Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
Combinatorial Representation Theory –Old and New
Georgia BenkartUniversity of Wisconsin-Madison
Combinatorial Representation Theory – Old and New – p.1/29
Group Representations
Combinatorial Representation Theory – Old and New – p.2/29
Group Representations
Ferdinand Georg Frobenius
Combinatorial Representation Theory – Old and New – p.2/29
Group Representations
Ferdinand Georg Frobenius
group representation (1897):
Combinatorial Representation Theory – Old and New – p.2/29
Group Representations
Ferdinand Georg Frobenius
group representation (1897):
ϕ : G→ GL(V ) (invertible transformations on V )
Combinatorial Representation Theory – Old and New – p.2/29
Group Representations
Ferdinand Georg Frobenius
group representation (1897):
ϕ : G→ GL(V ) (invertible transformations on V )
ϕ(gh) = ϕ(g)ϕ(h)
Combinatorial Representation Theory – Old and New – p.2/29
Group Representations
Ferdinand Georg Frobenius
group representation (1897):
ϕ : G→ GL(V ) (invertible transformations on V )
ϕ(gh) = ϕ(g)ϕ(h)
g.v = ϕ(g)(v) makes V a G-module
Combinatorial Representation Theory – Old and New – p.2/29
Group Representations
Ferdinand Georg Frobenius
group representation (1897):
ϕ : G→ GL(V ) (invertible transformations on V )
ϕ(gh) = ϕ(g)ϕ(h)
g.v = ϕ(g)(v) makes V a G-module
Irreducible repns. of 1−1←→ λ ` k
symmetric group Sk over C partitions of k
(1900)
Combinatorial Representation Theory – Old and New – p.2/29
Representations ofGLn
Combinatorial Representation Theory – Old and New – p.3/29
Representations ofGLn
Isaai Schur (1901)
Combinatorial Representation Theory – Old and New – p.3/29
Representations ofGLn
Isaai Schur (1901)
GLn acts on V = Cn via matrix multiplication g.v
Combinatorial Representation Theory – Old and New – p.3/29
Representations ofGLn
Isaai Schur (1901)
GLn acts on V = Cn via matrix multiplication g.v
GLn acts on V ⊗k viag.(v1 ⊗ · · · ⊗ vk) = g.v1 ⊗ · · · ⊗ g.vk
Combinatorial Representation Theory – Old and New – p.3/29
Representations ofGLn
Isaai Schur (1901)
GLn acts on V = Cn via matrix multiplication g.v
GLn acts on V ⊗k viag.(v1 ⊗ · · · ⊗ vk) = g.v1 ⊗ · · · ⊗ g.vk
Sk acts on V ⊗k via place permutations
Combinatorial Representation Theory – Old and New – p.3/29
Representations ofGLn
Isaai Schur (1901)
GLn acts on V = Cn via matrix multiplication g.v
GLn acts on V ⊗k viag.(v1 ⊗ · · · ⊗ vk) = g.v1 ⊗ · · · ⊗ g.vk
Sk acts on V ⊗k via place permutations
The two actions commute.
Combinatorial Representation Theory – Old and New – p.3/29
Representations ofGLn
Isaai Schur (1901)
GLn acts on V = Cn via matrix multiplication g.v
GLn acts on V ⊗k viag.(v1 ⊗ · · · ⊗ vk) = g.v1 ⊗ · · · ⊗ g.vk
Sk acts on V ⊗k via place permutations
The two actions commute.
Use Sk -repns. to study GLn-repns.
Combinatorial Representation Theory – Old and New – p.3/29
Dawn of Modern Age of Repn. Theory
Combinatorial Representation Theory – Old and New – p.4/29
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Combinatorial Representation Theory – Old and New – p.4/29
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Repns. of G over F⇐⇒ Repns. of group algebra FG
Combinatorial Representation Theory – Old and New – p.4/29
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Repns. of G over F⇐⇒ Repns. of group algebra FG
FG ∼= direct sum of matrix blocks (char(F) = 0)
Combinatorial Representation Theory – Old and New – p.4/29
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Repns. of G over F⇐⇒ Repns. of group algebra FG
FG ∼= direct sum of matrix blocks (char(F) = 0)
FG/rad(FG) ∼= direct sum of matrix blocks (char(F) = p)
Combinatorial Representation Theory – Old and New – p.4/29
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Repns. of G over F⇐⇒ Repns. of group algebra FG
FG ∼= direct sum of matrix blocks (char(F) = 0)
FG/rad(FG) ∼= direct sum of matrix blocks (char(F) = p)
One block for each irreducible repn. of G
Combinatorial Representation Theory – Old and New – p.4/29
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Repns. of G over F⇐⇒ Repns. of group algebra FG
FG ∼= direct sum of matrix blocks (char(F) = 0)
FG/rad(FG) ∼= direct sum of matrix blocks (char(F) = p)
One block for each irreducible repn. of G
dim. matrix block = (dim. of the irred. repn)2
Combinatorial Representation Theory – Old and New – p.4/29
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Repns. of G over F⇐⇒ Repns. of group algebra FG
FG ∼= direct sum of matrix blocks (char(F) = 0)
FG/rad(FG) ∼= direct sum of matrix blocks (char(F) = p)
One block for each irreducible repn. of G
dim. matrix block = (dim. of the irred. repn)2
Ex. FSk∼=
⊕λ`k Mλ (char(F) = 0)
Combinatorial Representation Theory – Old and New – p.4/29
Schur-Weyl Duality
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k)
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
EndGLn(V ⊗k) ∼= FSk/ ker ΦS
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
EndGLn(V ⊗k) ∼= FSk/ ker ΦS
EndSk(V⊗k) ∼= FGLn/ ker ΦG
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
EndGLn(V ⊗k) ∼= FSk/ ker ΦS
EndSk(V⊗k) ∼= FGLn/ ker ΦG =: SF(n, k)
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
EndGLn(V ⊗k) ∼= FSk/ ker ΦS
EndSk(V⊗k) ∼= FGLn/ ker ΦG =: SF(n, k) Schur algebra
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
EndGLn(V ⊗k) ∼= FSk/ ker ΦS
EndSk(V⊗k) ∼= FGLn/ ker ΦG =: SF(n, k) Schur algebra
(Polynomial) repns. of GLn over F⇐⇒ repns. of SF(n, k)
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
EndGLn(V ⊗k) ∼= FSk/ ker ΦS
EndSk(V⊗k) ∼= FGLn/ ker ΦG =: SF(n, k) Schur algebra
(Polynomial) repns. of GLn over F⇐⇒ repns. of SF(n, k)
k = 1, 2, . . .
Combinatorial Representation Theory – Old and New – p.5/29
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
(Here F can be any infinite field S. Doty ’06)
EndGLn(V ⊗k) ∼= FSk/ ker ΦS
∼= FSk/⊕
λ`k
#parts>n
Mλ (char(F) = 0)
EndSk(V⊗k) ∼= FGLn/ ker ΦG =: SF(n, k) Schur algebra
(Polynomial) repns. of GLn over F⇐⇒ repns. of SF(n, k)
k = 1, 2, . . .
Combinatorial Representation Theory – Old and New – p.5/29
Schur Algebras
Combinatorial Representation Theory – Old and New – p.6/29
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Combinatorial Representation Theory – Old and New – p.6/29
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Thm. (S. Doty & D. Nakano ’98)
Let F be algebraically closed.
Combinatorial Representation Theory – Old and New – p.6/29
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Thm. (S. Doty & D. Nakano ’98)
Let F be algebraically closed. SF(n, k) is semisimple iff
Combinatorial Representation Theory – Old and New – p.6/29
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Thm. (S. Doty & D. Nakano ’98)
Let F be algebraically closed. SF(n, k) is semisimple iff
(1) char(F) = 0
Combinatorial Representation Theory – Old and New – p.6/29
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Thm. (S. Doty & D. Nakano ’98)
Let F be algebraically closed. SF(n, k) is semisimple iff
(1) char(F) = 0
(2) char(F) = p > k
Combinatorial Representation Theory – Old and New – p.6/29
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Thm. (S. Doty & D. Nakano ’98)
Let F be algebraically closed. SF(n, k) is semisimple iff
(1) char(F) = 0
(2) char(F) = p > k
(3) char(F) = 2, n = 2, k = 3
Combinatorial Representation Theory – Old and New – p.6/29
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Thm. (S. Doty & D. Nakano ’98)
Let F be algebraically closed. SF(n, k) is semisimple iff
(1) char(F) = 0
(2) char(F) = p > k
(3) char(F) = 2, n = 2, k = 3
K. Erdmann (’93): Determined when SF(n, k) has finitely manyindecomposable modules.
Combinatorial Representation Theory – Old and New – p.6/29
Resulting Connections
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
(2) Cohomology Connections
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
(2) Cohomology Connections
ExtiSF(n,k)(M, N)
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
(2) Cohomology Connections
ExtiSF(n,k)(M, N) ∼=
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
(2) Cohomology Connections
ExtiSF(n,k)(M, N) ∼= Exti
FSk(F(M), F(N))
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
(2) Cohomology Connections
ExtiSF(n,k)(M, N) ∼= Exti
FSk(F(M), F(N))
SOMETIMES!
Combinatorial Representation Theory – Old and New – p.7/29
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
(2) Cohomology Connections
ExtiSF(n,k)(M, N) ∼= Exti
FSk(F(M), F(N))
SOMETIMES! – see Kleshchev & Nakano ’01
Combinatorial Representation Theory – Old and New – p.7/29
Partitions & Their Ups and Downs
Combinatorial Representation Theory – Old and New – p.8/29
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
Going Up and Down
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
On lattice of partitions: du− ud = I
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
On lattice of partitions: du− ud = I (Weyl alg. relation)
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
On lattice of partitions: du− ud = I (Weyl alg. relation)
Irreducible repns. for Sk1−1←→ λ ` k
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
On lattice of partitions: du− ud = I (Weyl alg. relation)
Irreducible repns. for Sk1−1←→ λ ` k
resSk
Sk−1λ =
∑
κ⊂λ|λ/κ|=1
κ
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
On lattice of partitions: du− ud = I (Weyl alg. relation)
Irreducible repns. for Sk1−1←→ λ ` k
resSk
Sk−1λ =
∑
κ⊂λ|λ/κ|=1
κ = d(λ)
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
On lattice of partitions: du− ud = I (Weyl alg. relation)
Irreducible repns. for Sk1−1←→ λ ` k
resSk
Sk−1λ =
∑
κ⊂λ|λ/κ|=1
κ = d(λ)
indSk+1
Skλ =
∑
ν⊃λ|ν/λ|=1
ν
Combinatorial Representation Theory – Old and New – p.9/29
Going Up and Down
R. Stanley ’88, S. Fomin ’94
On lattice of partitions: du− ud = I (Weyl alg. relation)
Irreducible repns. for Sk1−1←→ λ ` k
resSk
Sk−1λ =
∑
κ⊂λ|λ/κ|=1
κ = d(λ)
indSk+1
Skλ =
∑
ν⊃λ|ν/λ|=1
ν = u(λ)
Combinatorial Representation Theory – Old and New – p.9/29
Up and Down ? Paths
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ →
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → →
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → →
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → →
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → →
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 2
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 23
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 23 54
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 23 54
<
∧
standard tableau
Combinatorial Representation Theory – Old and New – p.10/29
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 23 54
<
∧
standard tableau
fλ: no. of standard tableaux of shape λ
= no. of paths up to λ
Combinatorial Representation Theory – Old and New – p.10/29
Counting Up and Down Paths
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
∑λ`k f2
λ = coefficient of ∅ in dkuk∅
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
∑λ`k f2
λ = coefficient of ∅ in dkuk∅
To compute this use: duk = ukd + kuk−1
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
∑λ`k f2
λ = coefficient of ∅ in dkuk∅
To compute this use: duk = ukd + kuk−1
dkuk∅ = dk−1(duk)∅
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
∑λ`k f2
λ = coefficient of ∅ in dkuk∅
To compute this use: duk = ukd + kuk−1
dkuk∅ = dk−1(duk)∅
= dk−1(ukd + kuk−1)∅
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
∑λ`k f2
λ = coefficient of ∅ in dkuk∅
To compute this use: duk = ukd + kuk−1
dkuk∅ = dk−1(duk)∅
= dk−1(ukd + kuk−1)∅
= k dk−1uk−1∅
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
∑λ`k f2
λ = coefficient of ∅ in dkuk∅
To compute this use: duk = ukd + kuk−1
dkuk∅ = dk−1(duk)∅
= dk−1(ukd + kuk−1)∅
= k dk−1uk−1∅
= k(k − 1) dk−2uk−2∅ = · · ·
Combinatorial Representation Theory – Old and New – p.11/29
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
∑λ`k f2
λ = coefficient of ∅ in dkuk∅
To compute this use: duk = ukd + kuk−1
dkuk∅ = dk−1(duk)∅
= dk−1(ukd + kuk−1)∅
= k dk−1uk−1∅
= k(k − 1) dk−2uk−2∅ = · · ·
= (k!)∅
Combinatorial Representation Theory – Old and New – p.11/29
Therefore:
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ and take dimensions
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ and take dimensions
k! =∑
λ`k
dim Mλ =
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ and take dimensions
k! =∑
λ`k
dim Mλ =
∑
λ`k
(dim λ)2
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ and take dimensions
k! =∑
λ`k
dim Mλ =
∑
λ`k
(dim λ)2 =∑
λ`k
f2λ
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ and take dimensions
k! =∑
λ`k
dim Mλ =
∑
λ`k
(dim λ)2 =∑
λ`k
f2λ
Problem: Determine all posets for which du− ud = rI.
Combinatorial Representation Theory – Old and New – p.12/29
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ and take dimensions
k! =∑
λ`k
dim Mλ =
∑
λ`k
(dim λ)2 =∑
λ`k
f2λ
Problem: Determine all posets for which du− ud = rI.
Connected with determining all combinatorial Hopf algs.(Bergeron-Lam-Li ’07)
Combinatorial Representation Theory – Old and New – p.12/29
Characteristic p
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
FSk irreds. 1−1←→ λ ` k p-regular
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
FSk irreds. 1−1←→ λ ` k p-regular
(no part repeated p or more times)
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
FSk irreds. 1−1←→ λ ` k p-regular
(no part repeated p or more times)
FSk irreds. Dλ = Sλ/Qλ, λ ` k p-regular (James ’76)
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
FSk irreds. 1−1←→ λ ` k p-regular
(no part repeated p or more times)
FSk irreds. Dλ = Sλ/Qλ, λ ` k p-regular (James ’76)
Outstanding problems
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
FSk irreds. 1−1←→ λ ` k p-regular
(no part repeated p or more times)
FSk irreds. Dλ = Sλ/Qλ, λ ` k p-regular (James ’76)
Outstanding problems
1. Find dim Dλ
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
FSk irreds. 1−1←→ λ ` k p-regular
(no part repeated p or more times)
FSk irreds. Dλ = Sλ/Qλ, λ ` k p-regular (James ’76)
Outstanding problems
1. Find dim Dλ
2. Find [Sλ : Dν ], λ, ν ` k
Combinatorial Representation Theory – Old and New – p.13/29
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
FSk irreds. 1−1←→ λ ` k p-regular
(no part repeated p or more times)
FSk irreds. Dλ = Sλ/Qλ, λ ` k p-regular (James ’76)
Outstanding problems
1. Find dim Dλ
2. Find [Sλ : Dν ], λ, ν ` k
3. Find [resSk
Sk−1Dλ : Dµ], λ ` k, µ ` k − 1
Combinatorial Representation Theory – Old and New – p.13/29
"Revolution in Representation Theory"
Combinatorial Representation Theory – Old and New – p.14/29
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Combinatorial Representation Theory – Old and New – p.14/29
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Lattice of p-regular partitions is
Combinatorial Representation Theory – Old and New – p.14/29
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Lattice of p-regular partitions is
the crystal base of an slp-module due to Misra-Miwa (’90)
Combinatorial Representation Theory – Old and New – p.14/29
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Lattice of p-regular partitions is
the crystal base of an slp-module due to Misra-Miwa (’90)
Ariki (’96) Kleshchev (’05)
Combinatorial Representation Theory – Old and New – p.14/29
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Lattice of p-regular partitions is
the crystal base of an slp-module due to Misra-Miwa (’90)
Ariki (’96) Kleshchev (’05)
Combinatorial Representation Theory – Old and New – p.14/29
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Lattice of p-regular partitions is
the crystal base of an slp-module due to Misra-Miwa (’90)
Ariki (’96) Kleshchev (’05)
The LLT algorithm gives the decomposition numbers.
Combinatorial Representation Theory – Old and New – p.14/29
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Lattice of p-regular partitions is
the crystal base of an slp-module due to Misra-Miwa (’90)
Ariki (’96) Kleshchev (’05)
The LLT algorithm gives the decomposition numbers.
James (’80), Erdmann (’96)
Knowing decomposition nos. for Sk is equivalent to knowingdecomposition nos. for GLn
Combinatorial Representation Theory – Old and New – p.14/29
Crystal base for sl3
Combinatorial Representation Theory – Old and New – p.15/29
Crystal base for sl3...
∅Combinatorial Representation Theory – Old and New – p.15/29
Crystal base for sl3...
∅Combinatorial Representation Theory – Old and New – p.15/29
Crystal base for sl3...
∅
Fill box (i, j)
with j − i mod 3
Combinatorial Representation Theory – Old and New – p.15/29
Crystal base for sl3...
∅
Fill box (i, j)
with j − i mod 3
0
200 1
210 0 12
120 10
210
02
1 20 1 2 0
Combinatorial Representation Theory – Old and New – p.15/29
The Invention of q
Combinatorial Representation Theory – Old and New – p.16/29
The Invention of q
Drinfeld (’85) and Jimbo (’85)
Combinatorial Representation Theory – Old and New – p.16/29
The Invention of q
Drinfeld (’85) and Jimbo (’85)
Uq(gln) acts on V = C(q)n and also on V ⊗k
Combinatorial Representation Theory – Old and New – p.16/29
The Invention of q
Drinfeld (’85) and Jimbo (’85)
Uq(gln) acts on V = C(q)n and also on V ⊗k
EndUq(gln)(V⊗k) =??
Combinatorial Representation Theory – Old and New – p.16/29
The Invention of q
Drinfeld (’85) and Jimbo (’85)
Uq(gln) acts on V = C(q)n and also on V ⊗k
EndUq(gln)(V⊗k) =??
Sk has gens. si = (i i + 1), i = 1, . . . , k − 1, and relns.
Combinatorial Representation Theory – Old and New – p.16/29
The Invention of q
Drinfeld (’85) and Jimbo (’85)
Uq(gln) acts on V = C(q)n and also on V ⊗k
EndUq(gln)(V⊗k) =??
Sk has gens. si = (i i + 1), i = 1, . . . , k − 1, and relns.
sisj = sjsi if |i− j| ≥ 2
sisi+1si = si+1sisi+1
s2i = 1
Combinatorial Representation Theory – Old and New – p.16/29
Hecke Algebra
Combinatorial Representation Theory – Old and New – p.17/29
Hecke Algebra
Hk(q): Hecke algebra (q ∈ F×)
Combinatorial Representation Theory – Old and New – p.17/29
Hecke Algebra
Hk(q): Hecke algebra (q ∈ F×)
F-algebra with gens. Ti, i = 1, . . . , k − 1, and relns.
Combinatorial Representation Theory – Old and New – p.17/29
Hecke Algebra
Hk(q): Hecke algebra (q ∈ F×)
F-algebra with gens. Ti, i = 1, . . . , k − 1, and relns.
TiTj = TjTi if |i− j| ≥ 2
TiTi+1Ti = Ti+1TiTi+1
(Ti + 1)(Ti − q) = 0
Combinatorial Representation Theory – Old and New – p.17/29
The Invention of R
Combinatorial Representation Theory – Old and New – p.18/29
The Invention of R
U = Uq(gln) has an R-matrix
R =∑
j xj ⊗ yj ∈ U⊗U (invertible)
Combinatorial Representation Theory – Old and New – p.18/29
The Invention of R
U = Uq(gln) has an R-matrix
R =∑
j xj ⊗ yj ∈ U⊗U (invertible)
1. It gives a soln. to the quantum Yang-Baxter eqn.
Combinatorial Representation Theory – Old and New – p.18/29
The Invention of R
U = Uq(gln) has an R-matrix
R =∑
j xj ⊗ yj ∈ U⊗U (invertible)
1. It gives a soln. to the quantum Yang-Baxter eqn.
2. R : M ⊗N → N ⊗M , R(m⊗ n) =∑
j yjn⊗ xjm,
Combinatorial Representation Theory – Old and New – p.18/29
The Invention of R
U = Uq(gln) has an R-matrix
R =∑
j xj ⊗ yj ∈ U⊗U (invertible)
1. It gives a soln. to the quantum Yang-Baxter eqn.
2. R : M ⊗N → N ⊗M , R(m⊗ n) =∑
j yjn⊗ xjm,
is a U -module isom. for M, N fin. dim’l U -mods,
Combinatorial Representation Theory – Old and New – p.18/29
The Invention of R
U = Uq(gln) has an R-matrix
R =∑
j xj ⊗ yj ∈ U⊗U (invertible)
1. It gives a soln. to the quantum Yang-Baxter eqn.
2. R : M ⊗N → N ⊗M , R(m⊗ n) =∑
j yjn⊗ xjm,
is a U -module isom. for M, N fin. dim’l U -mods,
On V ⊗k: Ri = id⊗(i−1) ⊗ R⊗ id⊗(k−i−1)
Combinatorial Representation Theory – Old and New – p.18/29
The Invention of R
U = Uq(gln) has an R-matrix
R =∑
j xj ⊗ yj ∈ U⊗U (invertible)
1. It gives a soln. to the quantum Yang-Baxter eqn.
2. R : M ⊗N → N ⊗M , R(m⊗ n) =∑
j yjn⊗ xjm,
is a U -module isom. for M, N fin. dim’l U -mods,
On V ⊗k: Ri = id⊗(i−1) ⊗ R⊗ id⊗(k−i−1)
(i) Ri ∈ EndU (V ⊗k)
(ii) Ri, i = 1, . . . , k − 1, satisfy the braid relations.Combinatorial Representation Theory – Old and New – p.18/29
q-Schur-Weyl Duality
Combinatorial Representation Theory – Old and New – p.19/29
q-Schur-Weyl Duality
ΦU : Uq(gln)→ End(V ⊗k) & ΦH : Hk(q)→ End(V ⊗k)
Combinatorial Representation Theory – Old and New – p.19/29
q-Schur-Weyl Duality
ΦU : Uq(gln)→ End(V ⊗k) & ΦH : Hk(q)→ End(V ⊗k)
Ti 7→ Ri
Combinatorial Representation Theory – Old and New – p.19/29
q-Schur-Weyl Duality
ΦU : Uq(gln)→ End(V ⊗k) & ΦH : Hk(q)→ End(V ⊗k)
Ti 7→ Ri
EndUq(gln)(V⊗k) ∼= Hk(q)/ ker ΦH
Combinatorial Representation Theory – Old and New – p.19/29
q-Schur-Weyl Duality
ΦU : Uq(gln)→ End(V ⊗k) & ΦH : Hk(q)→ End(V ⊗k)
Ti 7→ Ri
EndUq(gln)(V⊗k) ∼= Hk(q)/ ker ΦH
EndHk(q)(V⊗k) ∼= Uq(gln)/ kerΦU =: Sq,F(n, k)
Combinatorial Representation Theory – Old and New – p.19/29
q-Schur-Weyl Duality
ΦU : Uq(gln)→ End(V ⊗k) & ΦH : Hk(q)→ End(V ⊗k)
Ti 7→ Ri
EndUq(gln)(V⊗k) ∼= Hk(q)/ ker ΦH
EndHk(q)(V⊗k) ∼= Uq(gln)/ kerΦU =: Sq,F(n, k)
q-Schur algebra
Combinatorial Representation Theory – Old and New – p.19/29
More revolutions - more revelations
Combinatorial Representation Theory – Old and New – p.20/29
More revolutions - more revelations
Hk(q) is semisimple & irred. reps. 1−1←→ λ ` k
(q not root of 1)
Combinatorial Representation Theory – Old and New – p.20/29
More revolutions - more revelations
Hk(q) is semisimple & irred. reps. 1−1←→ λ ` k
(q not root of 1)
Irred. reps. of Hk(q), q` = 1, 1−1←→ λ ` k, `-regular
Combinatorial Representation Theory – Old and New – p.20/29
More revolutions - more revelations
Hk(q) is semisimple & irred. reps. 1−1←→ λ ` k
(q not root of 1)
Irred. reps. of Hk(q), q` = 1, 1−1←→ λ ` k, `-regular
Lattice of `-regular partitions is
the crystal base of an sl`-module due to Misra-Miwa (’90)
Combinatorial Representation Theory – Old and New – p.20/29
More revolutions - more revelations
Hk(q) is semisimple & irred. reps. 1−1←→ λ ` k
(q not root of 1)
Irred. reps. of Hk(q), q` = 1, 1−1←→ λ ` k, `-regular
Lattice of `-regular partitions is
the crystal base of an sl`-module due to Misra-Miwa (’90)
Ariki (’96)
Combinatorial Representation Theory – Old and New – p.20/29
More revolutions - more revelations
Hk(q) is semisimple & irred. reps. 1−1←→ λ ` k
(q not root of 1)
Irred. reps. of Hk(q), q` = 1, 1−1←→ λ ` k, `-regular
Lattice of `-regular partitions is
the crystal base of an sl`-module due to Misra-Miwa (’90)
Ariki (’96)
Combinatorial Representation Theory – Old and New – p.20/29
More revolutions - more revelations
Hk(q) is semisimple & irred. reps. 1−1←→ λ ` k
(q not root of 1)
Irred. reps. of Hk(q), q` = 1, 1−1←→ λ ` k, `-regular
Lattice of `-regular partitions is
the crystal base of an sl`-module due to Misra-Miwa (’90)
Ariki (’96)
The LLT algorithm gives the decomposition numbers.Combinatorial Representation Theory – Old and New – p.20/29
Affine Hecke Algebra
Combinatorial Representation Theory – Old and New – p.21/29
Affine Hecke Algebra
Haffk (q) ∼= Hk(q)⊗ F[X±1
1 , . . . , X±1k ]
Combinatorial Representation Theory – Old and New – p.21/29
Affine Hecke Algebra
Haffk (q) ∼= Hk(q)⊗ F[X±1
1 , . . . , X±1k ]
TiXiTi = qXi+1, TiXj = XjTi |i− j| > 2
Combinatorial Representation Theory – Old and New – p.21/29
Affine Hecke Algebra
Haffk (q) ∼= Hk(q)⊗ F[X±1
1 , . . . , X±1k ]
TiXiTi = qXi+1, TiXj = XjTi |i− j| > 2
Thm. (Grojnowski-Vazirani ’01)
M irred. Haffk (q)-module. Consider its restriction
reskk−1(M) to Haff
k−1(q). Then socle( reskk−1(M)) is
multiplicity-free.
Combinatorial Representation Theory – Old and New – p.21/29
Affine Hecke Algebra
Haffk (q) ∼= Hk(q)⊗ F[X±1
1 , . . . , X±1k ]
TiXiTi = qXi+1, TiXj = XjTi |i− j| > 2
Thm. (Grojnowski-Vazirani ’01)
M irred. Haffk (q)-module. Consider its restriction
reskk−1(M) to Haff
k−1(q). Then socle( reskk−1(M)) is
multiplicity-free.
Cor. socle(
resSk
Sk−1Dλ
)is multiplicity free.
(Kleshchev ’95)
Combinatorial Representation Theory – Old and New – p.21/29
Orthogonal Schur-Weyl Duality
Combinatorial Representation Theory – Old and New – p.22/29
Orthogonal Schur-Weyl Duality
( , ) nondegenerate symmetric bilinear form on V = Cn
Combinatorial Representation Theory – Old and New – p.22/29
Orthogonal Schur-Weyl Duality
( , ) nondegenerate symmetric bilinear form on V = Cn
On = {g ∈ GLn | (g.u, g.v) = (u, v) ∀ u, v ∈ V }
= {g ∈ GLn | ggt = I } orthogonal group
Combinatorial Representation Theory – Old and New – p.22/29
Orthogonal Schur-Weyl Duality
( , ) nondegenerate symmetric bilinear form on V = Cn
On = {g ∈ GLn | (g.u, g.v) = (u, v) ∀ u, v ∈ V }
= {g ∈ GLn | ggt = I } orthogonal group
EndOn(V ⊗k) =??
Combinatorial Representation Theory – Old and New – p.22/29
Orthogonal Schur-Weyl Duality
( , ) nondegenerate symmetric bilinear form on V = Cn
On = {g ∈ GLn | (g.u, g.v) = (u, v) ∀ u, v ∈ V }
= {g ∈ GLn | ggt = I } orthogonal group
EndOn(V ⊗k) =??
ci,j(v1 ⊗ · · · ⊗ vk) =
(vi, vj)∑n
`=1 v1 ⊗ · · · ⊗ e` ⊗ · · · ⊗ e` ⊗ · · · ⊗ vk
i j
{e`} orthonormal basis of V
Combinatorial Representation Theory – Old and New – p.22/29
Orthogonal Schur-Weyl Duality
( , ) nondegenerate symmetric bilinear form on V = Cn
On = {g ∈ GLn | (g.u, g.v) = (u, v) ∀ u, v ∈ V }
= {g ∈ GLn | ggt = I } orthogonal group
EndOn(V ⊗k) =??
ci,j(v1 ⊗ · · · ⊗ vk) =
(vi, vj)∑n
`=1 v1 ⊗ · · · ⊗ e` ⊗ · · · ⊗ e` ⊗ · · · ⊗ vk
i j
{e`} orthonormal basis of V
Thm. (R. Brauer ’37) EndOn(V ⊗k) is gen. by Sk and the ci,j
Combinatorial Representation Theory – Old and New – p.22/29
Brauer’s Algebra
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams:
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) =
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!
• • •d1 =
• •
• • • • •
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!
• • •d1 =
• •
• • • • •
d2 =• • • • •
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!
• • •d1 =
• •
• • • • •
d2 =• • • • •
• • • • •d1d2 =
• • • • •
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!
• • •d1 =
• •
• • • • •
d2 =• • • • •
• • • • •d1d2 =
• • • • •n
Combinatorial Representation Theory – Old and New – p.23/29
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!
• • •d1 =
• •
• • • • •
d2 =• • • • •
• • • • •d1d2 =
• • • • •n1
Combinatorial Representation Theory – Old and New – p.23/29
Brauer Generators
Combinatorial Representation Theory – Old and New – p.24/29
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
Combinatorial Representation Theory – Old and New – p.24/29
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
• · · · •ei =
i i + 1• •
• • • •
· · · •
• · · · • • · · · •
Combinatorial Representation Theory – Old and New – p.24/29
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
• · · · •ei =
i i + 1• •
• • • •
· · · •
• · · · • • · · · •ΦO : FOn → GL(V ⊗k)
Combinatorial Representation Theory – Old and New – p.24/29
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
• · · · •ei =
i i + 1• •
• • • •
· · · •
• · · · • • · · · •ΦO : FOn → GL(V ⊗k) and ΦB : Bk(n)→ GL(V ⊗k)
si 7→ (i i + 1) ei 7→ ci,i+1
Combinatorial Representation Theory – Old and New – p.24/29
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
• · · · •ei =
i i + 1• •
• • • •
· · · •
• · · · • • · · · •ΦO : FOn → GL(V ⊗k) and ΦB : Bk(n)→ GL(V ⊗k)
si 7→ (i i + 1) ei 7→ ci,i+1
EndOn(V ⊗k) ∼= Bk(n)/ ker ΦB
Combinatorial Representation Theory – Old and New – p.24/29
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
• · · · •ei =
i i + 1• •
• • • •
· · · •
• · · · • • · · · •ΦO : FOn → GL(V ⊗k) and ΦB : Bk(n)→ GL(V ⊗k)
si 7→ (i i + 1) ei 7→ ci,i+1
EndOn(V ⊗k) ∼= Bk(n)/ ker ΦB
EndBk(n)(V⊗k) ∼= FOn/ kerΦO
Combinatorial Representation Theory – Old and New – p.24/29
Special Orthogonal Group
Combinatorial Representation Theory – Old and New – p.25/29
Special Orthogonal Group
SOn = {g ∈ On | det(g) = 1}
Combinatorial Representation Theory – Old and New – p.25/29
Special Orthogonal Group
SOn = {g ∈ On | det(g) = 1}
Thm. (R. Brauer, ’37)
EndSOn(V ⊗k) ∼=
{Bk(n)/ ker ΦB if n oddCk(n)/ ker ΦC if n = 2r even
Combinatorial Representation Theory – Old and New – p.25/29
Special Orthogonal Group
SOn = {g ∈ On | det(g) = 1}
Thm. (R. Brauer, ’37)
EndSOn(V ⊗k) ∼=
{Bk(n)/ ker ΦB if n oddCk(n)/ ker ΦC if n = 2r even
Ck(n) = Bk(n)⊕ spanC{d | (k, r)− diagram}(r ≤ k)
Combinatorial Representation Theory – Old and New – p.25/29
Special Orthogonal Group
SOn = {g ∈ On | det(g) = 1}
Thm. (R. Brauer, ’37)
EndSOn(V ⊗k) ∼=
{Bk(n)/ ker ΦB if n oddCk(n)/ ker ΦC if n = 2r even
Ck(n) = Bk(n)⊕ spanC{d | (k, r)− diagram}(r ≤ k)
• • •
d =
• • • • •
• • • • • • • •
Combinatorial Representation Theory – Old and New – p.25/29
Special Orthogonal Group
SOn = {g ∈ On | det(g) = 1}
Thm. (R. Brauer, ’37)
EndSOn(V ⊗k) ∼=
{Bk(n)/ ker ΦB if n oddCk(n)/ ker ΦC if n = 2r even
Ck(n) = Bk(n)⊕ spanC{d | (k, r)− diagram}(r ≤ k)
• • •
d =
• • • • •
• • • • • • • •
n = 2r unconnected dots
Combinatorial Representation Theory – Old and New – p.25/29
Special Orthogonal Group
SOn = {g ∈ On | det(g) = 1}
Thm. (R. Brauer, ’37)
EndSOn(V ⊗k) ∼=
{Bk(n)/ ker ΦB if n oddCk(n)/ ker ΦC if n = 2r even
Ck(n) = Bk(n)⊕ spanC{d | (k, r)− diagram}(r ≤ k)
• • •
d =
• • • • •
• • • • • • • •
n = 2r unconnected dots
(C. Grood ’98)Combinatorial Representation Theory – Old and New – p.25/29
Back to GLn
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C
)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C H
)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C H L
)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C H L L
)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C H L L S
)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C H I L L S
)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C H I L L S (’94)
)
Combinatorial Representation Theory – Old and New – p.26/29
Back to GLn
GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)
So GLn acts on V ⊗k ⊗ (V ∗)⊗`
Thm.(C H I L L S (’94)
)
EndGLn(V ⊗k ⊗ (V ∗)⊗`) ∼= Bk,`(n)/ ker ΦB
EndBk,`(n)(V⊗k ⊗ (V ∗)⊗`) ∼= CGLn/ ker ΦG
Combinatorial Representation Theory – Old and New – p.26/29
Putting Up a Wall
Combinatorial Representation Theory – Old and New – p.27/29
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Combinatorial Representation Theory – Old and New – p.27/29
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Horizontal lines must cross the wall, vertical lines shouldn’t
Combinatorial Representation Theory – Old and New – p.27/29
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Horizontal lines must cross the wall, vertical lines shouldn’t
• • • • •
d =
• • • •
• • • • • • • • •
Combinatorial Representation Theory – Old and New – p.27/29
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Horizontal lines must cross the wall, vertical lines shouldn’t
• • • • •
d =
? •♣ ♠
• • • • • • • • •
Combinatorial Representation Theory – Old and New – p.27/29
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Horizontal lines must cross the wall, vertical lines shouldn’t
• • • • •
d =
? •♣ ♠
• • • • • • • • •
• • • • •
d′ =
• • • •
? •♣ ♠• • • • •
Combinatorial Representation Theory – Old and New – p.27/29
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Horizontal lines must cross the wall, vertical lines shouldn’t
• • • • •
d =
? •♣ ♠
• • • • • • • • •
• • • • •
d′ =
• • • •
? •♣ ♠• • • • •
So dim Bk,`(n) =
Combinatorial Representation Theory – Old and New – p.27/29
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Horizontal lines must cross the wall, vertical lines shouldn’t
• • • • •
d =
? •♣ ♠
• • • • • • • • •
• • • • •
d′ =
• • • •
? •♣ ♠• • • • •
So dim Bk,`(n) = (k + `)!
Combinatorial Representation Theory – Old and New – p.27/29
WHY?
Combinatorial Representation Theory – Old and New – p.28/29
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Combinatorial Representation Theory – Old and New – p.28/29
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
Combinatorial Representation Theory – Old and New – p.28/29
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
Combinatorial Representation Theory – Old and New – p.28/29
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
ci,j(v1 ⊗ · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ u∗`) =
Combinatorial Representation Theory – Old and New – p.28/29
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
ci,j(v1 ⊗ · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ u∗`) =
u∗j(vi)
n∑
r=1
v1 ⊗ · · · ⊗ er · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ e∗r ⊗ · · · ⊗ u∗`
where e∗s(er) = δs,r
Combinatorial Representation Theory – Old and New – p.28/29
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
ci,j(v1 ⊗ · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ u∗`) =
u∗j(vi)
n∑
r=1
v1 ⊗ · · · ⊗ er · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ e∗r ⊗ · · · ⊗ u∗`
where e∗s(er) = δs,r
• • • • •3 2
• • • •
• •• •• • • • •
Combinatorial Representation Theory – Old and New – p.28/29
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
ci,j(v1 ⊗ · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ u∗`) =
u∗j(vi)
n∑
r=1
v1 ⊗ · · · ⊗ er · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ e∗r ⊗ · · · ⊗ u∗`
where e∗s(er) = δs,r
• • • • •3 2
• • • •
• •• •• • • • •7→ c3,2
Combinatorial Representation Theory – Old and New – p.28/29
GOING A LITTLE CRAZY
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
GLn acts on sln via
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
GLn acts on sln via g.x = gxg−1
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
GLn acts on sln via g.x = gxg−1
EndGLn(sl⊗k
n ) = ???
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
GLn acts on sln via g.x = gxg−1
EndGLn(sl⊗k
n ) = ???
Thm: (B-Doty) EndGLn(sl⊗k
n ) = Dk(n)/ kerΦD where
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
GLn acts on sln via g.x = gxg−1
EndGLn(sl⊗k
n ) = ???
Thm: (B-Doty) EndGLn(sl⊗k
n ) = Dk(n)/ kerΦD where
Dk(n) ⊂ Bk,k(n) is the deranged algebra
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
GLn acts on sln via g.x = gxg−1
EndGLn(sl⊗k
n ) = ???
Thm: (B-Doty) EndGLn(sl⊗k
n ) = Dk(n)/ kerΦD where
Dk(n) ⊂ Bk,k(n) is the deranged algebra
dim Dk(n) = d2k, no. of derangements of {1, . . . , 2k}
Combinatorial Representation Theory – Old and New – p.29/29
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
GLn acts on sln via g.x = gxg−1
EndGLn(sl⊗k
n ) = ???
Thm: (B-Doty) EndGLn(sl⊗k
n ) = Dk(n)/ kerΦD where
Dk(n) ⊂ Bk,k(n) is the deranged algebra
dim Dk(n) = d2k, no. of derangements of {1, . . . , 2k}
Use fact as GLn-modules,
V ⊗ V ∗ ∼= sln ⊕ CI where∑n
j=1 ej ⊗ e∗j 7→ I
Combinatorial Representation Theory – Old and New – p.29/29