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Combinatorial Representation Theory –Old and New

Georgia BenkartUniversity of Wisconsin-Madison

Combinatorial Representation Theory – Old and New – p.1/29

Group Representations



Ferdinand Georg Frobenius




group representation (1897):





ϕ : G→ GL(V ) (invertible transformations on V )






ϕ(gh) = ϕ(g)ϕ(h)






ϕ(gh) = ϕ(g)ϕ(h)

g.v = ϕ(g)(v) makes V a G-module






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)


Representations ofGLn



Isaai Schur (1901)



Isaai Schur (1901)

GLn acts on V = Cn via matrix multiplication g.v



Isaai Schur (1901)


GLn acts on V ⊗k viag.(v1 ⊗ · · · ⊗ vk) = g.v1 ⊗ · · · ⊗ g.vk



Isaai Schur (1901)



Sk acts on V ⊗k via place permutations



Isaai Schur (1901)




The two actions commute.



Isaai Schur (1901)




The two actions commute.

Use Sk -repns. to study GLn-repns.


Dawn of Modern Age of Repn. Theory



Emmy Noether (1929)



Emmy Noether (1929)

Repns. of G over F⇐⇒ Repns. of group algebra FG



Emmy Noether (1929)


FG ∼= direct sum of matrix blocks (char(F) = 0)



Emmy Noether (1929)



FG/rad(FG) ∼= direct sum of matrix blocks (char(F) = p)



Emmy Noether (1929)




One block for each irreducible repn. of G



Emmy Noether (1929)





dim. matrix block = (dim. of the irred. repn)2



Emmy Noether (1929)





dim. matrix block = (dim. of the irred. repn)2

Ex. FSk∼=

⊕λ`k Mλ (char(F) = 0)


Schur-Weyl Duality


Schur-Weyl Duality

ΦG : FGLn → GL(V ⊗k)


Schur-Weyl Duality

ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)


Schur-Weyl Duality


(Here F can be any infinite field S. Doty ’06)


Schur-Weyl Duality



EndGLn(V ⊗k) ∼= FSk/ ker ΦS


Schur-Weyl Duality




EndSk(V⊗k) ∼= FGLn/ ker ΦG


Schur-Weyl Duality




EndSk(V⊗k) ∼= FGLn/ ker ΦG =: SF(n, k)


Schur-Weyl Duality




EndSk(V⊗k) ∼= FGLn/ ker ΦG =: SF(n, k) Schur algebra


Schur-Weyl Duality





(Polynomial) repns. of GLn over F⇐⇒ repns. of SF(n, k)


Schur-Weyl Duality






k = 1, 2, . . .


Schur-Weyl Duality




∼= FSk/⊕

λ`k

#parts>n

Mλ (char(F) = 0)



k = 1, 2, . . .


Schur Algebras


Schur Algebras

Schur algebras: J.A. Green (’80) S. Martin (’93)


Schur Algebras


Thm. (S. Doty & D. Nakano ’98)

Let F be algebraically closed.


Schur Algebras



Let F be algebraically closed. SF(n, k) is semisimple iff


Schur Algebras




(1) char(F) = 0


Schur Algebras




(1) char(F) = 0

(2) char(F) = p > k


Schur Algebras




(1) char(F) = 0

(2) char(F) = p > k

(3) char(F) = 2, n = 2, k = 3


Schur Algebras




(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.


Resulting Connections



(1) Schur functor:



(1) Schur functor:

SF(n, k)-modules



(1) Schur functor:

SF(n, k)-modules F−→



(1) Schur functor:

SF(n, k)-modules F−→ FSk-modules



(1) Schur functor:


(2) Cohomology Connections



(1) Schur functor:



ExtiSF(n,k)(M, N)



(1) Schur functor:



ExtiSF(n,k)(M, N) ∼=



(1) Schur functor:



ExtiSF(n,k)(M, N) ∼= Exti

FSk(F(M), F(N))



(1) Schur functor:




FSk(F(M), F(N))

SOMETIMES!



(1) Schur functor:




FSk(F(M), F(N))

SOMETIMES! – see Kleshchev & Nakano ’01


Partitions & Their Ups and Downs


Partitions & Their Ups and Downs...

∅Combinatorial Representation Theory – Old and New – p.8/29

Going Up and Down


Going Up and Down

R. Stanley ’88, S. Fomin ’94


Going Up and Down


On lattice of partitions: du− ud = I


Going Up and Down


On lattice of partitions: du− ud = I (Weyl alg. relation)


Going Up and Down



Irreducible repns. for Sk1−1←→ λ ` k


Going Up and Down




resSk

Sk−1λ =

∑

κ⊂λ|λ/κ|=1

κ


Going Up and Down




resSk

Sk−1λ =

∑

κ⊂λ|λ/κ|=1

κ = d(λ)


Going Up and Down




resSk

Sk−1λ =

∑

κ⊂λ|λ/κ|=1

κ = d(λ)

indSk+1

Skλ =

∑

ν⊃λ|ν/λ|=1

ν


Going Up and Down




resSk

Sk−1λ =

∑

κ⊂λ|λ/κ|=1

κ = d(λ)

indSk+1

Skλ =

∑

ν⊃λ|ν/λ|=1

ν = u(λ)


Up and Down ? Paths


Up and Down ? Paths

a path from ∅ to a partition ⇐⇒ standard tableau


Up and Down ? Paths


∅ →


Up and Down ? Paths


∅ → →


Up and Down ? Paths


∅ → → →


Up and Down ? Paths


∅ → → → →


Up and Down ? Paths


∅ → → → → →


Up and Down ? Paths


∅ → → → → → 1


Up and Down ? Paths


∅ → → → → → 1 2


Up and Down ? Paths


∅ → → → → → 1 23


Up and Down ? Paths


∅ → → → → → 1 23 54


Up and Down ? Paths


∅ → → → → → 1 23 54

<

∧

standard tableau


Up and Down ? Paths


∅ → → → → → 1 23 54

<

∧

standard tableau

fλ: no. of standard tableaux of shape λ

= no. of paths up to λ


Counting Up and Down Paths



f2λ : no. of paths up to λ & back down to ∅




∑λ`k f2

λ = coefficient of ∅ in dkuk∅




∑λ`k f2


To compute this use: duk = ukd + kuk−1




∑λ`k f2



dkuk∅ = dk−1(duk)∅




∑λ`k f2




= dk−1(ukd + kuk−1)∅




∑λ`k f2





= k dk−1uk−1∅




∑λ`k f2





= k dk−1uk−1∅

= k(k − 1) dk−2uk−2∅ = · · ·




∑λ`k f2





= k dk−1uk−1∅

= k(k − 1) dk−2uk−2∅ = · · ·

= (k!)∅


Therefore:


Therefore:∑

λ`k

f2λ = coefficient of ∅ in dkuk∅


Therefore:∑

λ`k


= k!


Therefore:∑

λ`k


= k!

Recall CSk∼=

⊕λ`k Mλ


Therefore:∑

λ`k


= k!

Recall CSk∼=

⊕λ`k Mλ and take dimensions


Therefore:∑

λ`k


= k!

Recall CSk∼=


k! =∑

λ`k

dim Mλ =


Therefore:∑

λ`k


= k!

Recall CSk∼=


k! =∑

λ`k

dim Mλ =

∑

λ`k

(dim λ)2


Therefore:∑

λ`k


= k!

Recall CSk∼=


k! =∑

λ`k

dim Mλ =

∑

λ`k

(dim λ)2 =∑

λ`k

f2λ


Therefore:∑

λ`k


= k!

Recall CSk∼=


k! =∑

λ`k

dim Mλ =

∑

λ`k

(dim λ)2 =∑

λ`k

f2λ

Problem: Determine all posets for which du− ud = rI.


Therefore:∑

λ`k


= k!

Recall CSk∼=


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)


Characteristic p


Characteristic p

Sλ, λ ` k, Specht modules


Characteristic p

Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)


Characteristic p


char(F) = p:


Characteristic p


char(F) = p:

FSk irreds. 1−1←→ λ ` k p-regular


Characteristic p


char(F) = p:


(no part repeated p or more times)


Characteristic p


char(F) = p:



FSk irreds. Dλ = Sλ/Qλ, λ ` k p-regular (James ’76)


Characteristic p


char(F) = p:




Outstanding problems


Characteristic p


char(F) = p:





1. Find dim Dλ


Characteristic p


char(F) = p:





1. Find dim Dλ

2. Find [Sλ : Dν ], λ, ν ` k


Characteristic p


char(F) = p:





1. Find dim Dλ

2. Find [Sλ : Dν ], λ, ν ` k

3. Find [resSk

Sk−1Dλ : Dµ], λ ` k, µ ` k − 1


"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.







The LLT algorithm gives the decomposition numbers.

James (’80), Erdmann (’96)

Knowing decomposition nos. for Sk is equivalent to knowingdecomposition nos. for GLn


Crystal base for sl3


Crystal base for sl3...

∅Combinatorial Representation Theory – Old and New – p.15/29


∅

Fill box (i, j)

with j − i mod 3



∅

Fill box (i, j)

with j − i mod 3

0

200 1

210 0 12

120 10

210

02

1 20 1 2 0


The Invention of q


The Invention of q

Drinfeld (’85) and Jimbo (’85)


The Invention of q


Uq(gln) acts on V = C(q)n and also on V ⊗k


The Invention of q



EndUq(gln)(V⊗k) =??


The Invention of q




Sk has gens. si = (i i + 1), i = 1, . . . , k − 1, and relns.


The Invention of q




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


Hecke Algebra


Hecke Algebra

Hk(q): Hecke algebra (q ∈ F×)


Hecke Algebra


F-algebra with gens. Ti, i = 1, . . . , k − 1, and relns.


Hecke Algebra


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


The Invention of R


The Invention of R

U = Uq(gln) has an R-matrix

R =∑

j xj ⊗ yj ∈ U⊗U (invertible)


The Invention of R


R =∑


1. It gives a soln. to the quantum Yang-Baxter eqn.


The Invention of R


R =∑



2. R : M ⊗N → N ⊗M , R(m⊗ n) =∑

j yjn⊗ xjm,


The Invention of R


R =∑



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,


The Invention of R


R =∑



2. R : M ⊗N → N ⊗M , R(m⊗ n) =∑

j yjn⊗ xjm,


On V ⊗k: Ri = id⊗(i−1) ⊗ R⊗ id⊗(k−i−1)


The Invention of R


R =∑



2. R : M ⊗N → N ⊗M , R(m⊗ n) =∑

j yjn⊗ xjm,


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



ΦU : Uq(gln)→ End(V ⊗k) & ΦH : Hk(q)→ End(V ⊗k)




Ti 7→ Ri




Ti 7→ Ri

EndUq(gln)(V⊗k) ∼= Hk(q)/ ker ΦH




Ti 7→ Ri


EndHk(q)(V⊗k) ∼= Uq(gln)/ kerΦU =: Sq,F(n, k)




Ti 7→ Ri


EndHk(q)(V⊗k) ∼= Uq(gln)/ kerΦU =: Sq,F(n, k)

q-Schur algebra


More revolutions - more revelations



Hk(q) is semisimple & irred. reps. 1−1←→ λ ` k

(q not root of 1)




(q not root of 1)

Irred. reps. of Hk(q), q` = 1, 1−1←→ λ ` k, `-regular




(q not root of 1)


Lattice of `-regular partitions is

the crystal base of an sl`-module due to Misra-Miwa (’90)




(q not root of 1)




Ariki (’96)




(q not root of 1)




Ariki (’96)

The LLT algorithm gives the decomposition numbers.Combinatorial Representation Theory – Old and New – p.20/29

Affine Hecke Algebra



Haffk (q) ∼= Hk(q)⊗ F[X±1

1 , . . . , X±1k ]




1 , . . . , X±1k ]

TiXiTi = qXi+1, TiXj = XjTi |i− j| > 2




1 , . . . , X±1k ]


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.




1 , . . . , X±1k ]


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)


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) =??






EndOn(V ⊗k) =??

ci,j(v1 ⊗ · · · ⊗ vk) =

(vi, vj)∑n

`=1 v1 ⊗ · · · ⊗ e` ⊗ · · · ⊗ e` ⊗ · · · ⊗ vk

i j

{e`} orthonormal basis of V






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


Brauer’s Algebra


Brauer’s Algebra

Bk(n) has basis the k-diagrams:


Brauer’s Algebra

Bk(n) has basis the k-diagrams: • • • • •

• • • • •


Brauer’s Algebra


• • • • •

dim Bk(n) =


Brauer’s Algebra


• • • • •

dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!


Brauer’s Algebra


• • • • •

dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!

• • •d1 =

• •

• • • • •


Brauer’s Algebra


• • • • •

dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!

• • •d1 =

• •

• • • • •

d2 =• • • • •


Brauer’s Algebra


• • • • •

dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!

• • •d1 =

• •

• • • • •

d2 =• • • • •

• • • • •d1d2 =

• • • • •


Brauer’s Algebra


• • • • •

dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!

• • •d1 =

• •

• • • • •

d2 =• • • • •

• • • • •d1d2 =

• • • • •n


Brauer’s Algebra


• • • • •

dim Bk(n) = (2k − 1)(2k − 3) · · · 3 · 1 = (2k − 1)!!

• • •d1 =

• •

• • • • •

d2 =• • • • •

• • • • •d1d2 =

• • • • •n1


Brauer Generators


Brauer Generators

• · · · •si =

i i + 1• •

• • • •

· · · •

• · · · • • • · · · •


Brauer Generators

• · · · •si =

i i + 1• •

• • • •

· · · •

• · · · • • • · · · •

• · · · •ei =

i i + 1• •

• • • •

· · · •

• · · · • • · · · •


Brauer Generators

• · · · •si =

i i + 1• •

• • • •

· · · •

• · · · • • • · · · •

• · · · •ei =

i i + 1• •

• • • •

· · · •

• · · · • • · · · •ΦO : FOn → GL(V ⊗k)


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


Brauer Generators

• · · · •si =

i i + 1• •

• • • •

· · · •

• · · · • • • · · · •

• · · · •ei =

i i + 1• •

• • • •

· · · •


si 7→ (i i + 1) ei 7→ ci,i+1

EndOn(V ⊗k) ∼= Bk(n)/ ker ΦB


Brauer Generators

• · · · •si =

i i + 1• •

• • • •

· · · •

• · · · • • • · · · •

• · · · •ei =

i i + 1• •

• • • •

· · · •


si 7→ (i i + 1) ei 7→ ci,i+1

EndOn(V ⊗k) ∼= Bk(n)/ ker ΦB

EndBk(n)(V⊗k) ∼= FOn/ kerΦO


Special Orthogonal Group



SOn = {g ∈ On | det(g) = 1}



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



SOn = {g ∈ On | det(g) = 1}


EndSOn(V ⊗k) ∼=


Ck(n) = Bk(n)⊕ spanC{d | (k, r)− diagram}(r ≤ k)



SOn = {g ∈ On | det(g) = 1}


EndSOn(V ⊗k) ∼=



• • •

d =

• • • • •

• • • • • • • •



SOn = {g ∈ On | det(g) = 1}


EndSOn(V ⊗k) ∼=



• • •

d =

• • • • •

• • • • • • • •

n = 2r unconnected dots



SOn = {g ∈ On | det(g) = 1}


EndSOn(V ⊗k) ∼=



• • •

d =

• • • • •

• • • • • • • •

n = 2r unconnected dots

(C. Grood ’98)Combinatorial Representation Theory – Old and New – p.25/29

Back to GLn


Back to GLn

GLn acts on V ∗ = HomC(V, C) via (g.u∗)(v) = u∗(g−1.v)


Back to GLn


So GLn acts on V ⊗k ⊗ (V ∗)⊗`


Back to GLn



Thm.(C

)


Back to GLn



Thm.(C H

)


Back to GLn



Thm.(C H L

)


Back to GLn



Thm.(C H L L

)


Back to GLn



Thm.(C H L L S

)


Back to GLn



Thm.(C H I L L S

)


Back to GLn



Thm.(C H I L L S (’94)

)


Back to GLn



Thm.(C H I L L S (’94)

)

EndGLn(V ⊗k ⊗ (V ∗)⊗`) ∼= Bk,`(n)/ ker ΦB

EndBk,`(n)(V⊗k ⊗ (V ∗)⊗`) ∼= CGLn/ ker ΦG


Putting Up a Wall


Putting Up a Wall

Bk,`(n) has a basis of walled (k + `)-diagrams


Putting Up a Wall


Horizontal lines must cross the wall, vertical lines shouldn’t


Putting Up a Wall



• • • • •

d =

• • • •

• • • • • • • • •


Putting Up a Wall



• • • • •

d =

? •♣ ♠

• • • • • • • • •


Putting Up a Wall



• • • • •

d =

? •♣ ♠

• • • • • • • • •

• • • • •

d′ =

• • • •

? •♣ ♠• • • • •


Putting Up a Wall



• • • • •

d =

? •♣ ♠

• • • • • • • • •

• • • • •

d′ =

• • • •

? •♣ ♠• • • • •

So dim Bk,`(n) =


Putting Up a Wall



• • • • •

d =

? •♣ ♠

• • • • • • • • •

• • • • •

d′ =

• • • •

? •♣ ♠• • • • •

So dim Bk,`(n) = (k + `)!


WHY?


WHY?

On V ⊗k ⊗ (V ∗)⊗` :


WHY?

On V ⊗k ⊗ (V ∗)⊗` :

Sk permutes the first k factors & S` the last ` factors


WHY?

On V ⊗k ⊗ (V ∗)⊗` :


ci,j(v1 ⊗ · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ u∗`) =


WHY?

On V ⊗k ⊗ (V ∗)⊗` :


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


WHY?

On V ⊗k ⊗ (V ∗)⊗` :


ci,j(v1 ⊗ · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ u∗`) =

u∗j(vi)

n∑

r=1

v1 ⊗ · · · ⊗ er · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ e∗r ⊗ · · · ⊗ u∗`


• • • • •3 2

• • • •

• •• •• • • • •


WHY?

On V ⊗k ⊗ (V ∗)⊗` :


ci,j(v1 ⊗ · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ u∗`) =

u∗j(vi)

n∑

r=1

v1 ⊗ · · · ⊗ er · · · ⊗ vk ⊗ u∗1 ⊗ · · · ⊗ e∗r ⊗ · · · ⊗ u∗`


• • • • •3 2

• • • •

• •• •• • • • •7→ c3,2


GOING A LITTLE CRAZY



sln (n× n) matrices of trace 0




GLn acts on sln via




GLn acts on sln via g.x = gxg−1





EndGLn(sl⊗k

n ) = ???





EndGLn(sl⊗k

n ) = ???

Thm: (B-Doty) EndGLn(sl⊗k

n ) = Dk(n)/ kerΦD where





EndGLn(sl⊗k

n ) = ???



Dk(n) ⊂ Bk,k(n) is the deranged algebra





EndGLn(sl⊗k

n ) = ???




dim Dk(n) = d2k, no. of derangements of {1, . . . , 2k}





EndGLn(sl⊗k

n ) = ???




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


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