sorting monoids on coxeter groups - bucknell...

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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

Underlying combinatorics: right permutahedron

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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

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

2 / 19

213 132

312231

2134 1324 1243

2314 3124 2143 1342 1423

2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132

3421 4231 4312

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 · · ·︸︷︷︸

= sjsi · · ·︸︷︷︸mi,j

, for i 6= j

Reduced words

4 / 19

0-Hecke monoid

213 132

312231

213 132

312231

π1 π2

π2 π1

π1 π2

213 132

312231

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19

0-Hecke monoid

213 132

312231

213 132

312231

π1 π2

π2 π1

π1 π2

213 132

312231

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19

0-Hecke monoid

213 132

312231

213 132

312231

π1 π2

π2 π1

π1 π2

213 132

312231

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19

0-Hecke monoid

213 132

312231

213 132

312231

π1 π2

π2 π1

π1 π2

213 132

312231

π1 π2

π2 π1

π1 π2

S3 H0(S3) H0(S3)

4 / 19

0-Hecke monoid

213 132

312231

213 132

312231

π1 π2

π2 π1

π1 π2

213 132

312231

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

213 132

312231

213 132

312231

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

213 132

312231

213 132

312231

Left order Left-Right order

Right order

Bruhat order

Suffix Factor Prefix Subword

6 / 19

213 132

312231

213 132

312231

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

213 132

312231

213 132

312231

Left order Left-Right order

Right order

Bruhat order

Suffix Factor

Prefix

Subword

6 / 19

213 132

312231

213 132

312231

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

213 132

312231

213 132

312231

Left order

Left-Right order

Right order

Bruhat order

Suffix

Factor

Prefix

Subword

6 / 19

213 132

312231

213 132

312231

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

213 132

312231

213 132

312231

Left order Left-Right order Right order

Bruhat order

Suffix Factor Prefix

Subword

6 / 19

213 132

312231

213 132

312231

s1×s1 s2×s2

×s2 ×s1s2× s1×

s1×s2× ×s1 ×s2

213 132

312231

213 132

312231

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

••

7 / 19

• Example: w := 36475812

••

7 / 19

• Example: w := 36475812

••

7 / 19

• Example: w := 36475812

••

7 / 19

• Example: w := 36475812

••

8 / 19

The cutting poset

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

u v w if u = wJ with J block

••

•••

••••

•••

••

•••

••••

••

•••

Theorem

• Intervals are lattices

• Mobius function: inclusion-exclusion along minimal blocks

• Meet-semi lattice?

8 / 19

The cutting poset

••

•••

••••

•••

••

•••

••••

••

•••

Theorem

8 / 19

The cutting poset

••

•••

••••

•••

••

•••

••••

••

•••

Theorem

8 / 19

The cutting poset

••

•••

••••

•••

••

•••

••••

••

•••

Theorem

8 / 19

The cutting poset

••

•••

••••

•••

••

•••

••••

••

•••

Theorem

• Intervals are (distributive?) lattices

8 / 19

The cutting poset

••

•••

••••

•••

••

•••

••••

••

•••

Theorem

• Intervals are (distributive?) lattices

9 / 19

The Big Picture

Hζ(W )

Hζ(Sn) ∧

H−1(W )

Hq(W )

Hq(Sn) ∧

H0(W )〈π1, π2, . . . 〉

H0(W )〈π0π1, π2, . . . 〉

H0(W )〈π1, π2, . . . 〉

H0(W )〈π0π1, π2, . . . 〉

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

Q[π0π1, π2, . . . ]

W〈s1, s2, . . . 〉

W〈s0s1, s2, . . . 〉

Sn ∧

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 )| =∑

dim Sw . dim Pw

10 / 19

The biHecke monoid

Question

Theorem (HST08)

• Why do we care?

|M(W )| =∑

dim Sw . dim Pw

10 / 19

The biHecke monoid

Question

Theorem (HST08)

• Why do we care?

|M(W )| =∑

dim Sw . dim Pw

10 / 19

The biHecke monoid

Question

Theorem (HST08)

• Why do we care?

|M(W )| =∑

dim Sw . dim Pw

10 / 19

The biHecke monoid

Question

Theorem (HST08)

• Why do we care?

|M(W )| =∑

dim Sw . dim Pw

10 / 19

The biHecke monoid

Question

Theorem (HST08)

• Why do we care?

|M(W )| =∑

dim Sw . dim Pw

10 / 19

The biHecke monoid

Question

Theorem (HST08)

• Why do we care?

|M(W )| =∑

dim Sw . dim Pw

11 / 19

Key combinatorial lemma

. 7−→

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

Proof.

Exchange property / associativity

11 / 19

. ••.

. •.

. 7−→

Proof.

11 / 19

. ••.

. •.

. 7−→

. .•.

. .•

Proof.

11 / 19

. ••.

. •.

. 7−→

. .•.

. .•

Proof.

11 / 19

. ••.

. •.

. 7−→

. .•.

. .•

Proof.

11 / 19

. ••.

. •.

. 7−→

. .•.

. .•

Proof.

11 / 19

. ••.

. •.

. 7−→

. .•.

. .•

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

. ••.

. •.

. 7−→

. .•.

. .•

Corollary

11 / 19

. ••.

. •.

. 7−→

. .•.

. .•

Corollary

11 / 19

. ••.

. •.

. 7−→

. .•.

. .•

Corollary

11 / 19

. 7−→

. .•.

. .•

Corollary

12 / 19

Some elements of the monoid1

. 7→

.•.• .

.• .•

. 7→

.•• .

•. .

. 7→

. .•.

. .•

. 7→

•. .

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

12 / 19

. 7→

.•.• .

.• .•

. 7→

.•• .

•. .

. 7→

. .•.

. .•

. 7→

•. .

12 / 19

. 7→

.•.• .

.• .•

. 7→

.•• .

•. .

. 7→

. .•.

. .•

. 7→

•. .

12 / 19

. 7→

.•.• .

.• .•

. 7→

.•• .

•. .

. 7→

. .•.

. .•

. 7→

•. .

12 / 19

. 7→

.•.• .

.• .•

. 7→

.•• .

•. .

. 7→

. .•.

. .•

. 7→

•. .

13 / 19

Green relations

2×2×1

2×21×1

×2×1

×1 ×2

×1×2

2×2 2×

••

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

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

3124 1342

4123 1432

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

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

3124 1342

4123 1432

17 / 19

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

3124 1342

4123 1432

17 / 19

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

3124 1342

4123 1432

17 / 19

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

3124 1342

4123 1432

17 / 19

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

3124 1342

4123 1432

17 / 19

1324 1243

3124 1342 1423

3142 1432 4123

3412 4132

3124 1342

4123 1432

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

sorting monoids on coxeter groups - bucknell...

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