Download pdf - Sorting monoids on Coxeter groups - Bucknell …linux.bucknell.edu/~pm040/Slides/Schilling.pdfSorting monoids on Coxeter groups Florent Hivert1 Anne Schilling2 Nicolas M. Thi ery2;3

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Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory

Sorting monoids on Coxeter groups

Florent Hivert1 Anne Schilling2 Nicolas M. Thiery2,3

1LITIS/LIFAR, Universite Rouen, France

2University of California at Davis, USA

3Laboratoire de Mathematiques d’Orsay, Universite Paris Sud, France

FPSAC’10, San Francisco, August 3rd of 2010

arXiv:0711.1561 [math.RT] (FPSAC’06)arXiv:0804.3781 [math.RT] (FPSAC’08)arXiv:0912.2212 [math.CO] (FPSAC’10)

+ research in progress

2 / 19


Bubble (anti) sort algorithm

1234

Underlying combinatorics: right permutahedron

123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321

Elementary transpositions: s1, s2, s3, . . .Relations: s2

i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



1234


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



1243


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



1423


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4123


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4123


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4132


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4312


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4312


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4321


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4321


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4321


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4321


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4321


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321

Elementary transpositions: s1, s2, s3, . . .

Relations: s2i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

2 / 19



4321


123

213 132

312231

321

1234

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

4321


i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1

3 / 19


Coxeter groups

Definition (Coxeter group W )

Generators : s1, s2, . . . (simple reflections)

Relations: s2i = 1 and si sj · · ·︸︷︷︸

mi,j

= sjsi · · ·︸︷︷︸mi,j

, for i 6= j

Reduced words

4 / 19


0-Hecke monoid

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

π1 π2

π1 π2

π2 π1

π2 π1

π1 π2

π1 π2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19


0-Hecke monoid

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

π1 π2

π1 π2

π2 π1

π2 π1

π1 π2

π1 π2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19


0-Hecke monoid

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19


0-Hecke monoid

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19


0-Hecke monoid

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

123

213 132

312231

321

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

5 / 19


0-Hecke monoid

Definition (0-Hecke monoid H0(W ) of a Coxeter group W )

Generators : 〈π1, π2, . . . 〉 (simple reflections)Relations: π2

i = πi and braid relations

Theorem

|H0(W )| = |W |+ lots of nice properties

Motivation: simple combinatorial model (bubble sort)appears in Iwahori-Hecke algebras, Schur symmetric functions,Schubert, Macdonald, Kazhdan-Lusztig polynomials,(affine) Stanley symmetric functions, mathematical physics,Schur-Weyl duality for quantum groups, representations of GL(Fq),. . .

5 / 19


0-Hecke monoid




Theorem



5 / 19


0-Hecke monoid




Theorem



6 / 19


Classical orders on Coxeter groups

123

213 132

312231

321

s1 s2

s2 s1

s1s2

123

213 132

312231

321

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

Left order Left-Right order

Right order

Bruhat order

Suffix Factor Prefix Subword

6 / 19



123

213 132

312231

321

s1 s2

s2 s1

s1s2

123

213 132

312231

321

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

Left order Left-Right order

Right order

Bruhat order

Suffix Factor

Prefix

Subword

6 / 19



123

213 132

312231

321

s1 s2

s2 s1

s1s2

123

213 132

312231

321

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

Left order

Left-Right order

Right order

Bruhat order

Suffix

Factor

Prefix

Subword

6 / 19



123

213 132

312231

321

s1 s2

s2 s1

s1s2

123

213 132

312231

321

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

Left order Left-Right order Right order

Bruhat order

Suffix Factor Prefix

Subword

6 / 19



123

213 132

312231

321

s1 s2

s2 s1

s1s2

123

213 132

312231

321

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

123

213 132

312231

321

s1 s2

s2 s1

s1 s2

123

213 132

312231

321

Left order Left-Right order Right order Bruhat order

Suffix Factor Prefix Subword

7 / 19


Blocks of permutations

Definition (Block of a permutation w)

• Type A: sub-permutation matrix

• Type free: J,K such that WJw = wWK

• Example: w := 36475812

••

••

••

••

• Simple permutation: cf. [Albert, Atkinson 05] + dim 2 posets• {blocks of w}: sub-lattice of the Boolean lattice

7 / 19






• Example: w := 36475812

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7 / 19






• Example: w := 36475812

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7 / 19






• Example: w := 36475812

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7 / 19






• Example: w := 36475812

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7 / 19






• Example: w := 36475812

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8 / 19


The cutting poset

Definition (HST09: Cutting poset (W ,v))

u v w if u = wJ with J block

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Theorem

• Intervals are lattices

• Mobius function: inclusion-exclusion along minimal blocks

• Meet-semi lattice?

8 / 19


The cutting poset



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Theorem




8 / 19


The cutting poset



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Theorem




8 / 19


The cutting poset



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Theorem




8 / 19


The cutting poset



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Theorem

• Intervals are (distributive?) lattices



8 / 19


The cutting poset



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Theorem

• Intervals are (distributive?) lattices



9 / 19


The Big Picture

Hζ(W )

Hζ(W )

Hζ(Sn) ∧

H−1(W )

H−1(W )

TLn

Hq(W )

Hq(W )

Hq(Sn) ∧

H0(W )〈π1, π2, . . . 〉

H0(W )〈π0π1, π2, . . . 〉

NDPFn

H0(W )〈π1, π2, . . . 〉

H0(W )〈π0π1, π2, . . . 〉

NDPFn

NDPFB

HWQ[π1, π2, . . . , s1, s2, . . . ]Q[π1, π2, . . . , π1, π2, . . . ]

Q[π0π1, π2, . . . ]

NDFn

W〈s1, s2, . . . 〉

W〈s0s1, s2, . . . 〉

Sn ∧

M1

NDPF (Bruhat(W )) End(<L (W ))

〈π1, π2, . . . , π1, π2, . . . 〉〈π1, π2, . . . , s1, s2, . . . 〉〈π0π1, π2, . . . 〉

End(BooleanLattice)

10 / 19


The biHecke monoid

Question

Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?

• How to attack such a problem?

• Generators and relations?

• Representation theory?

Theorem (HST08)

M(W ) admits |W | simple / indecomposable projective modules

• Why do we care?

|M(W )| =∑

w∈W

dim Sw . dim Pw

10 / 19


The biHecke monoid

Question





Theorem (HST08)


• Why do we care?

|M(W )| =∑

w∈W

dim Sw . dim Pw

10 / 19


The biHecke monoid

Question





Theorem (HST08)


• Why do we care?

|M(W )| =∑

w∈W

dim Sw . dim Pw

10 / 19


The biHecke monoid

Question





Theorem (HST08)


• Why do we care?

|M(W )| =∑

w∈W

dim Sw . dim Pw

10 / 19


The biHecke monoid

Question





Theorem (HST08)


• Why do we care?

|M(W )| =∑

w∈W

dim Sw . dim Pw

10 / 19


The biHecke monoid

Question





Theorem (HST08)


• Why do we care?

|M(W )| =∑

w∈W

dim Sw . dim Pw

10 / 19


The biHecke monoid

Question





Theorem (HST08)


• Why do we care?

|M(W )| =∑

w∈W

dim Sw . dim Pw

11 / 19


Key combinatorial lemma

132

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Lemma

For f ∈ M(W ) and w ∈W : (siw).f = w .f or si (w .f )

Proof.

Exchange property / associativity

11 / 19



132

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Lemma


Proof.


11 / 19



132

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Lemma


Proof.


11 / 19



132

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Lemma


Proof.


11 / 19



132

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Lemma


Proof.


11 / 19



132

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Lemma


Proof.


11 / 19



132

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Corollary

• Preservation of left order: u ≤L v =⇒ u.f ≤L v .f

• Preservation of Bruhat order: u ≤B v =⇒ u.f ≤B v .f

• M(W ) is aperiodic

• f in M(W ) is determined by its fibers and f (1)

11 / 19



132

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Corollary





11 / 19



132

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Corollary





11 / 19



132

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Corollary





11 / 19



132

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Corollary





12 / 19


Some elements of the monoid1

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. 7→

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•

.•

.• .•

.•

.•

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.•

.•

2

.

.

.

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.•

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132

.

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.•

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. .•

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.•

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2123

.

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. .

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•. .

.

.

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.

.

.•

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•.

Lemma

The image set of an idempotent is an interval in left order

12 / 19



.

.

.

. .

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.

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.

.

.

. .

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.

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.

.

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. 7→

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2

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. 7→

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132

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2123

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Lemma


12 / 19



.

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.

. .

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. .

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2

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132

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2123

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Lemma


12 / 19



.

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2

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132

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2123

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Lemma


12 / 19



.

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. .

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. .

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2

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132

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2123

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Lemma


13 / 19


Green relations

×1

2×

×1

×2

2×

×2

×2

2×2×1

×2

2×21×1

2×

×2

2×

1×

2×

×2×1

×2

×1

1×

×1 ×2

×1

×1

×1

×2

×2

1×

×1

×2

×1

1×

×2

1×

×1

1×

×2

×1×2

2×

×1

×1

2×

×1

×1

×1

×2

1×

1×1

×1

×1

2×

×2

1×1

×2

2×2 2×

2×

×1

×2

1×1

×1

2×2

1×

1×

×2

×2

1×

×2

1×

×2

×2

×1

2×

2×

1×

×1

×2

1×

1×

2×

×2

×1

2×

×2

×1

.

.

.

.

.

.7→

••

••

••

1

.

.

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.7→

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.•

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2

.

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.7→

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2

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.7→

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.•

1

.

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.7→

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•.

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12

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.7→

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.•

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12

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.7→

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21

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21

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21

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21

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12

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12

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121

.

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.7→

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.•

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121

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.7→

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.•

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121

.

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•.

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•.

212

.

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.•

212

.

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.7→

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212

.

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.7→

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.•

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212

.

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.7→

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.

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•.

212

.

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.7→

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121

.

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.7→

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.

.•

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121

.

.

.

.

.

.7→

.•

.

.

.

.

14 / 19


Green relations

Theorem

• Regular J -classes are indexed by W

• J -order on regular classes: left-right order on W

• R-classes: intervals in right order on W

• R-order on regular R-classes: ≈ right order on W

• L-order on regular L-classes: ≈ left order on W

Problems

• L,R,J -order between non regular classes?

• L-classes?

14 / 19


Green relations

Theorem






Problems


• L-classes?

14 / 19


Green relations

Theorem






Problems


• L-classes?

14 / 19


Green relations

Theorem






Problems


• L-classes?

14 / 19


Green relations

Theorem






Problems


• L-classes?

14 / 19


Green relations

Theorem






Problems


• L-classes?

15 / 19


Representation theory of M(W )

Corollary

M(W ) admits |W | simple modules / indecomposable projectivemodules

Problem

Dimension of simple and indecomposable projective modules?

15 / 19


Representation theory of M(W )

Corollary

M(W ) admits |W | simple modules / indecomposable projectivemodules

Problem

Dimension of simple and indecomposable projective modules?

16 / 19


The “Borel” submonoid M1

Definition

Submonoid M1 := {f ∈ M, f (1) = 1}

Properties

• Weakly increasing and contracting on Bruhat =⇒ J -trivial

• Idempotents: (ew )w∈W

• Generated by ew for w grassmanian, e.g. atom for (W ,∨L)

• |W | simple modules of dimension 1

• Semi-simple quotient: monoid algebra of (W ,∨L)

• Conjugacy order among idempotents: <L

• dim Pw = |{f ∈ M1, f (w) ≤L w}| =?

Problem

Inducing these results to M?

16 / 19



Definition

Submonoid M1 := {f ∈ M, f (1) = 1}

Properties







• dim Pw = |{f ∈ M1, f (w) ≤L w}| =?

Problem


16 / 19



Definition

Submonoid M1 := {f ∈ M, f (1) = 1}

Properties







• dim Pw = |{f ∈ M1, f (w) ≤L w}| =?

Problem


17 / 19


R-classes and translation algebras

1234

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

4312

1234

1324

3124 1342

3142

3412

1243

1423

4123 1432

4132

4312

Definition (Translation algebra)

HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R

• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ

• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?

17 / 19



1234

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

4312

1234

1324

3124 1342

3142

3412

1243

1423

4123 1432

4132

4312





17 / 19



1234

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

4312

1234

1324

3124 1342

3142

3412

1243

1423

4123 1432

4132

4312





17 / 19



1234

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

4312

1234

1324

3124 1342

3142

3412

1243

1423

4123 1432

4132

4312





17 / 19



1234

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

4312

1234

1324

3124 1342

3142

3412

1243

1423

4123 1432

4132

4312





17 / 19



1234

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

4312

1234

1324

3124 1342

3142

3412

1243

1423

4123 1432

4132

4312





17 / 19



1234

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

4312

1234

1324

3124 1342

3142

3412

1243

1423

4123 1432

4132

4312





18 / 19


Work in progress

• L-classes? Projective modules? Cartan Matrix?

• Generalization to R-trivial and aperiodic monoids(collaboration with Denton and Berg, Bergeron, Saliola)

• Fast implementation in Sage(interface with Semigroupe, ...)

19 / 19


Sage-Combinat meeting tonight

Sage’s mission:

“To create a viable high-quality and open-source alternativeto MapleTM, MathematicaTM, MagmaTM, and MATLABTM”

...“and to foster a friendly community of users and developers”

Tonight, Thurston Hall, Room 236

• 7pm-8pm: Introduction to Sage and Sage-Combinat

• 8pm-10pm: Help on installation & getting startedBring your laptop!

• Design discussions