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Combinatorics of generalized exponents
Cédric Lecouvey
Institut Denis Poisson Tours
Dias de combinatórica 2018
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 1 / 20
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I. Generalized exponents in type A
Zn =nLi=1
Zei .
Let Q be the sublattice of Zn generated by the vectors ei � ei+1, 1 � i < n.
For β = (β1, . . . , βn) 2 Zn , set xβ = xβ11 � � � x
βnn .
The q-Kostant partition function for gln(C) is de�ned by
∏1�i<j�n
11� q xixj
= ∑β2Zn
Pq(β)xβ.
Pq(β) 2 Z�0 [q]
P1(β) gives the number of nonnegative decompositions of β as a sum ofei � ej , 1 � i < j � n.Pq(β) = 0 when β /2 Q.
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 2 / 20
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I. Generalized exponents in type A
Zn =nLi=1
Zei .
Let Q be the sublattice of Zn generated by the vectors ei � ei+1, 1 � i < n.
For β = (β1, . . . , βn) 2 Zn , set xβ = xβ11 � � � x
βnn .
The q-Kostant partition function for gln(C) is de�ned by
∏1�i<j�n
11� q xixj
= ∑β2Zn
Pq(β)xβ.
Pq(β) 2 Z�0 [q]
P1(β) gives the number of nonnegative decompositions of β as a sum ofei � ej , 1 � i < j � n.
Pq(β) = 0 when β /2 Q.
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 2 / 20
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I. Generalized exponents in type A
Zn =nLi=1
Zei .
Let Q be the sublattice of Zn generated by the vectors ei � ei+1, 1 � i < n.
For β = (β1, . . . , βn) 2 Zn , set xβ = xβ11 � � � x
βnn .
The q-Kostant partition function for gln(C) is de�ned by
∏1�i<j�n
11� q xixj
= ∑β2Zn
Pq(β)xβ.
Pq(β) 2 Z�0 [q]
P1(β) gives the number of nonnegative decompositions of β as a sum ofei � ej , 1 � i < j � n.Pq(β) = 0 when β /2 Q.
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 2 / 20
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A partition is a sequence λ = (λ1 � � � � � λn � 0) 2 Zn .Each partition is encoded by its Young diagram. For example
λ = (4, 3, 2, 1)$ and jλj = 10.
Let ρ = (n, n� 1, . . . , 2, 1).
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 3 / 20
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Let Sn be the symmetric group of rank n.The group Sn acts on Zn by permutation σ � (β1, . . . , βn) = (βσ(1), . . . , βσ(n)).Consider λ and µ two partitions such that jλj = jµj.
De�nitionThe Kostka polynomial Kλ,µ(q) is the polynomial of Z[q] s.t.
Kλ,µ(q) = ∑σ2Sn
ε(σ)Pq(σ(λ+ ρ)� ρ� µ)
De�nition
In the particular case µ =�jλjn , . . . , jλjn
�, the polynomial Kλ(q) = Kλ,µ(q) is the
generalized exponent associated to λ.
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 4 / 20
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A semistandard tableau T of shape λ is a �lling of λ by letters in f1, . . . , ng withstrictly increasing columns from top to bottom
weakly increasing rows from left to right.
Its weight is wt(T ) = (µ1, . . . , µn) with µi = # letters i in T
Example
T =
1 1 2 42 3 53 45
with weight wt(T ) = (2, 2, 2, 2, 2) and reading
w(T ) = 4211532435
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Theorem
1 Kλ,µ(1) is equal to the number of SST of shape λ and weight µ.2 Kλ(1) is equal to the number of homogeneous SST of shape λ, i.e. with
weight µ =�jλjn , . . . , jλjn
�.
3 Kλ(q) 2 Z�0 [q].
For Assertion 3 :
sophisticated geometric or algebraic proofs (Intersection cohomology ofnilpotent orbits, a¢ ne Kazhdan-Lusztig polynomials, Brylinsky-Kostant�ltration).
a combinatorial proof and description by Lascoux & Schützenberger...
...which extends in fact to any partition µ (Kostka polynomial).
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 6 / 20
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Theorem
1 Kλ,µ(1) is equal to the number of SST of shape λ and weight µ.2 Kλ(1) is equal to the number of homogeneous SST of shape λ, i.e. with
weight µ =�jλjn , . . . , jλjn
�.
3 Kλ(q) 2 Z�0 [q].
For Assertion 3 :
sophisticated geometric or algebraic proofs (Intersection cohomology ofnilpotent orbits, a¢ ne Kazhdan-Lusztig polynomials, Brylinsky-Kostant�ltration).
a combinatorial proof and description by Lascoux & Schützenberger...
...which extends in fact to any partition µ (Kostka polynomial).
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 6 / 20
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Theorem
1 Kλ,µ(1) is equal to the number of SST of shape λ and weight µ.2 Kλ(1) is equal to the number of homogeneous SST of shape λ, i.e. with
weight µ =�jλjn , . . . , jλjn
�.
3 Kλ(q) 2 Z�0 [q].
For Assertion 3 :
sophisticated geometric or algebraic proofs (Intersection cohomology ofnilpotent orbits, a¢ ne Kazhdan-Lusztig polynomials, Brylinsky-Kostant�ltration).
a combinatorial proof and description by Lascoux & Schützenberger...
...which extends in fact to any partition µ (Kostka polynomial).
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 6 / 20
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II. The charge statistics
Examples of row insertions
1 1 2 42 3 53 4
� 5 =1 1 2 4 52 3 53 4
1 1 2 4 52 5 53 6
� 3 =
1 1 2 3 52 4 53 56
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 7 / 20
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Cocyclage from T of weight µ:
T =
1 1 2 42 3 53 45
!1 1 2 4 52 3 53 4
!1 1 2 3 52 3 44 5
!1 1 2 3 42 3 4 55
! � � � ! 1 1 2 2 3 3 4 4 5 5
TheoremCocyclage operations eventually ends to the unique row Rµ of weight µ
De�nitionSet chn(T ) = kµk � l = chn(Rµ)� l � 0 where l the number of cocyclageoperations needed to get Rµ and
kµk =n�1∑i=1(n� i)µi .
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 8 / 20
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Theorem (LS 1980)We have
Kλ,µ(q) = ∑T2SST (λ)µ
qchn(T ).
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 9 / 20
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ExampleFor λ = (2, 1, 0) and µ = (1, 1, 1) we have
1 23
! 1 2 3
1 32
! 1 23
! 1 2 3
hence
chn
�1 32
�= 1 and chn
�1 23
�= 2
thusK(2,1,0)(q) = q + q
2.
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 10 / 20
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Let w = x1 � � � x` be a word on f1 < � � � < ng.For each i = 1, . . . , n� 1, form wi the subword of w contained only the letters iand i + 1.w redi = (i + 1)εi (w )i ϕi (w ) is obtained by recursive deletion of factors i(i + 1) inwi .Example: w = 2421153243135 with n = 5
1 w1 = 21(12)1 and w red1 = 211. Thus ε1(w) = 1 and ϕ1(w) = 2
2 w2 = 2(23)(23)3 and w red2 = ∅. Thus ε2(w) = ϕ2(w) = 0.
3 w3 = 4(34)3 and w red3 = 43. Thus ε3(w) = ϕ3(w) = 1.
4 w4 = (45)(45) and w red4 = ∅. Thus ε4(w) = ϕ4(w) = 0.
Theorem (LLT 1995)When T is homogeneous, we have
chn(T ) =n�1∑i=1(n� i)εi (w(T )).
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 11 / 20
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ExampleFor λ = (2, 1, 0) and µ = (1, 1, 1) we have
chn
�1 23
�= chn(213) = (3� 1)� 1+ (3� 2)� 0 = 2
chn
�1 32
�= chn(312) = (3� 1)� 0+ (3� 2)� 1 = 1
thusK(2,1,0)(q) = q + q
2.
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 12 / 20
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Generalized exponents in type C and beyond
Start from
∏1�i�j�n
11� qxi xj ∏
1�i<j�n
11� q xixj
= ∑β2Zn
PCnq (β)xβ.
and replace Sn by the group Wn of signed permutations on f1, 1, 2, 2, . . . , n, ng
w(x) = y () w(x) = y .
It acts on Zn by
w � (β1, . . . , βn) = (β01, . . . , β0n) with β0i =
(βw (i ) if w(i) > 0�β�w (i ), otherwise
De�nitionFor any partitions λ and µ
KCnλ,µ(q) = ∑w2Wn
εCn (w)PCnq (w(λ+ ρ)� ρ� µ) and KCnλ (q) = KCnλ,0(q)
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 13 / 20
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In fact KCnλ (1) gives the dimension of the zero weight space in the irreduciblesp2n(C)-representation indexed by λ. Thus :
KCnλ (1) = # King tableaux of shape λ and zero weight,
KCnλ (1) = # Kashiwara-Nakashima tableaux of shape λ and zero weight
KCnλ (1) = # Littelmann paths of shape λ starting and ending at 0.
This generalizes to any weight.
ProblemFind a charge for type Cn which proves the positivity of the coe¢ cients.
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 14 / 20
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From Cauchy and Littlewood identities on symmetric functions one gets:
Theorem (C.L., C. Lenart)1
Kλ(q)
∏ni=1(1� qi )
= ∑γ2Pn
qjγjcλγ,γ� .
2
KCnλ (q)
∏ni=1(1� q2i )
= ∑ν22P2n
qjνj/2cλν
Here
γ� = (0,γ1 � γ2, . . . ,γ1 � γn),
the cλγ,γ��s are tensor product multiplicities,
the cλν �s are the branching coe¢ cients appearing in the restrictions
V (ν) #gl2nsp2n
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 15 / 20
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From Cauchy and Littlewood identities on symmetric functions one gets:
Theorem (C.L., C. Lenart)1
Kλ(q)
∏ni=1(1� qi )
= ∑γ2Pn
qjγjcλγ,γ� .
2
KCnλ (q)
∏ni=1(1� q2i )
= ∑ν22P2n
qjνj/2cλν
Here
γ� = (0,γ1 � γ2, . . . ,γ1 � γn),
the cλγ,γ��s are tensor product multiplicities,
the cλν �s are the branching coe¢ cients appearing in the restrictions
V (ν) #gl2nsp2n
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The combinatorial description of Kλ(q) (for gln) with the εi�s easily followsfrom 1.
Very more di¢ cult for sp2n because cλν has a simple description only when
νk = 0 for k > n wherecλ
ν = ∑δ2P�n
cνλ,δ .
Only gets a description of the formal series KC∞λ (q) = limn!+∞ K
Cnλ (q).
Example
KC∞� (q) = ∑
k�1q2k =
q2
1� q2 .
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The combinatorial description of Kλ(q) (for gln) with the εi�s easily followsfrom 1.
Very more di¢ cult for sp2n because cλν has a simple description only when
νk = 0 for k > n wherecλ
ν = ∑δ2P�n
cνλ,δ .
Only gets a description of the formal series KC∞λ (q) = limn!+∞ K
Cnλ (q).
Example
KC∞� (q) = ∑
k�1q2k =
q2
1� q2 .
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 16 / 20
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The combinatorial description of Kλ(q) (for gln) with the εi�s easily followsfrom 1.
Very more di¢ cult for sp2n because cλν has a simple description only when
νk = 0 for k > n wherecλ
ν = ∑δ2P�n
cνλ,δ .
Only gets a description of the formal series KC∞λ (q) = limn!+∞ K
Cnλ (q).
Example
KC∞� (q) = ∑
k�1q2k =
q2
1� q2 .
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 16 / 20
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A �King tableau�T of shape λ = (λ1, . . . ,λn) is a �lling of λ by letters of
f1 < 2 < 3 < 4 < � � � < 2n� 1 < 2ng
s.t.
T is semistandard,
The letters in row i are greater to 2i � 1Such a tableau is distinguished when we have moreover
ϕi (T ) = 0 for any odd i
εi (T ) is even for any odd i .
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A �King tableau�T of shape λ = (λ1, . . . ,λn) is a �lling of λ by letters of
f1 < 2 < 3 < 4 < � � � < 2n� 1 < 2ng
s.t.
T is semistandard,
The letters in row i are greater to 2i � 1Such a tableau is distinguished when we have moreover
ϕi (T ) = 0 for any odd i
εi (T ) is even for any odd i .
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 17 / 20
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Theorem (C. L. and C. Lenart)We have
KCnλ (q) = ∑T distinguished of shape λ
qchCn (T )
where
chCn (T ) =2n�1∑i=1
(2n� i)�
εi (T )2
�.
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Corollary
1 KCn+1λ (q)�KCnλ (q) has nonnegative coe¢ cients.
2 Determination of the highest and lowest monomials in KCnλ .3 Simple formulas for particular partitions λ.
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Futures directions
1 Other classical types for �nite n.2 Generalization to any dominant weight µ 6= 0.3 Connect to a conjectural charge statistics de�ned on KN tableaux fromcyclage operation.
4 Parabolic cases (positivity not even known in general).
C. Lecouvey (Institut Denis Poisson Tours) Combinatorics of generalized exponents Dias de combinatórica 2018 20 / 20