combinatorial representation theory – old and newjessica2.msri.org/attachments/12618/12618.pdf ·...
TRANSCRIPT
![Page 1: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/1.jpg)
Combinatorial Representation Theory –Old and New
Georgia BenkartUniversity of Wisconsin-Madison
Combinatorial Representation Theory – Old and New – p.1/29
![Page 2: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/2.jpg)
Group Representations
Combinatorial Representation Theory – Old and New – p.2/29
![Page 3: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/3.jpg)
Group Representations
Ferdinand Georg Frobenius
Combinatorial Representation Theory – Old and New – p.2/29
![Page 4: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/4.jpg)
Group Representations
Ferdinand Georg Frobenius
group representation (1897):
Combinatorial Representation Theory – Old and New – p.2/29
![Page 5: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/5.jpg)
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
![Page 6: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/6.jpg)
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
![Page 7: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/7.jpg)
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
![Page 8: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/8.jpg)
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
![Page 9: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/9.jpg)
Representations ofGLn
Combinatorial Representation Theory – Old and New – p.3/29
![Page 10: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/10.jpg)
Representations ofGLn
Isaai Schur (1901)
Combinatorial Representation Theory – Old and New – p.3/29
![Page 11: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/11.jpg)
Representations ofGLn
Isaai Schur (1901)
GLn acts on V = Cn via matrix multiplication g.v
Combinatorial Representation Theory – Old and New – p.3/29
![Page 12: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/12.jpg)
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
![Page 13: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/13.jpg)
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
![Page 14: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/14.jpg)
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
![Page 15: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/15.jpg)
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
![Page 16: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/16.jpg)
Dawn of Modern Age of Repn. Theory
Combinatorial Representation Theory – Old and New – p.4/29
![Page 17: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/17.jpg)
Dawn of Modern Age of Repn. Theory
Emmy Noether (1929)
Combinatorial Representation Theory – Old and New – p.4/29
![Page 18: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/18.jpg)
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
![Page 19: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/19.jpg)
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
![Page 20: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/20.jpg)
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
![Page 21: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/21.jpg)
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
![Page 22: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/22.jpg)
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
![Page 23: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/23.jpg)
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
![Page 24: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/24.jpg)
Schur-Weyl Duality
Combinatorial Representation Theory – Old and New – p.5/29
![Page 25: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/25.jpg)
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k)
Combinatorial Representation Theory – Old and New – p.5/29
![Page 26: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/26.jpg)
Schur-Weyl Duality
ΦG : FGLn → GL(V ⊗k) and ΦS : FSk → GL(V ⊗k)
Combinatorial Representation Theory – Old and New – p.5/29
![Page 27: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/27.jpg)
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
![Page 28: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/28.jpg)
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
![Page 29: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/29.jpg)
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
![Page 30: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/30.jpg)
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
![Page 31: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/31.jpg)
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
![Page 32: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/32.jpg)
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
![Page 33: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/33.jpg)
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
![Page 34: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/34.jpg)
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
![Page 35: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/35.jpg)
Schur Algebras
Combinatorial Representation Theory – Old and New – p.6/29
![Page 36: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/36.jpg)
Schur Algebras
Schur algebras: J.A. Green (’80) S. Martin (’93)
Combinatorial Representation Theory – Old and New – p.6/29
![Page 37: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/37.jpg)
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
![Page 38: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/38.jpg)
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
![Page 39: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/39.jpg)
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
![Page 40: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/40.jpg)
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
![Page 41: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/41.jpg)
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
![Page 42: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/42.jpg)
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
![Page 43: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/43.jpg)
Resulting Connections
Combinatorial Representation Theory – Old and New – p.7/29
![Page 44: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/44.jpg)
Resulting Connections
(1) Schur functor:
Combinatorial Representation Theory – Old and New – p.7/29
![Page 45: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/45.jpg)
Resulting Connections
(1) Schur functor:
SF(n, k)-modules
Combinatorial Representation Theory – Old and New – p.7/29
![Page 46: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/46.jpg)
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→
Combinatorial Representation Theory – Old and New – p.7/29
![Page 47: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/47.jpg)
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
Combinatorial Representation Theory – Old and New – p.7/29
![Page 48: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/48.jpg)
Resulting Connections
(1) Schur functor:
SF(n, k)-modules F−→ FSk-modules
(2) Cohomology Connections
Combinatorial Representation Theory – Old and New – p.7/29
![Page 49: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/49.jpg)
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
![Page 50: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/50.jpg)
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
![Page 51: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/51.jpg)
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
![Page 52: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/52.jpg)
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
![Page 53: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/53.jpg)
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
![Page 54: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/54.jpg)
Partitions & Their Ups and Downs
Combinatorial Representation Theory – Old and New – p.8/29
![Page 55: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/55.jpg)
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
![Page 56: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/56.jpg)
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
![Page 57: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/57.jpg)
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
![Page 58: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/58.jpg)
Partitions & Their Ups and Downs...
∅Combinatorial Representation Theory – Old and New – p.8/29
![Page 59: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/59.jpg)
Going Up and Down
Combinatorial Representation Theory – Old and New – p.9/29
![Page 60: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/60.jpg)
Going Up and Down
R. Stanley ’88, S. Fomin ’94
Combinatorial Representation Theory – Old and New – p.9/29
![Page 61: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/61.jpg)
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
![Page 62: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/62.jpg)
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
![Page 63: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/63.jpg)
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
![Page 64: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/64.jpg)
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
![Page 65: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/65.jpg)
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
![Page 66: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/66.jpg)
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
![Page 67: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/67.jpg)
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
![Page 68: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/68.jpg)
Up and Down ? Paths
Combinatorial Representation Theory – Old and New – p.10/29
![Page 69: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/69.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
Combinatorial Representation Theory – Old and New – p.10/29
![Page 70: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/70.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ →
Combinatorial Representation Theory – Old and New – p.10/29
![Page 71: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/71.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → →
Combinatorial Representation Theory – Old and New – p.10/29
![Page 72: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/72.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → →
Combinatorial Representation Theory – Old and New – p.10/29
![Page 73: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/73.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → →
Combinatorial Representation Theory – Old and New – p.10/29
![Page 74: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/74.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → →
Combinatorial Representation Theory – Old and New – p.10/29
![Page 75: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/75.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1
Combinatorial Representation Theory – Old and New – p.10/29
![Page 76: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/76.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 2
Combinatorial Representation Theory – Old and New – p.10/29
![Page 77: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/77.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 23
Combinatorial Representation Theory – Old and New – p.10/29
![Page 78: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/78.jpg)
Up and Down ? Paths
a path from ∅ to a partition ⇐⇒ standard tableau
∅ → → → → → 1 23 54
Combinatorial Representation Theory – Old and New – p.10/29
![Page 79: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/79.jpg)
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
![Page 80: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/80.jpg)
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
![Page 81: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/81.jpg)
Counting Up and Down Paths
Combinatorial Representation Theory – Old and New – p.11/29
![Page 82: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/82.jpg)
Counting Up and Down Paths
f2λ : no. of paths up to λ & back down to ∅
Combinatorial Representation Theory – Old and New – p.11/29
![Page 83: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/83.jpg)
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
![Page 84: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/84.jpg)
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
![Page 85: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/85.jpg)
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
![Page 86: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/86.jpg)
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
![Page 87: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/87.jpg)
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
![Page 88: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/88.jpg)
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
![Page 89: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/89.jpg)
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
![Page 90: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/90.jpg)
Therefore:
Combinatorial Representation Theory – Old and New – p.12/29
![Page 91: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/91.jpg)
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
Combinatorial Representation Theory – Old and New – p.12/29
![Page 92: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/92.jpg)
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Combinatorial Representation Theory – Old and New – p.12/29
![Page 93: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/93.jpg)
Therefore:∑
λ`k
f2λ = coefficient of ∅ in dkuk∅
= k!
Recall CSk∼=
⊕λ`k Mλ
Combinatorial Representation Theory – Old and New – p.12/29
![Page 94: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/94.jpg)
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
![Page 95: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/95.jpg)
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
![Page 96: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/96.jpg)
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
![Page 97: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/97.jpg)
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
![Page 98: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/98.jpg)
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
![Page 99: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/99.jpg)
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
![Page 100: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/100.jpg)
Characteristic p
Combinatorial Representation Theory – Old and New – p.13/29
![Page 101: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/101.jpg)
Characteristic p
Sλ, λ ` k, Specht modules
Combinatorial Representation Theory – Old and New – p.13/29
![Page 102: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/102.jpg)
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
Combinatorial Representation Theory – Old and New – p.13/29
![Page 103: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/103.jpg)
Characteristic p
Sλ, λ ` k, Specht modules (Irred. Sk-mods. char. 0)
char(F) = p:
Combinatorial Representation Theory – Old and New – p.13/29
![Page 104: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/104.jpg)
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
![Page 105: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/105.jpg)
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
![Page 106: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/106.jpg)
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
![Page 107: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/107.jpg)
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
![Page 108: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/108.jpg)
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
![Page 109: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/109.jpg)
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
![Page 110: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/110.jpg)
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
![Page 111: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/111.jpg)
"Revolution in Representation Theory"
Combinatorial Representation Theory – Old and New – p.14/29
![Page 112: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/112.jpg)
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Combinatorial Representation Theory – Old and New – p.14/29
![Page 113: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/113.jpg)
"Revolution in Representation Theory"
Lascoux- Leclerc-Thibon (’96)
Lattice of p-regular partitions is
Combinatorial Representation Theory – Old and New – p.14/29
![Page 114: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/114.jpg)
"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
![Page 115: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/115.jpg)
"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
![Page 116: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/116.jpg)
"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
![Page 117: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/117.jpg)
"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
![Page 118: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/118.jpg)
"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
![Page 119: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/119.jpg)
Crystal base for sl3
Combinatorial Representation Theory – Old and New – p.15/29
![Page 120: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/120.jpg)
Crystal base for sl3...
∅Combinatorial Representation Theory – Old and New – p.15/29
![Page 121: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/121.jpg)
Crystal base for sl3...
∅Combinatorial Representation Theory – Old and New – p.15/29
![Page 122: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/122.jpg)
Crystal base for sl3...
∅
Fill box (i, j)
with j − i mod 3
Combinatorial Representation Theory – Old and New – p.15/29
![Page 123: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/123.jpg)
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
![Page 124: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/124.jpg)
The Invention of q
Combinatorial Representation Theory – Old and New – p.16/29
![Page 125: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/125.jpg)
The Invention of q
Drinfeld (’85) and Jimbo (’85)
Combinatorial Representation Theory – Old and New – p.16/29
![Page 126: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/126.jpg)
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
![Page 127: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/127.jpg)
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
![Page 128: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/128.jpg)
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
![Page 129: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/129.jpg)
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
![Page 130: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/130.jpg)
Hecke Algebra
Combinatorial Representation Theory – Old and New – p.17/29
![Page 131: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/131.jpg)
Hecke Algebra
Hk(q): Hecke algebra (q ∈ F×)
Combinatorial Representation Theory – Old and New – p.17/29
![Page 132: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/132.jpg)
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
![Page 133: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/133.jpg)
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
![Page 134: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/134.jpg)
The Invention of R
Combinatorial Representation Theory – Old and New – p.18/29
![Page 135: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/135.jpg)
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
![Page 136: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/136.jpg)
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
![Page 137: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/137.jpg)
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
![Page 138: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/138.jpg)
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
![Page 139: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/139.jpg)
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
![Page 140: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/140.jpg)
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
![Page 141: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/141.jpg)
q-Schur-Weyl Duality
Combinatorial Representation Theory – Old and New – p.19/29
![Page 142: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/142.jpg)
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
![Page 143: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/143.jpg)
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
![Page 144: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/144.jpg)
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
![Page 145: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/145.jpg)
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
![Page 146: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/146.jpg)
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
![Page 147: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/147.jpg)
More revolutions - more revelations
Combinatorial Representation Theory – Old and New – p.20/29
![Page 148: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/148.jpg)
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
![Page 149: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/149.jpg)
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
![Page 150: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/150.jpg)
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
![Page 151: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/151.jpg)
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
![Page 152: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/152.jpg)
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
![Page 153: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/153.jpg)
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
![Page 154: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/154.jpg)
Affine Hecke Algebra
Combinatorial Representation Theory – Old and New – p.21/29
![Page 155: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/155.jpg)
Affine Hecke Algebra
Haffk (q) ∼= Hk(q)⊗ F[X±1
1 , . . . , X±1k ]
Combinatorial Representation Theory – Old and New – p.21/29
![Page 156: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/156.jpg)
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
![Page 157: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/157.jpg)
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
![Page 158: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/158.jpg)
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
![Page 159: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/159.jpg)
Orthogonal Schur-Weyl Duality
Combinatorial Representation Theory – Old and New – p.22/29
![Page 160: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/160.jpg)
Orthogonal Schur-Weyl Duality
( , ) nondegenerate symmetric bilinear form on V = Cn
Combinatorial Representation Theory – Old and New – p.22/29
![Page 161: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/161.jpg)
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
![Page 162: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/162.jpg)
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
![Page 163: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/163.jpg)
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
![Page 164: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/164.jpg)
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
![Page 165: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/165.jpg)
Brauer’s Algebra
Combinatorial Representation Theory – Old and New – p.23/29
![Page 166: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/166.jpg)
Brauer’s Algebra
Bk(n) has basis the k-diagrams:
Combinatorial Representation Theory – Old and New – p.23/29
![Page 167: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/167.jpg)
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
Combinatorial Representation Theory – Old and New – p.23/29
![Page 168: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/168.jpg)
Brauer’s Algebra
Bk(n) has basis the k-diagrams: • • • • •
• • • • •
dim Bk(n) =
Combinatorial Representation Theory – Old and New – p.23/29
![Page 169: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/169.jpg)
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
![Page 170: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/170.jpg)
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
![Page 171: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/171.jpg)
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
![Page 172: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/172.jpg)
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
![Page 173: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/173.jpg)
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
![Page 174: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/174.jpg)
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
![Page 175: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/175.jpg)
Brauer Generators
Combinatorial Representation Theory – Old and New – p.24/29
![Page 176: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/176.jpg)
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
Combinatorial Representation Theory – Old and New – p.24/29
![Page 177: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/177.jpg)
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
• · · · •ei =
i i + 1• •
• • • •
· · · •
• · · · • • · · · •
Combinatorial Representation Theory – Old and New – p.24/29
![Page 178: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/178.jpg)
Brauer Generators
• · · · •si =
i i + 1• •
• • • •
· · · •
• · · · • • • · · · •
• · · · •ei =
i i + 1• •
• • • •
· · · •
• · · · • • · · · •ΦO : FOn → GL(V ⊗k)
Combinatorial Representation Theory – Old and New – p.24/29
![Page 179: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/179.jpg)
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
![Page 180: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/180.jpg)
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
![Page 181: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/181.jpg)
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
![Page 182: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/182.jpg)
Special Orthogonal Group
Combinatorial Representation Theory – Old and New – p.25/29
![Page 183: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/183.jpg)
Special Orthogonal Group
SOn = {g ∈ On | det(g) = 1}
Combinatorial Representation Theory – Old and New – p.25/29
![Page 184: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/184.jpg)
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
![Page 185: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/185.jpg)
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
![Page 186: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/186.jpg)
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
![Page 187: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/187.jpg)
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
![Page 188: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/188.jpg)
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
![Page 189: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/189.jpg)
Back to GLn
Combinatorial Representation Theory – Old and New – p.26/29
![Page 190: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/190.jpg)
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
![Page 191: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/191.jpg)
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
![Page 192: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/192.jpg)
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
![Page 193: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/193.jpg)
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
![Page 194: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/194.jpg)
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
![Page 195: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/195.jpg)
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
![Page 196: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/196.jpg)
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
![Page 197: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/197.jpg)
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
![Page 198: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/198.jpg)
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
![Page 199: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/199.jpg)
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
![Page 200: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/200.jpg)
Putting Up a Wall
Combinatorial Representation Theory – Old and New – p.27/29
![Page 201: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/201.jpg)
Putting Up a Wall
Bk,`(n) has a basis of walled (k + `)-diagrams
Combinatorial Representation Theory – Old and New – p.27/29
![Page 202: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/202.jpg)
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
![Page 203: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/203.jpg)
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
![Page 204: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/204.jpg)
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
![Page 205: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/205.jpg)
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
![Page 206: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/206.jpg)
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
![Page 207: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/207.jpg)
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
![Page 208: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/208.jpg)
WHY?
Combinatorial Representation Theory – Old and New – p.28/29
![Page 209: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/209.jpg)
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Combinatorial Representation Theory – Old and New – p.28/29
![Page 210: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/210.jpg)
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
Combinatorial Representation Theory – Old and New – p.28/29
![Page 211: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/211.jpg)
WHY?
On V ⊗k ⊗ (V ∗)⊗` :
Sk permutes the first k factors & S` the last ` factors
Combinatorial Representation Theory – Old and New – p.28/29
![Page 212: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/212.jpg)
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
![Page 213: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/213.jpg)
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
![Page 214: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/214.jpg)
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
![Page 215: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/215.jpg)
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
![Page 216: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/216.jpg)
GOING A LITTLE CRAZY
Combinatorial Representation Theory – Old and New – p.29/29
![Page 217: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/217.jpg)
GOING A LITTLE CRAZY
sln (n× n) matrices of trace 0
Combinatorial Representation Theory – Old and New – p.29/29
![Page 218: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/218.jpg)
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
![Page 219: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/219.jpg)
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
![Page 220: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/220.jpg)
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
![Page 221: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/221.jpg)
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
![Page 222: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/222.jpg)
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
![Page 223: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/223.jpg)
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
![Page 224: Combinatorial Representation Theory – Old and Newjessica2.msri.org/attachments/12618/12618.pdf · Combinatorial Representation Theory – Old and New Georgia Benkart University](https://reader033.vdocuments.us/reader033/viewer/2022053015/5f15842a7202c833ee7ff8ac/html5/thumbnails/224.jpg)
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