q, t-Fuß-Catalan numbers for finite reflection groups
Christian Stump
University of Vienna, Austria
December 4, 2008
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Overview
Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
The extended Shi arrangement
Conjectures on the specializations t = 1 and t = q −1
Connections to rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Overview
Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
The extended Shi arrangement
Conjectures on the specializations t = 1 and t = q −1
Connections to rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Finite real reflection groups
Let V be a finite-dimensional real vector space.
I For α ∈ V \ {0}, define sα ∈ O(V ) to be the reflection in V
sending α to its negative while fixing point-wise the
hyperplane Hα orthogonal to α.
I A (finite) real reflection group W ⊆ O(V ) is a finite group
generated by reflections.
Example
s
@
s
s
A
A
Q
Q A
Q
A
Q s
Q
A
Q
Q A
QA
QQ
@
s
@
@
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
@
s
@
@
@
@
@ s
@
@
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Root systems
I
A root system Φ is a finite set of non-zero vectors (called
roots) in V satisfying the conditions
i. Φ ∩ Rα = {±α} for all α ∈ Φ,
ii. the group WΦ := hsα : α ∈ Φi permutes Φ among itself, i.e.
sα (β) ∈ Φ
for all α, β ∈ Φ.
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Root systems
I
A root system Φ is a finite set of non-zero vectors (called
roots) in V satisfying the conditions
i. Φ ∩ Rα = {±α} for all α ∈ Φ,
ii. the group WΦ := hsα : α ∈ Φi permutes Φ among itself, i.e.
sα (β) ∈ Φ
for all α, β ∈ Φ.
I
Every root system contains a (not unique) positive system
Φ+ ⊆ Φ and
I
every positive system contains a unique simple system
∆ ⊆ Φ+ ⊆ Φ.
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Root systems
I
A root system Φ is a finite set of non-zero vectors (called
roots) in V satisfying the conditions
i. Φ ∩ Rα = {±α} for all α ∈ Φ,
ii. the group WΦ := hsα : α ∈ Φi permutes Φ among itself, i.e.
sα (β) ∈ Φ
for all α, β ∈ Φ.
Example
@
αs1
α1 +
s α2
A
A
Q A Q
A
αs2
Q s
s
Q
A
Q
Q A
QA
QQ
Q
s
α0
s
@
s
s
@s
@
s
2α0 + α1
@
@
s
@
α0 + α1
@
@
s
s
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
s
@ s
@
α
1@
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Irreducible real reflection groups
I
Every (finite) real reflection group W is equal to WΦ for some
root system Φ.
I
The following root systems determine (up to isomorphisms)
the irreducible finite real reflection groups:
An , Bn , Dn and
I2 (k), H3 , H4 , F4 , E6 , E7 , E8 .
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Example: The symmetric group Sn
The most classical example of a reflection group is the symmetric
group Sn of all permutations (bijections)
{1, . . . , n} −→
˜ {1, . . . , n}.
This group can be seen as the reflection group of type An−1 :
transposition ↔
(i, j)
=
simple transposition ↔
(i, i + 1)
=
reflection
sej −ei
simple reflection
sei+1 −ei
and where the positive system Φ+ and the simple system ∆ are
given by
Φ+ = {ej − ei : 1 ≤ i < j ≤ n},
∆ = {ei+1 − ei : 1 ≤ i < n}.
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Finite complex reflection groups
The concept of reflection groups can be generalized to complex
reflection groups:
Let V be a finite-dimensional complex vector space.
I A complex reflection s ∈ U(V ) is an orthogonal
transformation of a finite dimensional complex vector space V
such that
i. s has finite order and
ii. the fixed-point space of s has codimension 1.
I
A (finite) complex reflection group is a finite group of
W ⊆ O(V ) generated by complex reflections.
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Example: The cyclic group Ck = hζi
The most simple example of a complex reflection group which is
not real is the cyclic group Ck = hζi of order k ≥ 3, where ζ is a
primitive k-th root of unity, acting on C by z 7→ ζz.
ζ
t
ζ2 t+
I
t
t
1 = ζ5
ζ3
W t
- t
ζ4
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Reflections ↔ Reflecting hyperplanes
Remark
I
In real reflection groups, any reflection t = sα has order two
and there is a 1 : 1-correspondence
reflections
↔
reflecting hyperplanes
t
I
7→
Hα .
In complex reflection groups, any reflection t = sα has order
k ≥ 2 and there is a correspondence
reflections
↔
reflecting hyperplanes
t, t 2 , . . . , t k−1 7→
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
Hα .
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Overview
Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
The extended Shi arrangement
Conjectures on the specializations t = 1 and t = q −1
Connections to rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
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Alternating polynomials of type A
Let W = An−1 = Sn be the reflection group of type An−1 . Define
a diagonal action on
C[x, y] := C[x1 , y1 , . . . , xn , yn ]
for σ ∈ Sn by
σ p(x1 , y1 , . . . , xn , yn ) := p(xσ(1) , yσ(1) , . . . , xσ(n) , yσ(n) ).
A polynomial f ∈ C[x, y] is called
I
invariant if σ(f ) = f for all σ ∈ Sn ,
I
alternating if σ(f ) = sgn(σ) f for all σ ∈ Sn .
Example
I
x1 y2 + x2 y1 is invariant but not alternating,
I
x1 y2 − x2 y1 is not invariant but alternating.
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Alternating polynomials of type B
Let W = Bn be the reflection group of type Bn (group of signed
permutations). For σ ∈ Bn , define
σ p(x1 , y1 , . . . , xn , yn ) = ±p(x|σ(1)| , y|σ(1)| , . . . , x|σ(n)| , y|σ(n)| ).
A polynomial f ∈ C[x, y] is called
I
invariant if σ(f ) = f for all σ ∈ Bn ,
I
alternating if σ(f ) = sgn(σ) f for all σ ∈ Bn ,
where sgn(σ) := sgn(|σ|) · (−1)#{σ(i)<0} .
Example
I
x12 y22 + x22 y12 is invariant but not alternating,
I
x1 y2 − x2 y1 is not invariant but alternating.
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A generalization of alternating polynomials
The definition of alternating polynomials can be generalized as
follows:
Let W be a real reflection group acting on V . The
contragredient action of W on V ∗ = Hom(V , C) is given by
ω(ρ) := ρ ◦ ω −1 .
This gives an action of W on the symmetric algebra S(V ∗ ) = C[x]
and ’doubling up’ this action gives a diagonal action on
C[x, y] := C[V ⊕ V ].
A polynomial f ∈ C[x, y] is called
I
invariant if ω(f ) = f for all ω ∈ W ,
I
alternating if ω(f ) = det−1 (ω) f for all ω ∈ W .
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q, t-Fuß-Catalan numbers as a bigraded Hilbert series
Let W be a reflection group now acting on C[x, y].
Define I (resp. A) to be the ideal generated by all invariant
polynomials without constant term (resp. alternating polynomials).
The space of generalized diagonal coinvariants DR (m) (W ) is
defined to be the module
DR (m) (W ) := Am−1 /IAm−1 .
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q, t-Fuß-Catalan numbers as a bigraded Hilbert series
Let W be a reflection group now acting on C[x, y].
Define I (resp. A) to be the ideal generated by all invariant
polynomials without constant term (resp. alternating polynomials).
The space of generalized diagonal coinvariants DR (m) (W ) is
defined to be the module
DR (m) (W ) := Am−1 /IAm−1 .
Its alternating component M (m) (W ) := edet (DR (m) (W )) can be
shown to be equal to the minimal generating space of Am ,
M (m) (W ) = Am /hx, yiAm ∼
= CB,
where B is any homogeneous minimal generating set for Am .
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q, t-Fuß-Catalan numbers as a bigraded Hilbert series
Definition
For any real reflection group W , define q, t-Fuß-Catalan
numbers to be the bigraded Hilbert series of M (m) (W ), this is the
generating function of the dimensions of its bigraded components,
Cat(m) (W ; q, t) := H(M (m) (W ); q, t).
I
Cat(m) (W ; q, t) is a symmetric polynomial in q and t,
I
it reduces in type An−1 to the classical q, t-Fuß-Catalan
numbers,
(m)
Cat(m) (Sn ; q, t) = Catn (q, t).
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Example: Cat(1) (S3 ; q, t)
For W = S3 , one can show that
M (1) (S3 ) = A/hx, yiA
∆{(1,0),(2,0)} , ∆{(1,0),(1,1)} , ∆{(0,1),(1,1)} ,
∼
,
= C
∆{(0,1),(0,2)} , ∆{(1,0),(0,1)}
where
1
i1 j1

∆{(i1 ,j1 ),(i2 ,j2 )} (x, y) := det x1 y1
x1i2 y1j2

1
i1 j1
x2 y2
x2i2 y2j2

1
x3i1 y3j1  .
x3i2 y3j2
This gives
Cat(1) (S3 ; q, t) = H(M (1) (S3 ); q, t)
= q 3 + q 2 t + qt 2 + t 3 + qt.
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A conjectured formula for the dimension of M (m) (W )
Computations of the dimensions of M (m) (W ) were the first
motivation for further investigations:
Conjecture
Let W be a real reflection group. Then
Cat(m) (W ; 1, 1) =
l
Y
di + mh
,
di
i=1
where
I
l is the rank of W ,
I
h is the Coxeter number and
I
d1 , . . . , dl are the degrees of W .
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A conjectured formula for the dimension of M (m) (W )
I
These numbers, called Fuß-Catalan numbers, count many
combinatorial objects, for example
I
positive regions in the generalized Shi arrangement
(Athanasiadis, Postnikov),
I
m-divisible non-crossing partitions (Armstrong, Bessis,
Reiner),
facets in the generalized Cluster complex (Fomin, Reading,
Zelevinsky).
I
I
They reduce for m = 1 to the well-known Catalan numbers
associated to real reflection groups:
An−1
1 2n
n+1 n
I2 (k)
k +2
H3
32
Bn
2n
n
H4
280
F4
105
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
2n
n
Dn
− 2(n−1)
n−1
E6
833
E7
4160
E8
25080
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The classical q, t-Fuß-Catalan numbers
In type A, Cat(m) (Sn ; q, t) occurred within the past 15 years in
various fields of mathematics:
I they were first introduced as a complicated rational function
in the context of modified Macdonald polynomials (Garsia,
Haiman),
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The classical q, t-Fuß-Catalan numbers
In type A, Cat(m) (Sn ; q, t) occurred within the past 15 years in
various fields of mathematics:
I they were first introduced as a complicated rational function
in the context of modified Macdonald polynomials (Garsia,
Haiman),
I later they were shown to be equal to the Hilbert series of the
space of diagonal coinvariants (Haiman)
I and of some cohomology module in the theory of Hilbert
schemes of points in the plane (Haiman),
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The classical q, t-Fuß-Catalan numbers
In type A, Cat(m) (Sn ; q, t) occurred within the past 15 years in
various fields of mathematics:
I they were first introduced as a complicated rational function
in the context of modified Macdonald polynomials (Garsia,
Haiman),
I later they were shown to be equal to the Hilbert series of the
space of diagonal coinvariants (Haiman)
I and of some cohomology module in the theory of Hilbert
schemes of points in the plane (Haiman),
I they have a conjectured combinatorial interpretation in terms
of two statistics on partitions fitting inside the partition
µ := ((n − 1)m, . . . , 2m, m),
X
(m)
Catn (q, t) =
q area(λ) t dinv(λ) .
λ⊆µ
I
Proved for m = 1 (Garsia, Haglund) and for t = 1 (Haiman).
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Example: Cat(1) (S3 ; q, t) (continued)
Let λ be a partition that fits inside the partition µ := (2, 1).
Then
area(λ) := |(2, 1)| − |λ| = 3 − |λ| and
dinv(λ) := # c ∈ λ : arm(c) − leg(c) ∈ {0, 1} .
=⇒ Cat(1) (S3 ; q, t) =
λ=∅
P
λ = (1)
λ⊆µ q
area(λ) t dinv(λ)
λ = (2)
is given by
λ = (2, 1)
λ = (1, 1)
Cat(1) (S3 ; q, t) = q 3 + q 2 t + qt 2 + t 3 + qt.
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Overview
Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
The extended Shi arrangement
Conjectures on the specializations t = 1 and t = q −1
Connections to rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
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The extended Shi arrangement
Let W be a crystallographic reflection group and let Φ+ be a set
of positive roots.
The extended Shi arrangement Shi(m) (W ) is defined to be the
collection of hyperplanes in V given by
k
Hα : α ∈ Φ+ , −1 ≤ k ≤ m ,
where Hαk = {x ∈ V : (x, α) = k}.
I
A region of Shi(m) (W ) is a connected component of
[
V\
Hαk .
k ∈Shi(m) (W )
Hα
Remark
The extended Shi arrangement contains the Coxeter
arrangement associated to W .
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Example: Shi(1) (A2 )
regions
Hα13
Q
Q
Q
Q Q
Q
Hα−1
Q
Q
3
s Q
Q QQ
Q
Q
Hα12 QQ
?
Q Q
Q
Q
Q
Q
Hα02
Q
Q
Q
−1
Hα03
Hα 2
Q
Q
Hα−1
Hα01 Hα11
1
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The coheight of a region in the extended Shi arrangement
Let R ∞ be the region given by
(x, α) > m for all α ∈ Φ+ .
For any region R define the coheight of R, coh(R), to be the
number of hyperplanes separating R from R ∞ .
We combinatorially define q-Fuß-Catalan numbers by
X
Cat(m) (W ; q) :=
q coh(R) ,
where the sum ranges over all regions of Shi(m) (Φ) which lie in the
fundamental chamber of the associated Coxeter arrangement.
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Example: Shi(1) (A2 ) and Cat(1) (A2 ; q)
R∞
Hα13
Hα03
Q
Q
0
Q
Q 1 1 Q
2
Q Q Hα−1
Q
3 Q
3
s Q
Q QQ
QQ
Hα12 QQ
Q Q
Q
Q
Q
Q
Hα02
Q
Q
Q
−1
Q
Hα 2
Hα−1
Hα01 Hα11
1
X
Cat(1) (A2 ; q) =
q coh(R) = 1 + 2q + q 2 + q 3 .
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Overview
Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
The extended Shi arrangement
Conjectures on the specializations t = 1 and t = q −1
Connections to rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
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The specialization t = 1 in Cat(m) (W ; q, t)
The definition of q-Fuß-Catalan numbers is motivated by the
following conjecture:
Conjecture
Let W be a crystallographic reflection group or let W be the
dihedral group of type I2 (k) for any k. Then
Cat(m) (W ; q, 1) = Cat(m) (W ; q).
I
The conjecture is known to be true for type A,
I
was validated by computations for several types.
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The specialization t = q −1 in Cat(m) (W ; q, t)
Conjecture
Let W be a well-generated complex reflection group. Then
Cat(m) (W ; q, q −1 ) = q −mN
l
Y
[di + mh]q
,
[di ]q
i=1
where
I
N is the number of reflections in W ,
I
[n]q := q n−1 + . . . + q + 1 is the usual q-analogue of n.
Remark
This conjecture is stronger than the previous conjecture about the
dimension of M (m) (W ).
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The dihedral group I2 (k)
Theorem
Let W be the dihedral group of type I2 (k). Then
(m)
Cat
(W ; q, t) =
m
X
q m−j t m−j [ jk + 1]q,t ,
j=0
where
[n]q,t := q n−1 + q n−2 t + . . . + qt n−2 + t n−1 .
Corollary
All shown conjectures hold for the dihedral groups.
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Overview
Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
The extended Shi arrangement
Conjectures on the specializations t = 1 and t = q −1
Connections to rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
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Some facts about rational Cherednik algebras
Let W be a real reflection group. The rational Cherednik algebra
Hc = Hc (W )
is an associative algebra generated by the vector spaces h, its dual
h∗ and the group W subject to defining relations depending on a
rational parameter c, such that
I
the polynomial rings C[V ], C[V ∗ ] and
I
the group algebra CW
are subalgebras of Hc .
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A simple Hc -module
For c = m + h1 there exists a unique simple Hc -module L which
carries a natural filtration.
Theorem (Berest, Etingof, Ginzburg)
Let W be a real reflection group. Then
H(e(gr(L)); q) = q −mN
l
Y
[di + mh]q
,
[di ]q
i=1
where e(gr(L)) is the trivial component of associated graded
module of L.
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The connection between L and the space of generalized
diagonal coinvariants
Theorem
Let W be a real reflection group and let
DR (m) (W ) = Am−1 /IAm−1
be the generalized diagonal coinvariants graded by degree in x
minus degree in y. Then there exists a natural surjection of graded
modules,
DR (m) (W ) ⊗ det gr(L),
where det is the sign representation.
Remark
This theorem generalizes a theorem by Gordon, who proved the
m = 1 case, following mainly his approach.
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The connection between L and the space of generalized
diagonal coinvariants
Conjecture
The W -stable kernel of the surjection in the previous theorem does
not contain a copy of the trivial representation.
Corollary
If the previous conjecture holds, then
M (m) (W ) ∼
= e(gr(L))
as (bi-)graded modules. In particular,
Cat(W ; q, q −1 ) = q −mN
l
Y
[di + mh]q
.
[di ]q
i=1
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Overview
Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
The extended Shi arrangement
Conjectures on the specializations t = 1 and t = q −1
Connections to rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
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Example: The cyclic group Ck = hζi (continued)
The cyclic group Ck = hζi would act on C[x, y ] := C[C ⊕ C] by
ζ(x a y b ) = ζ a · x a ζ b · y b = ζ a+b · x a y b .
This would give
C[x, y ]Ck
= span x a y b : a + b ≡ 0 mod k ,
C[x, y ]det = span x a y b : a + b ≡ 1 mod k
= xC[x, y ]Ck + y C[x, y ]Ck ,
C[x, y ]det
−1
= span x a y b : a + b ≡ k − 1 mod k
X
=
x i y j C[x, y ]Ck .
i+j=k−1
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Example: The cyclic group Ck = hζi (continued)
C[x, y ]det = xC[x, y ]Ck + y C[x, y ]Ck ,
C[x, y ]det
−1
=
X
x i y j C[x, y ]Ck .
i+j=k−1
We would have two possible choices to define q, t-Catalan numbers
for the cyclic group Ck :
Cat(1) (Ck ; q, t) = q + t or
Cat(1) (Ck ; q, t) = q k−1 + q k−2 t + . . . + qt k−2 + t k−1 .
I
Both choices are in contradiction to the previously
shown conjectures!
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A generalization of alternating polynomials
Let W be a real reflection group acting on V .
The contragredient action of W on V ∗ = Hom(V , C) is given by
ω(ρ) := ρ ◦ ω −1 .
This gives an action of W on the symmetric algebra S(V ∗ ) = C[x]
and ’doubling up’ this action gives a diagonal action on
C[x, y] := C[V ⊕ V ].
A polynomial f ∈ C[x, y] is called
I
invariant if ω(f ) = f for all ω ∈ W ,
I
alternating if ω(f ) = det−1 (ω) f for all ω ∈ W .
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A generalization of alternating polynomials
Let W be a complex reflection group acting on V .
The contragredient action of W on V ∗ = Hom(V , C) is given by
ω(ρ) := ρ ◦ ω −1 .
This gives an action of W on the symmetric algebra S(V ∗ ) = C[x]
and ’doubling up’ this action gives a diagonal action on
C[x, y] := C[V ⊕ V ∗ ].
A polynomial f ∈ C[x, y] is called
I
invariant if ω(f ) = f for all ω ∈ W ,
I
alternating if ω(f ) = det−1 (ω) f for all ω ∈ W .
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Example: The cyclic group Ck = hζi (continued)
The cyclic group Ck = hζi would act on C[x, y ] := C[C ⊕ C] by
ζ(x a y b ) = ζ a · x a ζ b · y b = ζ a+b · x a y b .
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Example: The cyclic group Ck = hζi (continued)
The cyclic group Ck = hζi
acts
on C[x, y ] := C[C ⊕ C∗ ] by
ζ(x a y b ) = ζ a · x a ζ −b · y b = ζ a−b · x a y b .
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Example: The cyclic group Ck = hζi (continued)
The cyclic group Ck = hζi
acts
on C[x, y ] := C[C ⊕ C∗ ] by
ζ(x a y b ) = ζ a · x a ζ −b · y b = ζ a−b · x a y b .
This gives
C[x, y ]Ck
= span x a y b : a ≡ b mod k ,
C[x, y ]det = span x a y b : a + 1 ≡ b mod k
= xC[x, y ]Ck + y k−1 C[x, y ]Ck ,
C[x, y ]det
−1
= span x a y b : a ≡ b + 1 mod k
= x k−1 C[x, y ]Ck + y C[x, y ]Ck .
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Example: The cyclic group Ck = hζi (continued)
The cyclic group Ck = hζi
acts
on C[x, y ] := C[C ⊕ C∗ ] by
ζ(x a y b ) = ζ a · x a ζ −b · y b = ζ a−b · x a y b .
C[x, y ]det = xC[x, y ]Ck + y k−1 C[x, y ]Ck ,
−1
C[x, y ]det
= x k−1 C[x, y ]Ck + y C[x, y ]Ck .
Now, we have (beside interchanging q and t) only the following
choice:
Cat(1) (Ck ; q, t) := q + t k−1 .
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Example: The cyclic group Ck = hζi (continued)
The cyclic group Ck = hζi
acts
on C[x, y ] := C[C ⊕ C∗ ] by
ζ(x a y b ) = ζ a · x a ζ −b · y b = ζ a−b · x a y b .
Now, we have (beside interchanging x and y ) only the following
choice:
Cat(1) (Ck ; q, t) := q + t k−1 .
I
The q-power equals the number of reflecting hyperplanes
N ∗ = 1 and the t-power equal the number of reflections
N = k − 1. We have
∗
q N Cat(1) (Ck ; q, q −1 ) = q N Cat(1) (Ck ; q −1 , q)
= 1 + q k = [2k]q /[k]q .
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Final conjecture
Defining q, t-Fuß-Catalan numbers for complex reflection groups in
the same way as for real reflection groups with
C[x, y] := C[V ⊕ V ∗ ],
we conjecture the following:
Conjecture
Let W be a well-generated complex reflection group. Then
∗
q mN Cat(m) (W ; q, q −1 ) = q mN Cat(m) (W ; q −1 , q)
=
l
Y
[di + mh]q
,
[di ]q
i=1
where
I
N is the number of reflections in W and where
I
N ∗ is the number of reflecting hyperplanes.
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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Thank you very much!
Christian Stump – q, t-Fuß-Catalan numbers for finite reflection groups
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