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Categorical actionson unipotent representations
of finite classical groups
Olivier Dudas(with P. Shan, M. Varagnolo and E. Vasserot)
Paris 7 - Paris Diderot
March 2015
O. Dudas (Paris 7) Categorical actions Mar. 2015 1 / 1
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Motivation
Finite classical groups
Gn(q) = GLn(q), GUn(q), Sp2n(q) . . .
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Motivation
Finite classical groups
Gn(q) = GLn(q), GUn(q), Sp2n(q) . . .
Study representations over a field k of characteristic ℓ ≥ 0 using higher Lie
theory, i.e. categorical actions of Lie algebras on
C =⊕
n≥0
kGn(q)-modu
coming from (parabolic) induction and restriction
O. Dudas (Paris 7) Categorical actions Mar. 2015 2 / 1
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Motivation
Finite classical groups
Gn(q) = GLn(q), GUn(q), Sp2n(q) . . .
Study representations over a field k of characteristic ℓ ≥ 0 using higher Lie
theory, i.e. categorical actions of Lie algebras on
C =⊕
n≥0
kGn(q)-modu
coming from (parabolic) induction and restriction
Hoping for the following applications:
O. Dudas (Paris 7) Categorical actions Mar. 2015 2 / 1
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Motivation
Finite classical groups
Gn(q) = GLn(q), GUn(q), Sp2n(q) . . .
Study representations over a field k of characteristic ℓ ≥ 0 using higher Lie
theory, i.e. categorical actions of Lie algebras on
C =⊕
n≥0
kGn(q)-modu
coming from (parabolic) induction and restriction
Hoping for the following applications:
◮ Branching graph for induction/restriction [Gerber-Hiss-Jacon]
O. Dudas (Paris 7) Categorical actions Mar. 2015 2 / 1
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Motivation
Finite classical groups
Gn(q) = GLn(q), GUn(q), Sp2n(q) . . .
Study representations over a field k of characteristic ℓ ≥ 0 using higher Lie
theory, i.e. categorical actions of Lie algebras on
C =⊕
n≥0
kGn(q)-modu
coming from (parabolic) induction and restriction
Hoping for the following applications:
◮ Branching graph for induction/restriction [Gerber-Hiss-Jacon]
◮ Derived equivalences [Broue]
O. Dudas (Paris 7) Categorical actions Mar. 2015 2 / 1
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Motivation
Finite classical groups
Gn(q) = GLn(q), GUn(q), Sp2n(q) . . .
Study representations over a field k of characteristic ℓ ≥ 0 using higher Lie
theory, i.e. categorical actions of Lie algebras on
C =⊕
n≥0
kGn(q)-modu
coming from (parabolic) induction and restriction
Hoping for the following applications:
◮ Branching graph for induction/restriction [Gerber-Hiss-Jacon]
◮ Derived equivalences [Broue]
◮ Decomposition numbers
O. Dudas (Paris 7) Categorical actions Mar. 2015 2 / 1
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Unipotent characters of GLn(q)
There is a distinguished set of complex irreducible characters of GLn(q),called the unipotent characters, labelled by partitions
Uch(GLn(q)) ←→ {Partitions of n}χµ ←−[ µ
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Unipotent characters of GLn(q)
There is a distinguished set of complex irreducible characters of GLn(q),called the unipotent characters, labelled by partitions
Uch(GLn(q)) ←→ {Partitions of n} (←→ IrrSn)χµ ←−[ µ
O. Dudas (Paris 7) Categorical actions Mar. 2015 3 / 1
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Unipotent characters of GLn(q)
There is a distinguished set of complex irreducible characters of GLn(q),called the unipotent characters, labelled by partitions
Uch(GLn(q)) ←→ {Partitions of n} (←→ IrrSn)χµ ←−[ µ
Examples
◮ χ(n) = 1GLn(q) the trivial character
O. Dudas (Paris 7) Categorical actions Mar. 2015 3 / 1
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Unipotent characters of GLn(q)
There is a distinguished set of complex irreducible characters of GLn(q),called the unipotent characters, labelled by partitions
Uch(GLn(q)) ←→ {Partitions of n} (←→ IrrSn)χµ ←−[ µ
Examples
◮ χ(n) = 1GLn(q) the trivial character
◮ χ(1n) = StGLn(q) the Steinberg character
O. Dudas (Paris 7) Categorical actions Mar. 2015 3 / 1
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Unipotent characters of GLn(q)
There is a distinguished set of complex irreducible characters of GLn(q),called the unipotent characters, labelled by partitions
Uch(GLn(q)) ←→ {Partitions of n} (←→ IrrSn)χµ ←−[ µ
Examples
◮ χ(n) = 1GLn(q) the trivial character
◮ χ(1n) = StGLn(q) the Steinberg character
Under this parametrization, parabolic induction and restriction onunipotent characters coincide with induction and restriction on irreduciblecharacters of symmetric groups.
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Unipotent characters of GLn(q)
There is a distinguished set of complex irreducible characters of GLn(q),called the unipotent characters, labelled by partitions
Uch(GLn(q)) ←→ {Partitions of n} (←→ IrrSn)χµ ←−[ µ
Examples
◮ χ(n) = 1GLn(q) the trivial character
◮ χ(1n) = StGLn(q) the Steinberg character
Under this parametrization, parabolic induction and restriction onunipotent characters coincide with induction and restriction on irreduciblecharacters of symmetric groups.
In fact there is a Hecke algebra Hq(An−1) hiding there...
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
e
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f =
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f =
O. Dudas (Paris 7) Categorical actions Mar. 2015 4 / 1
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f = +
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f = + +
O. Dudas (Paris 7) Categorical actions Mar. 2015 4 / 1
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f = + + +
O. Dudas (Paris 7) Categorical actions Mar. 2015 4 / 1
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f
0 1 2 3-1 0-2 -1-3
=
0 1 2 3 4-1 0-2 -1-3
+
0 1 2 3-1 0 1-2 -1-3
+
0 1 2 3-1 0-2 -1-3 -2
+
0 1 2 3-1 0-2 -1-3-4
Use content to split f =∑
fa
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f
0 1 2 3-1 0-2 -1-3
=
0 1 2 3 4-1 0-2 -1-3
+
0 1 2 3-1 0 1-2 -1-3
+
0 1 2 3-1 0-2 -1-3 -2
+
0 1 2 3-1 0-2 -1-3-4
Use content to split f =∑
fa fa(χµ) is either zero or irreducible
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i -induction and i -restriction
Parabolic induction and restriction
Uch(GLn(q)) Uch(GLn+1(q))
f
ef add boxes, and e remove boxes
f
0 1 2 3-1 0-2 -1-3
=
0 1 2 3 4-1 0-2 -1-3
+
0 1 2 3-1 0 1-2 -1-3
+
0 1 2 3-1 0-2 -1-3 -2
+
0 1 2 3-1 0-2 -1-3-4
Use content to split f =∑
fa fa(χµ) is either zero or irreducible
〈ea, fa〉a∈Z induce an action of sl∞ on⊕
CUch(GLn(q)) isomorphic to aFock space representation of level 1.
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Positive characteristic
fa is constructed using the qa-eigenspace of a Jucys-Murphy element
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Positive characteristic
fa is constructed using the qa-eigenspace of a Jucys-Murphy element
Now assume char k = ℓ > 0, ℓ ∤ q and set
d = order of q ∈ k×
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Positive characteristic
fa is constructed using the qa-eigenspace of a Jucys-Murphy element
Now assume char k = ℓ > 0, ℓ ∤ q and set
d = order of q ∈ k×
Identification of qa and qa+d eigenspaces for a ∈ Z/dZ, set
fa =∑
a∈a
fa and ea =∑
a∈a
ea
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Positive characteristic
fa is constructed using the qa-eigenspace of a Jucys-Murphy element
Now assume char k = ℓ > 0, ℓ ∤ q and set
d = order of q ∈ k×
Identification of qa and qa+d eigenspaces for a ∈ Z/dZ, set
fa =∑
a∈a
fa and ea =∑
a∈a
ea
(content modulo d)
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Positive characteristic
fa is constructed using the qa-eigenspace of a Jucys-Murphy element
Now assume char k = ℓ > 0, ℓ ∤ q and set
d = order of q ∈ k×
Identification of qa and qa+d eigenspaces for a ∈ Z/dZ, set
fa =∑
a∈a
fa and ea =∑
a∈a
ea
(content modulo d)
Through the decomposition map Uch(GLn(q)) −→ K0(kGLn(q)-modu),the action of 〈ea, fa〉a∈Z/dZ induces an action of sld on
K0(C) =⊕
n≥0
K0(kGLn(q)-modu)
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Categorical action
Action of sld on V = C⊗Z K0(C)
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
◮ Sd acts on V and two blocks are in the same orbit iff they have thesame defect.
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
◮ Sd acts on V and two blocks are in the same orbit iff they have thesame defect.
Action on C:
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
◮ Sd acts on V and two blocks are in the same orbit iff they have thesame defect.
Action on C:
◮ adjoint pair of exact functors (Ea,Fa) with [Ea] = ea and [Fa] = fa
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
◮ Sd acts on V and two blocks are in the same orbit iff they have thesame defect.
Action on C:
◮ adjoint pair of exact functors (Ea,Fa) with [Ea] = ea and [Fa] = fa
◮ weight decomposition C ≃⊕Cλ with C⊗ K0(Cλ) = Vλ
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
◮ Sd acts on V and two blocks are in the same orbit iff they have thesame defect.
Action on C:
◮ adjoint pair of exact functors (Ea,Fa) with [Ea] = ea and [Fa] = fa
◮ weight decomposition C ≃⊕Cλ with C⊗ K0(Cλ) = Vλ
◮ extra structure on End Fma
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
◮ Sd acts on V and two blocks are in the same orbit iff they have thesame defect.
Action on C:
◮ adjoint pair of exact functors (Ea,Fa) with [Ea] = ea and [Fa] = fa
◮ weight decomposition C ≃⊕Cλ with C⊗ K0(Cλ) = Vλ
◮ extra structure on End Fma ← very important!
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Categorical action
Action of sld on V = C⊗Z K0(C)
Lascoux-Leclerc-Thibon
◮ The weight space decomposition V =⊕
Vλ coincide with the blockdecomposition
◮ Sd acts on V and two blocks are in the same orbit iff they have thesame defect.
Action on C:
◮ adjoint pair of exact functors (Ea,Fa) with [Ea] = ea and [Fa] = fa
◮ weight decomposition C ≃⊕Cλ with C⊗ K0(Cλ) = Vλ
◮ extra structure on End Fma ← very important!
Chuang-Rouquier
Elements of Sd lift to derived equivalences of C. In particular, two blockswith same defect are derived equivalent.
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-mod
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
◮ X ∈ End(F ) (Jucys-Murphy element)
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
◮ X ∈ End(F ) (Jucys-Murphy element)
◮ T ∈ End(F 2)
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
◮ X ∈ End(F ) (Jucys-Murphy element)
◮ T ∈ End(F 2)
satisfying affine Hecke relations
• (T − qIdF 2) ◦ (T + IdF 2) = 0
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
◮ X ∈ End(F ) (Jucys-Murphy element)
◮ T ∈ End(F 2)
satisfying affine Hecke relations
• (T − qIdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
◮ X ∈ End(F ) (Jucys-Murphy element)
◮ T ∈ End(F 2)
satisfying affine Hecke relations
• (T − qIdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = qX IdF
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
◮ X ∈ End(F ) (Jucys-Murphy element)
◮ T ∈ End(F 2)
satisfying affine Hecke relations
• (T − qIdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = qX IdF
algebra homomorphism Hq(Am−1) −→ EndFm
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Structure of End Fm
Induction and restriction come from an adjoint pair of exact functors
kGLn(q)-mod kGLn+1(q)-modF
E
The structure of categorical action comes from natural transformations
◮ X ∈ End(F ) (Jucys-Murphy element)
◮ T ∈ End(F 2)
satisfying affine Hecke relations
• (T − qIdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = qX IdF
algebra homomorphism Hq(Am−1) −→ EndFm
Fa := qa-eigenspace of X on F
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Decategorifying
Draw the quiver with
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
For parabolic induction and restriction we get
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
For parabolic induction and restriction we get
in char 0
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
For parabolic induction and restriction we get
in char 0 · · · q−2 q−1 1 q q2 · · ·
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
For parabolic induction and restriction we get
in char 0 · · · q−2 q−1 1 q q2 · · · sl∞
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
For parabolic induction and restriction we get
in char 0 · · · q−2 q−1 1 q q2 · · · sl∞
in char ℓ
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
For parabolic induction and restriction we get
in char 0 · · · q−2 q−1 1 q q2 · · · sl∞
⇐
in char ℓ 1 = qd
q−1 q
q−2 q2
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Decategorifying
Draw the quiver with
◮ vertices: eigenvalues of X
◮ edges: a −→ qa
the Kac-Moody algebra associated to this quiver acts on K0(C)
For parabolic induction and restriction we get
in char 0 · · · q−2 q−1 1 q q2 · · · sl∞
⇐
in char ℓ 1 = qd
q−1 q
q−2 q2
sld
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Decategorifying
C ∈ Uch(GL0(q)) is the only cuspidal character (i.e. such that E (χ) = 0)
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Decategorifying
C ∈ Uch(GL0(q)) is the only cuspidal character (i.e. such that E (χ) = 0)
One recovers the standard construction of the Hecke algebra of type A as
the endomorphism algebra of the induced representation IndGLm(q)B C
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Decategorifying
C ∈ Uch(GL0(q)) is the only cuspidal character (i.e. such that E (χ) = 0)
One recovers the standard construction of the Hecke algebra of type A as
the endomorphism algebra of the induced representation IndGLm(q)B C
Hq(Am−1) = 〈X1, . . . ,Xm,T1, . . . ,Tm−1〉 EndFm
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Decategorifying
C ∈ Uch(GL0(q)) is the only cuspidal character (i.e. such that E (χ) = 0)
One recovers the standard construction of the Hecke algebra of type A as
the endomorphism algebra of the induced representation IndGLm(q)B C
Hq(Am−1) = 〈X1, . . . ,Xm,T1, . . . ,Tm−1〉 EndFm
EndGLm(q) (FmC)
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Decategorifying
C ∈ Uch(GL0(q)) is the only cuspidal character (i.e. such that E (χ) = 0)
One recovers the standard construction of the Hecke algebra of type A as
the endomorphism algebra of the induced representation IndGLm(q)B C
Hq(Am−1) = 〈X1, . . . ,Xm,T1, . . . ,Tm−1〉 EndFm
EndGLm(q) (FmC)
EndGLm(q) (IndGLm(q)B C)
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Decategorifying
C ∈ Uch(GL0(q)) is the only cuspidal character (i.e. such that E (χ) = 0)
One recovers the standard construction of the Hecke algebra of type A as
the endomorphism algebra of the induced representation IndGLm(q)B C
Hq(Am−1) = 〈X1, . . . ,Xm,T1, . . . ,Tm−1〉 EndFm
Hq(Am−1) = 〈T1, . . . ,Tm−1〉 EndGLm(q) (FmC)
EndGLm(q) (IndGLm(q)B C)
∼
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Decategorifying
C ∈ Uch(GL0(q)) is the only cuspidal character (i.e. such that E (χ) = 0)
One recovers the standard construction of the Hecke algebra of type A as
the endomorphism algebra of the induced representation IndGLm(q)B C
Hq(Am−1) = 〈X1, . . . ,Xm,T1, . . . ,Tm−1〉 EndFm
Hq(Am−1) = 〈T1, . . . ,Tm−1〉 EndGLm(q) (FmC)
EndGLm(q) (IndGLm(q)B C)
∼
X1=q
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The finite unitary groups
Underlying algebraic group GLn(Fq)
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The finite unitary groups
Underlying algebraic group GLn(Fq)
GUn(q) := {M ∈ GLn(Fq) |MtFr(M) = In} ⊂ GLn(q
2)
where Fr : (mi ,j) 7−→ (mqi ,j)
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The finite unitary groups
Underlying algebraic group GLn(Fq)
GUn(q) := {M ∈ GLn(Fq) |MtFr(M) = In} ⊂ GLn(q
2)
where Fr : (mi ,j) 7−→ (mqi ,j)
Parabolic induction and restriction yield an adjoint pair of exact functors
kGUn(q)-mod kGUn+2(q)-modF
E
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The finite unitary groups
Underlying algebraic group GLn(Fq)
GUn(q) := {M ∈ GLn(Fq) |MtFr(M) = In} ⊂ GLn(q
2)
where Fr : (mi ,j) 7−→ (mqi ,j)
Parabolic induction and restriction yield an adjoint pair of exact functors
kGUn(q)-mod kGUn+2(q)-modF
E
“GUn(q) = GLn(−q)”
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The finite unitary groups
Underlying algebraic group GLn(Fq)
GUn(q) := {M ∈ GLn(Fq) |MtFr(M) = In} ⊂ GLn(q
2)
where Fr : (mi ,j) 7−→ (mqi ,j)
Parabolic induction and restriction yield an adjoint pair of exact functors
kGUn(q)-mod kGUn+2(q)-modF
E
“GUn(q) = GLn(−q)”
In particular, unipotent characters are still labelled by partitions of n
Uch(GUn(q)) ←→ {Partitions of n}χµ ←− [ µ
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The finite unitary groups
Underlying algebraic group GLn(Fq)
GUn(q) := {M ∈ GLn(Fq) |MtFr(M) = In} ⊂ GLn(q
2)
where Fr : (mi ,j) 7−→ (mqi ,j)
Parabolic induction and restriction yield an adjoint pair of exact functors
kGUn(q)-mod kGUn+2(q)-modF
E
“GUn(q) = GLn(−q)”
In particular, unipotent characters are still labelled by partitions of n
Uch(GUn(q)) ←→ {Partitions of n}χµ ←− [ µ
But branching rules are different!O. Dudas (Paris 7) Categorical actions Mar. 2015 10 / 1
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
f =
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
f =
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
f = +
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
f = + +
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
f = + + +
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
f
0 1 2 3-1 0-2 -1-3
=
0 1 2 3 4 5-1 0-2 -1-3
+
0 1 2 3-1 0 1 2-2 -1-3
+
0 1 2 3-1 0 1-2 -1 0-3
+
0 1 2 3-1 0-2 -1-3-4-5
Again, one can split f =∑
fa using contents
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Induction and restriction for GUn(q)
e = [E ] and f = [F ] act by removing/adding 2-hooks or on
unipotent characters
f
0 1 2 3-1 0-2 -1-3
=
0 1 2 3 4 5-1 0-2 -1-3
+
0 1 2 3-1 0 1 2-2 -1-3
+
0 1 2 3-1 0 1-2 -1 0-3
+
0 1 2 3-1 0-2 -1-3-4-5
Again, one can split f =∑
fa using contents
Up to a sign normalization,
fa = [f(GL)a , f
(GL)−qa ]
and 〈ea, fa〉a∈Z induce an action of sl⊕2∞ on
⊕CUch(GUn(q)) isomorphic
to a Fock space representation of level 2.
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Positive characteristic
fa is constructed using the (−q)a-eigenspace of a Jucys-Murphy element
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Positive characteristic
fa is constructed using the (−q)a-eigenspace of a Jucys-Murphy element
We setd = order of −q ∈ k×
and define fa =∑
a∈a fa and ea =∑
a∈a ea as before, for a ∈ Z/dZ
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Positive characteristic
fa is constructed using the (−q)a-eigenspace of a Jucys-Murphy element
We setd = order of −q ∈ k×
and define fa =∑
a∈a fa and ea =∑
a∈a ea as before, for a ∈ Z/dZ
Through the decomposition map Uch(GUn(q)) −→ K0(kGUn(q)-modu),the action of 〈ea, fa〉a∈Z/dZ induces an action of
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Positive characteristic
fa is constructed using the (−q)a-eigenspace of a Jucys-Murphy element
We setd = order of −q ∈ k×
and define fa =∑
a∈a fa and ea =∑
a∈a ea as before, for a ∈ Z/dZ
Through the decomposition map Uch(GUn(q)) −→ K0(kGUn(q)-modu),the action of 〈ea, fa〉a∈Z/dZ induces an action of
• sld if d is odd (unitary prime case)
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Positive characteristic
fa is constructed using the (−q)a-eigenspace of a Jucys-Murphy element
We setd = order of −q ∈ k×
and define fa =∑
a∈a fa and ea =∑
a∈a ea as before, for a ∈ Z/dZ
Through the decomposition map Uch(GUn(q)) −→ K0(kGUn(q)-modu),the action of 〈ea, fa〉a∈Z/dZ induces an action of
• sld if d is odd (unitary prime case)
• sld/2 ⊕ sld/2 if d is even (linear prime case)
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Positive characteristic
fa is constructed using the (−q)a-eigenspace of a Jucys-Murphy element
We setd = order of −q ∈ k×
and define fa =∑
a∈a fa and ea =∑
a∈a ea as before, for a ∈ Z/dZ
Through the decomposition map Uch(GUn(q)) −→ K0(kGUn(q)-modu),the action of 〈ea, fa〉a∈Z/dZ induces an action of
• sld if d is odd (unitary prime case)
• sld/2 ⊕ sld/2 if d is even (linear prime case)
onK0(C) =
⊕
n≥0
K0(kGUn(q)-modu)
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Positive characteristic
fa is constructed using the (−q)a-eigenspace of a Jucys-Murphy element
We setd = order of −q ∈ k×
and define fa =∑
a∈a fa and ea =∑
a∈a ea as before, for a ∈ Z/dZ
Through the decomposition map Uch(GUn(q)) −→ K0(kGUn(q)-modu),the action of 〈ea, fa〉a∈Z/dZ induces an action of
• sld if d is odd (unitary prime case)
• sld/2 ⊕ sld/2 if d is even (linear prime case)
onK0(C) =
⊕
n≥0
K0(kGUn(q)-modu)
Moreover, weight spaces coincide with blocks.
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Categorical action
As in the case of GLn(q), the action lifts to a categorical action on C
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Categorical action
As in the case of GLn(q), the action lifts to a categorical action on C
Theorem [DSVV]
There exist X ∈ End(F ) and T ∈ End(F 2) such that
• (T − q2IdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = q2X IdF
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Categorical action
As in the case of GLn(q), the action lifts to a categorical action on C
Theorem [DSVV]
There exist X ∈ End(F ) and T ∈ End(F 2) such that
• (T − q2IdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = q2X IdF
Moreover, the eigenvalues of X are powers of −q.
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Categorical action
As in the case of GLn(q), the action lifts to a categorical action on C
Theorem [DSVV]
There exist X ∈ End(F ) and T ∈ End(F 2) such that
• (T − q2IdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = q2X IdF
Moreover, the eigenvalues of X are powers of −q.
algebra homomorphism Hq2(Am−1) −→ EndFm
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Categorical action
As in the case of GLn(q), the action lifts to a categorical action on C
Theorem [DSVV]
There exist X ∈ End(F ) and T ∈ End(F 2) such that
• (T − q2IdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = q2X IdF
Moreover, the eigenvalues of X are powers of −q.
algebra homomorphism Hq2(Am−1) −→ EndFm
Fa := (−q)a-eigenspace of X on F
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Categorical action
As in the case of GLn(q), the action lifts to a categorical action on C
Theorem [DSVV]
There exist X ∈ End(F ) and T ∈ End(F 2) such that
• (T − q2IdF 2) ◦ (T + IdF 2) = 0
• (T IdF ) ◦ (IdFT ) ◦ (T IdF ) = (IdFT ) ◦ (T IdF ) ◦ (IdFT )
• T ◦ (IdFX ) ◦ T = q2X IdF
Moreover, the eigenvalues of X are powers of −q.
algebra homomorphism Hq2(Am−1) −→ EndFm
Fa := (−q)a-eigenspace of X on F
+ similar results for classical types B , C , D and 2D
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Decategorifying
χµ ∈ Uch(GUn(q)) is a cuspidal character
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Decategorifying
χµ ∈ Uch(GUn(q)) is a cuspidal character
⇐⇒ E (χµ) = 0 (highest weight vector)
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Decategorifying
χµ ∈ Uch(GUn(q)) is a cuspidal character
⇐⇒ E (χµ) = 0 (highest weight vector)
⇐⇒ one cannot remove any 2-hook from µ
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Decategorifying
χµ ∈ Uch(GUn(q)) is a cuspidal character
⇐⇒ E (χµ) = 0 (highest weight vector)
⇐⇒ one cannot remove any 2-hook from µ
⇐⇒ µ = (t, t − 1, t − 2, . . . , 1) is a triangular partition
(in particular n = t(t + 1)/2)
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Decategorifying
χµ ∈ Uch(GUn(q)) is a cuspidal character
⇐⇒ E (χµ) = 0 (highest weight vector)
⇐⇒ one cannot remove any 2-hook from µ
⇐⇒ µ = (t, t − 1, t − 2, . . . , 1) is a triangular partition
(in particular n = t(t + 1)/2)
Given t ≥ 0 and µ = (t, t − 1, t − 2, . . . , 1) one recovers that theendomorphism algebra of Fmχµ has type Bm and parameters (q2t+1, q2)
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Decategorifying
χµ ∈ Uch(GUn(q)) is a cuspidal character
⇐⇒ E (χµ) = 0 (highest weight vector)
⇐⇒ one cannot remove any 2-hook from µ
⇐⇒ µ = (t, t − 1, t − 2, . . . , 1) is a triangular partition
(in particular n = t(t + 1)/2)
Given t ≥ 0 and µ = (t, t − 1, t − 2, . . . , 1) one recovers that theendomorphism algebra of Fmχµ has type Bm and parameters (q2t+1, q2)
Hq2(Am−1) = 〈X1, . . . ,Xm,T1, . . . ,Tm−1〉 EndFm
Hq2t+1,q2(Bm) = 〈X1,T1, . . . ,Tm−1〉 EndGUn+m(q) (Fmχµ)
∼
(X1−(−q)t )(X1−(−q)−t−1)=0
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Applications to Broue’s conjecture
Given a finite group Γ and a block b with abelian defect group D, Broue’sconjecture predicts the existence of a derived equivalence between
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Applications to Broue’s conjecture
Given a finite group Γ and a block b with abelian defect group D, Broue’sconjecture predicts the existence of a derived equivalence between
◮ the block b (global, a block of Γ)
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Applications to Broue’s conjecture
Given a finite group Γ and a block b with abelian defect group D, Broue’sconjecture predicts the existence of a derived equivalence between
◮ the block b (global, a block of Γ)
◮ the Brauer correspondent of b (local, a block of NΓ(D))
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Applications to Broue’s conjecture
Given a finite group Γ and a block b with abelian defect group D, Broue’sconjecture predicts the existence of a derived equivalence between
◮ the block b (global, a block of Γ)
◮ the Brauer correspondent of b (local, a block of NΓ(D))
Categorical actions provide derived equivalences between blocks: “global↔ global”.
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Applications to Broue’s conjecture
Given a finite group Γ and a block b with abelian defect group D, Broue’sconjecture predicts the existence of a derived equivalence between
◮ the block b (global, a block of Γ)
◮ the Brauer correspondent of b (local, a block of NΓ(D))
Categorical actions provide derived equivalences between blocks: “global↔ global”.To go to the local block one needs good blocks.
Theorem [Livesey, 12]
Assume d is even. There is a family of good blocks of GUn(q) for whichBroue’s conjecture holds.
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Applications to Broue’s conjecture
Given a finite group Γ and a block b with abelian defect group D, Broue’sconjecture predicts the existence of a derived equivalence between
◮ the block b (global, a block of Γ)
◮ the Brauer correspondent of b (local, a block of NΓ(D))
Categorical actions provide derived equivalences between blocks: “global↔ global”.To go to the local block one needs good blocks.
Theorem [Livesey, 12]
Assume d is even. There is a family of good blocks of GUn(q) for whichBroue’s conjecture holds.
Theorem [DSVV]
Assume d is even. Each orbit of the action of the Weyl group on Ccontains a good block. Consequently, Broue’s conjecture holds in thatcase.
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Applications to Broue’s conjecture
Given a finite group Γ and a block b with abelian defect group D, Broue’sconjecture predicts the existence of a derived equivalence between
◮ the block b (global, a block of Γ)
◮ the Brauer correspondent of b (local, a block of NΓ(D))
Categorical actions provide derived equivalences between blocks: “global↔ global”.To go to the local block one needs good blocks.
Theorem [Livesey, 12]
Assume d is even. There is a family of good blocks of GUn(q) for whichBroue’s conjecture holds.
Theorem [DSVV]
Assume d is even. Each orbit of the action of the Weyl group on Ccontains a good block. Consequently, Broue’s conjecture holds in thatcase.
+ work in progress for type B and C (d odd)O. Dudas (Paris 7) Categorical actions Mar. 2015 15 / 1
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Applications to the branching graph
As a module for the Lie algebra
C⊗ K0(C) ≃⊕
t≥0
F((−q)t , (−q)−1−t)
O. Dudas (Paris 7) Categorical actions Mar. 2015 16 / 1
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Applications to the branching graph
As a module for the Lie algebra
C⊗ K0(C) ≃⊕
t≥0
F((−q)t , (−q)−1−t)
where F(q1, q2) is a level-2 Fock space representation corresponding tothe weight (q1, q2).
O. Dudas (Paris 7) Categorical actions Mar. 2015 16 / 1
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Applications to the branching graph
As a module for the Lie algebra
C⊗ K0(C) ≃⊕
t≥0
F((−q)t , (−q)−1−t)
where F(q1, q2) is a level-2 Fock space representation corresponding tothe weight (q1, q2).
Modular branching graph: given a simple module S , Fa(S) is either zero orhas a simple quotient
O. Dudas (Paris 7) Categorical actions Mar. 2015 16 / 1
![Page 110: Categorical actions on unipotent representations of finite classical groups · 2015-03-27 · Motivation Finite classical groups G n(q) = GL n(q), GU n(q), Sp2 (q)... Study representations](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7017c666484f573b29e69b/html5/thumbnails/110.jpg)
Applications to the branching graph
As a module for the Lie algebra
C⊗ K0(C) ≃⊕
t≥0
F((−q)t , (−q)−1−t)
where F(q1, q2) is a level-2 Fock space representation corresponding tothe weight (q1, q2).
Modular branching graph: given a simple module S , Fa(S) is either zero orhas a simple quotient, which one?
O. Dudas (Paris 7) Categorical actions Mar. 2015 16 / 1
![Page 111: Categorical actions on unipotent representations of finite classical groups · 2015-03-27 · Motivation Finite classical groups G n(q) = GL n(q), GU n(q), Sp2 (q)... Study representations](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7017c666484f573b29e69b/html5/thumbnails/111.jpg)
Applications to the branching graph
As a module for the Lie algebra
C⊗ K0(C) ≃⊕
t≥0
F((−q)t , (−q)−1−t)
where F(q1, q2) is a level-2 Fock space representation corresponding tothe weight (q1, q2).
Modular branching graph: given a simple module S , Fa(S) is either zero orhas a simple quotient, which one?
Theorem [DSVV]
The modular branching graph coincide with the disjoint union of thecrystal graphs of the Fock spaces F((−q)t , (−q)−1−t).
O. Dudas (Paris 7) Categorical actions Mar. 2015 16 / 1
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Applications to the branching graph
As a module for the Lie algebra
C⊗ K0(C) ≃⊕
t≥0
F((−q)t , (−q)−1−t)
where F(q1, q2) is a level-2 Fock space representation corresponding tothe weight (q1, q2).
Modular branching graph: given a simple module S , Fa(S) is either zero orhas a simple quotient, which one?
Theorem [DSVV]
The modular branching graph coincide with the disjoint union of thecrystal graphs of the Fock spaces F((−q)t , (−q)−1−t).
This is a weak version of a conjecture of Gerber-Hiss-Jacon.
O. Dudas (Paris 7) Categorical actions Mar. 2015 16 / 1
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Applications to the branching graph
As a module for the Lie algebra
C⊗ K0(C) ≃⊕
t≥0
F((−q)t , (−q)−1−t)
where F(q1, q2) is a level-2 Fock space representation corresponding tothe weight (q1, q2).
Modular branching graph: given a simple module S , Fa(S) is either zero orhas a simple quotient, which one?
Theorem [DSVV]
The modular branching graph coincide with the disjoint union of thecrystal graphs of the Fock spaces F((−q)t , (−q)−1−t).
This is a weak version of a conjecture of Gerber-Hiss-Jacon.The strongversion predicts a isomorphism of graphs inducing the identity onpartitions (labelling the vertices of the two graphs).
O. Dudas (Paris 7) Categorical actions Mar. 2015 16 / 1
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Decomposition numbers
For GLn(q), and for GUn(q) with d even, the basis of simple modules inK0(C) corresponds to the (dual) canonical basis of the Fock spaces.
O. Dudas (Paris 7) Categorical actions Mar. 2015 17 / 1
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Decomposition numbers
For GLn(q), and for GUn(q) with d even, the basis of simple modules inK0(C) corresponds to the (dual) canonical basis of the Fock spaces.
Question
Is there any Lie-theoretic interpretation of the basis of simple moduleswhen d is odd?
O. Dudas (Paris 7) Categorical actions Mar. 2015 17 / 1