank r, and 0 =
~ n~w~
be its maximal root. Then there are exactly rh-partitions mm,. ,m~
‘~) of
l<i<r
t~(m~))
<n~(l <1< r), andtheycan be numerated in
weight 8 satisfying maxn
such a way that max L~( m’ k)) = 65k (where the ~
and n~ are linear forms from
~.
Theorem 2.1 and Conjecture 2.2).
The proof is straighforward. We shall only give the list of ~
not mentioned are understood to be 0).
A,. : 0
=
1
=
m~ (all parts
w
=
—
. . . ,
(mjk+1
+
= mk+lk+2
=
...
=
m,.,.~1= 1).
m23
=
m~3= 1),
mk+2k÷3=
1= (m
B,. :0
=
+
m~
=
=
(ml,k+I
rn~= (m1
C,.: 0
2e1
=
m~
=
w2
=
=
m~’
mzk+i
=
rn2
=
=
=
l)(2 ~ k ~ r— 1),
1).
m~ = (mlk+1
2w1
(m1
=
12
=
mik+1
l)(1
=
~k
<r
—
1),
2).
1~
are the same as for B,.,
Dr(r
~
4) : 0
=
= ~i
(mi,.
+ 62 =
=
w2
rn
2,.
=
1).
mm,... , m~’~
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TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE
PROPOSITION 2.4. Conjecture 2.2 is true for the case when
£2 = £3 =
...
=
=
£,.
0.
The proof is based on the <<involution method>> of [6]. We postpone it till another
publication.
Finally, consider the case
= n
2 =
= 0.
...
COROLLARY 2.5 (OF CONJECTURE 2.2). a) For
(02,.,)
~
,.
m1
—
—
...
>
mm (n,., 2( g,.
—
g,._1)) we have
—
—
(at)
Ct,..’~,_,;m2_m,
=
£,.
—
m,—m,
,(91
—rn1,g2 —m1 —m2,...,g,._1—
m,.1)
t such that 0 < rn
the sum over all (m1,... , m,.) E Z
m1+...+rn,.=g,..
b) ForE,.
min(n,.,2g,.
—
1 <
g,._1) we have
(sp2,)
—
—
.t~oo ~
C11
~ m,. ~ n,. and
...
)
..4~;m2 —m1
(at
~
=
—
m1
—
...
—
m,—m,_1
(gi
—
rn1 ,92
—
m1
—
m2,.
.
,
9,.—i —
rn,.1)
the sum over all (rn1,... ,m,.) E Z~ such that 0
m1 <
...
<
m,. ~ 2r,,,.,
m1
—
rn1+...+ m,.2g,.,and rn1,...,rn,. are even.
c) ForE,.~min(r~,.,g,.+g,._1
(02,)
—
—
~
=
~
~
(at
)
m,_m,_,(9i
+
—
m,._2,g,._1
the sum overall (rn1,...
m,.
= 2g,.
and rn,.
=
wehave
9,.—2)
—
..
.
—
rn1, .
..
,9,.—2
—
. .
.
—
+
2
m,.) E Z” such that 0 <m1
m,. ~ n,., m1 + .
rn,.~I,...,rn,._2k= m,._2k_I (withthe convention m0 = 0).
,
~
...
<
.
We shall outline the proof of a). Consider an o2,.~1-partition m = (m13, m~,rn1)
contributing to ~
(g~,...,g,.) by Conjecture 2.2. It is easy to see that the
inequalities max r4t~(w) < n~imply that m~ = 0 for all
and 0 < rn1 <
i,j,
<m,. < n,.. Furthermore, the condition on weight of m implies ~ rn1 = g,.. Now
fix rn1,... rn,. satisfying these conditions and consider the remaining part (m13) of
,
464
AD. BERENSTEIN, A.V. ZELEVINSKY
m as an sE,.-partition m’. One can verify that the conditions of Conjecture 2.2 except
those related to
are equivalent to the fact that m’ contributes to
(at )
£,.
Cti..~,i;m
2_mi
m~
—
— ...
—
mm(9i
—
m1,g2
—
m1
—
m2,..
-
m,.1)
by Theorem 2.1. It remains to observe that the inequality max £~8)( rn) <
follows
from each one of the inequalities
n~and E,. 2(g,. g,._i) The proof of b) is
entirely similar, and the proofof c) follows the same lines although is more complicated:
in this case the parameters rn are chosen so that m,._2k = m,._2k_I = mr_2k1,,..
Consider now the special case of Corollary 2.5 when n,. = + oo. According to Proposition 1.2 the multiplicity in question is the reduction multiplicity from B = °2,.+I sp2,.
or ~2,. to the Levi subalgebra gE,..
Let A = A1 e~+
+ A,.~,.,[3= [3161 +
+ /3,.e,. be partitions in sense of [8], i.e.
A1, /3~ ~ Z, A1
A,. 0, [3~
0. Let VA be the irreducible
g-module with highest weight A (note that for g = 02,.+ and 02,. this excludes spinor
modules), and V~the gE,.-module with highest weight [3.
£,.
£,.
..
—
-
...
..
PROPOSITION 2.6. For B =
.
..
~2,.+ 1
>
/3,.
the multiplicity of V~in VA Igt~ is equal to ~ LR~
(see Remark 1 to the Theorem 2.1), where p runs over all partitions Pi
...
t’,.
0 such that I~I= IAI — 1/31. For B = ~P2,. this multi~plicityisequal to ~ LR~V, the
sum overall partitions v with evenpartssuchthat IziI = IAI— 1/31 .Finally, forg= °2,.
the multiplicity is equal to
and
ii’
>j~LR~V,,the sum over the same partitions
u as above,
stands for the conjugatepartition [8].
This follows atonce from Prorposition 1.2, Corollary 2.5 and the well-known symmetry LR~ = LR~ where
is the involution defined in Remark 3 to Theorem 2.1
(the coefficient LR~ should be understood as the multiplicity of V~in
Vi). On
the other hand, Proposition can be deduced from the results of [17], which gives an
independent affirmation of Conjecture 2.2.
Using Corollary 2.5 for £1 = £2 =
= £,._~ = 0 ,
= +oo and Proposition 1.2
we obtain
,
/3—p
/3
VA’
...
®
£,.
COROLLARY 2.7 (OF CONJECTURE 2.2). Let B be one ofthe algebras
°2,.~For any
°2,.+1, sp2,., or
n ~ 0 the restriction ofthe B-module V,..~ to the Levi subalgebra 91,.
is isomorphic to ~ V1~,where /3= /3~ +
+ f3,.c,. runs over all weights satisfying:
2 /3i /~2
i3,. _n/2, n/2 /3~E 7.
a) ForB= °2,.+l~!
b) For B ~P2,.
n ~
13~
/3,~ —a, n— /3~E 2Z.
c) FOrg = 02,. f/2
[3~ /3~
[3,. —n/2, n/2 i3~ E 7, /3,. =
= /3,._2k_1 (with the convention /3~= n/2).
- ..
—
-.-
..~
..
—
465
TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE
REMARK. Part a) is equivalent to [8, ch. I, §5, Ex. 16, formula (2)].
3. CLASSICAL LIE ALGEBRAS OF SMALL RANK
The next theorem summarizes the statements of Theorem 1.2 and Conjecture 2.2 in
some small rank cases.
THEOREM 3.1. In each of the following cases the multiplicity ~
is equal to the number of g-partitions m satisfying the conditions:
(9i,.-.
,
g,.)
a) A
1,g=sE2:
(3.1) ~E1 ~ m12
r~~ m~2
g1rn12
b) A2,g= sE3
(3.2)
£~ m12
£2
m13
£2
m13 + m23
n1 ~ m13
~i ~
—
m12
~2
9i
rn12
+
m13
92 =
m23
+
m13
rn~3+ m12
—
rn23
> m23
c) A3,g= &E4
£~ m12
£2 > m13
(3 3)
£2
1n13
+
rn23
—
£3
rn14
rn14
rn14
+
rn24
—
£3
£3
m12
n1 > m14
n1 ~
+
m13
n1 ~ rn14
+
m13 — rn24
~2
+
—
9i
92
rn13
rn13 + rn34
m12 + rn13
= m23 + m13
= m3~+ m24
=
m23
+
m14
+
+
m14
m14
713
+
m~
> m24
~ ~ rn~+ m23
—
—
—
m34
> m34
m24
d) B2,B=o5:
£2
(3.4)
£2
£2
rn1
rn1
rn2
—
1~1
m~ + m1
m12 + m1
—
~2
rn2
2(rnt
+
+
V2 ~
2 rn12)
2(rn~2 m12)
92
m~+ rn~2+ m1
= 2mT2 + m1 + m2
—
—
m2
m2
+
m12
—
rn23
466
A.D. BERENSTEIN, A.V.ZELEVINSKY
e) C
2,B= .sp4: (m3, rn2 are even)
£1rn12
£2
rni!2
2+m~
£2 m1!
2—rni2
12+ (rn~
£2
rn21
2—m12)
(3.5)
91 =
92 =
m~2+ m1 rn2
rni~+mi —rn2
V2
m2!2
fl1 ~
—
V1
m12 + m~2+ m1
rn~2+ m1/2+ m2,/2
I) D2,B=o4:
(3.6)
(£i_
rn~2
n~~ rn~
~ £2
rn12
V2
rn12
g) D3,B=o6:
£1rn12
£2
rn13
—
rn~2— rn12
£2
rn13
£2
m13 + m~2 m12 + m23
rn~3
m~3+ m~2 m12
£3
(3.7)
£3
£3
+
—
—
—
m13
> rn~3+ rn~2 rn12 + rn~3 rn13
—
=
=
Parts a),
—
—
—
92 =
Proof
Thrn~
~i ~ m~+ m~3
n~> m~+ m13
n1 ~ rn~+ m13
n1 ~ m12 + rn13
V2 m23
—
m23
rn~3
rn~3+ rn~3 rn23
m~3+ mt3
rn23
—
—
rn~3
713
m12 + m~2+ rn13 + rn13
rn~2+ m13 + rn23
rn12 + m~3+ m~3
b), c) are special cases of Theorem 2.1. To prove e) we use the result
of [13]:
PRoPosmoN 3.2 ([13], THEOREM 3). ~
(91,92) is equal to the number of 8tuples (P2, p~-,Pi-~P12, Poo ,PT2 p~, Po) ofnonnegative integers satisfying the condi,
tions:
(1) If a and t are two nonadjacent vertices of the graph
~
12
00
12
TENSOR PRODUCT MIJLTIPL1CITIES AND CONVEX POLYTOPES IN PARTITION SPACE
then min(pa,pt)
(2)
467
0.
=
J1Eip2+p~pi+2po
P1POOf2~Pi~
1
9i
P2 Py~2Po
2Pr2~Poo
V1P2~Pr~
V2
PPo~Pj~Pj~
£2
=
+
+ 2p
1-+ 2Pi7:
2131-2 + p~ + 21317:
g2P~PoPlr2~Poo~
By this proposition to prove e) it suffices to construct a bijection between 8-tuples
(p
8)
satisfying (1) and (2), and
3P4 -partitions
m satisfying (3.5). It is straightforward
to verify that the desired bijection can be defined by formulas
m12
rn12
P2 + 73~+p~+2P~
Py + Poo
=
=
rn1 = 2(pj7:+
2(P~-~
PP~2)
o +p
rn2=
1~+pj7:)
mln(m12,rn12)
p~= min(rn1!2,m2!2)
p-j-—
Poo
=
[_A12]~
P12 =
P2 = [min(A12,A12 +
p1~-=[min(_A1~72,—A1~,/2 —A12)]~
P0
=
p~-
where A12
=
[min(—A1~/2,A1~-!2+ A12)]~
[min(A12,—A12—A1~)]~,
rn12
—
rn~2,A1~= rn1
Proof of d). Since the root systems
exceptional) isomorphism between
CA~(’y) = C~A(’7) (see §1) imply
(05)
~-
—
—
£2
05
rn2 (see (2.3)), and [x]~
=
max(0, x)
and C2 are isomorphic we have a (so called
and sp4. This isomorphism and the property
(.~p.,)
C11fl,.~~‘.91~~2)
— C,,2,,,.tt
,.
‘.92,91
-
into account the already proven part e) it remains to construct a bijection between 05-partitions (rn12 m~2,m1, m2) satisfying (3.4) and 8134-partitions (~12,
m12, rn1 rn2) satisfying
Taken
,
Li
rn2/2
£2
rn12
£2
rn12
n~~
V1 mi!2+m12
2÷mi
+
rn1
91
92
—
V1m2!
V2 m12
rn2
-m~~
—
m12
+
m12
+
m1
m12
2m12
468
A.D. BERENSTEIN, A.V. ZELEVINSKY
(cf. (3.5)). Evidently, the desired bijection can be defined by (rn12, rnT2, rn1 m2)
,
=
(~2!2,~l!2,~i2,~l2)
Part f) is evident from the isomorphism D2
A1 x A1
exceptional isomorphism D3 = A3 which implies
(06)
-
To prove g) we use the
(844)
Ct,t2t5(9l,92,93)=C44t.~ny.~(92,9I,93)
Using c) it remains to construct a bijection between o6-partitions (m12, rn23, rn13,
rnT2 rn~3,rnT3) satisfying (3.7) and .sL4 -partitions (~12,rn13, rn14, rn23, rn24, rn34)
satisfying
,
Thrn24
£1
m13
£3
rn14
rn14 + in24
m14 + m24
£3
£3
+
in23
—
rn12
m13
— m13 + rn34
92 =
=
rn13
—
rn23
m14
+
m23
m12 + rn13
m1~+ rn24
+
rn14
+
rn34
+
71124 + 7fl23
V2
rn14 + rn13 — m24
m14 + m13 —m24 + m12
m34
V2
—
=
V1
V1
+
—
rn34
m23
rn24
(cf. (3.3)). One can verify that the following formulas define the desired bijection:
+ [—A12]~
min(rn24 rn13 + [A.12]~)
rn14 + [~I3 + [A12]~]÷
~l2
=
rn23
=
rn13
rn13
m23 =
m~3= rn34
rn34
m13
=
rn23
m13 = min(~12,~23)
rn~3= rn14 + [—A13— [A12]~]~
rn24
in14
rn12
m12
(39)
where A12
A13 =
=
=
=
—
m13
,
+
+
[—A12+
[A12
—
min(rn12,m12
m~2+ [A1~]~
= min(m~3
=
[—A11I~)
+
=
,
m12 — m~2, A1~- = rn~3— m23 (cf.
rn24 (cf. (2.1)).
(2.3)), A12
=
rn12
—
rn23
4. REDUCTION MULTIPLICITIES AND GENERALIZEDGELFAND-TSETLIN
PATTERNS
THEOREM 4.1. Let B be a simple classical Lie algebra of rank r, and A
=
~
1<i.Z,.
a highest weight for g. Then the weight multiplicity KAs is equal to the number of
B-pa.1z’itions m of weight
A
—
/3 satisfying max
the L~/~
were defined in (2.2), (2.4), (2.6).
( rn)
£~8)
<£, for
5
=
I
,...
,
r, where
TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE
469
According to Proposition 1.2 this Theorem is a special case ofTheorem 2.1 and Conjecture 2.2 (cf. Corollary to Conjecture 0.1).
We shall reformulate Theorem 4.1 in terms of generalized Gelfand-Tsetlin patterns
which will be defined separately for each type. First we recall familiar Gelfand-Tsetlin
patterns.
A GT-pattem (or gE,.-pauern) is an array A = (As,) (1 i <j <r) ofnonnegative integers A1, satisfying A~~_1 A1~1~
> A1, for all 1 ~ 1<5 ~ r. It is usually
drawn as follows:
A12
A11
A22
A1,.
A23
A2,.
A,.,.
The vector A = (A11, A12,... A1,.) E Z~will be called the highest weight of A
andtheweight /~=(f3I,--•,/~,.) ofA isdefinedby /32=1A11—1A2+11,where 1A11=
A11+A~~1+...+A2,.,IA,.+11=0.
We define an sE,.-pattem as an equivalence class of gEr-patterns modulo the following equivalence relation: A
A’if and only if A1, = A~,+ C for some C c Z
The highest weight and weight of an sL,.-pattem will be thought of as elements of
Z~/Z (1, 1,..., 1) i.e. as integral sL,.-weights.
We define an °2,.+1
as an array of numbers (A17, i~.,,) (1
I
j < r)
satisfying the following conditions:
,
.
,
(1) For all i,j we have ~
E Z
-
~-;
moreover, these numbers except
~,.,
,‘r~,.,. either all lie in Z orall lie in
+ Z
(2) min(A~~_1,A1+1~1) ~
max(A11,A2÷1,) for 1 ~ 1<5< r,and
min(A1,.,A1~1,.) m~ 0 for I <1< r (with the convention A4~12= +oo).
It is convenient to draw an o2,.~1-patternas follows:
~-
A12
A11
‘lii
A=
A1,.
‘112
‘li,.
A22...... A2,.
A,.,.
‘l,.,.
An sp2,.-pattem is an o2,.÷1-pattemsuch that all A1, and
~
are integers.
470
AD. BERENSTEIN, Ay. ZELEVINSKY
Finally, an o2,.-pattem is an array (A1, (1 < I < .f < r)
satisfying.
~ (1 < 1
<5 < r 1))
—
(1) Either all A11 and
are integers or all lie in
+Z
(2) min(A~,1_1,A1+1~_1) ~
max(A11,A1~1,,) for 1 < I < 5 < r, and
A1,. + A1~1,.+ min(A1,._1,A~~1,.~1)~
for 1 < i < r I (withtheconvention
A1~1~
= +oo).
The highest weightof each of these patterns is the weight A11e1 + A 12c2
A1,.r,.,
andtheweight /3 = /31e1 + .+/3,.e,. ofapatternisdefinedby /3~= IA1l—2lmH-IA~+iI.
It is easy to see that the highest weight of a g-pattem A is a highest g-weight, and the
weight of A is an integral g-weight.
It is also convenient to identify a gL,.-pattern A = (A13) with an °2,.+1
or o2,.-pattern(A11,’113) ,where ‘11j = A1~,1~1(1 < 1 <j < r
1) and rj~,.= 0.
Clearly, this identification preserves highest weight and weight.
We shall show that Theorem 4.1 is equivalent to
~-
—
+..
.+
. -
-,
—
THEOREM 4.2. For any simple classical Lie algebra B the weight multiplicity KAfi is
equal to the number of all B-patterns A ofhighest weight A and weight /3.
To show that Theorems 4.1 and 4.2 are equivalent it suffices to construct a bijection
between B-partitions rn from Theorem 4.1 and B-patterns A from Theorem 4.2. It is
easy to verify that such bijection in each case can be defined by formulas
(4.1)
rn11
= ‘1i,j_l
rn~=
— A~5,
‘1i.j_I
—
in1 =
A~~11,
2~,..
For B = sE,. the statement of Theorem 4.2 is well-known (the simple proof using
only (0.1) is given in [6]). Before considering other algebras we shall outline the proof
of Theorem 2.1.
Proof of Theorem 2.1. For B = sE,. the correspondence (4.1) takes form m13
A11. It is easy to see that it transforms the fl~t)( m) from (2.2) into
(4.2)
n~(A) = A1+1~ A1~+
~
—
=
A 1+1 ~
—
(2A1÷1~—
A1~ ~
—
1p~,.
t+
Therefore, Theorem 2.1 is equivalent to
THEOREM 4.3. Let A,p and
ii =
~
n,w
1 be highest .sL,.-weights. Then c~ is
1<i<,.—1
equal to the number of sL,. -patterns A of highest weight A, weight
that ~t)(A)
<n~for
1<1<
r—
I, i< t< r.
(~ v)
—
and such
TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE
471
But Theorem 4.3 is essentially proven in [2], [6] (note that definition of the weight of a
GT-pattern used there differs from the present one by the transformation (/3,... ,Ø,.)
(/3,.,... ,/3~)).
—~
REMARK. Correspondence (4.1) enables one to translate Conjecture 2.2 to the language
of B-patterns. We leave this reformulation to the reader.
The proof of Theorem 4.2 becomes easier if we generalize it to some reduction multiplicities. Fix one of the series B, C, D and denote the corresponding Lie algebra of rank r by g,.. For k = 0,... ,r consider the subset of simple roots H’ =
{ a~,... ca~,.}.Clearly, the corresponding semisimple Lie subalgebra go(’1’) is iso
morphic to B,.k , and the Levi subalgebra B( H’) naturally decomposes as ~ k ~ B,.k
(see §1). A~(fl’)-weightwillbethoughtofasapair (/3’, A’) ,where /3’ = (/3~,...,f3~),
and A’ = (A’k÷l,... A~) is a g,._~-weight. Denote by ~
the multiplicity of an
irreducible B(I1’)-module ~
in the reduction VAlIn~)(see §1); for k = r this is
just the weight multiplicity K~.
Forany k= 0,1,...,r byatnincatedg,.-patternA wemeananarray (A
11 (1 <
i < k + 1, 1 < 5 < r) ~ (1 < I < k, I < 5 < r)) which can be extended to a
,
,
,
B,.-pattern. For example, for k = 1 a truncated B,.-pattern looks as
A11
A12
A1,.
~lii
‘112
A22
A2,.
and for k = r it is a usual g,.-pattem. The highest weight of a truncated g,.-pattern A
is A = 2I~~1
(A11,A12,...
,A1,.) and the weight of A is the pair (f3’,A’) ,where /3~=
+ IA~+
A1~—
1I 1 <1< k) and A’ = (Ak+Ik+1,Ak+1k+z,...,Ak+1,.)
,
THEOREM 4.4. The reduction multiplicity K~j~~
is equal to the number of truncated
B,.-patterns of highest weight
A and weight (/3’, A’).
An obvious induction shows that it is sufficient to prove Theorem 4.4 for k = 1
i.e. to compute reduction of irreducible modules from B,. to ~ B,.~ The reduction
from ~P2,. to (ff~~ ~P2,._2) was computed by D.P. Zhelobenko [9], and the case of
orthogonal Lie algebras can be deduced from the results of [17]. Another way to prove
Theorem 4.4 for k = I is to use the involution method of [6]. Details will be given in
another publication.
~
.
REMARKS. 1. The similar result holds for g = aE,.. In this case it is well-known (see
e.g. [6]).
2. It is easy to verify that Theorem 4.4 is compatible with Conjecture 2.2.
472
AD. BERENSTEIN, Ay. ZELEVINSKY
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Manuscript received: October3, 1988