the 0-rook monoid€¦ · jo el gay joint work with florent hivert universit e paris-sud, lri &...

Symmetric Group and Rook Monoid The 0-Rook Monoid Representation theory Work in Progress

The 0-Rook Monoid

Joel Gayjoint work with Florent Hivert

Universite Paris-Sud, LRI & Ecole Polytechnique, LIX

Seminaire Lotharingien de Combinatoire - March 28, 2017

Joel Gay The 0-Rook Monoid

Contents

1 Symmetric Group and Rook Monoid

2 The 0-Rook Monoid

3 Representation theory

4 Work in Progress

q=1←− Hn(q)q=0−→ H0

Symmetric Group

Iwahori-Hecke Hecke monoid atalgebra q = 0

s2i = 1

si+1si si+1 = si si+1si

si sj = sjsi

T 2i = q1 + (q − 1)Ti

Ti+1TiTi+1 = TiTi+1Ti

TiTj = TjTi

π2i = πi

πi+1πiπi+1 = πiπi+1πi

πiπj = πjπiy x

si = Ti πi = Ti + 1

Snq=1←− Hn(q)

q=0−→ H0n

Symmetric Group Iwahori-Hecke

Hecke monoid at

algebra

s2i = 1

si sj = sjsi

T 2i = q1 + (q − 1)Ti

TiTj = TjTi

π2i = πi

πiπj = πjπiy x

Snq=1←− Hn(q)

q=0−→ H0n

Symmetric Group Iwahori-Hecke Hecke monoid atalgebra q = 0

s2i = 1

si sj = sjsi

T 2i = q1 + (q − 1)Ti

TiTj = TjTi

π2i = πi

πiπj = πjπiy x

Snq=1←− Hn(q)

q=0−→ H0n

s2i = 1

si sj = sjsi

T 2i = q1 + (q − 1)Ti

TiTj = TjTi

π2i = πi

πiπj = πjπi

Snq=1←− Hn(q)

q=0−→ H0n

s2i = 1

si sj = sjsi

T 2i = q1 + (q − 1)Ti

TiTj = TjTi

π2i = πi

πiπj = πjπi

Snq=1←− Hn(q)

q=0−→ H0n

s2i = 1

si sj = sjsi

T 2i = q1 + (q − 1)Ti

TiTj = TjTi

π2i = πi

πiπj = πjπiy x

Interesting properties of H0n

|H0n | = n!

H0n acts on Sn (bubble sort):

π5 · 3726145 = 3726415π5 · 3726415 = 3726415

An element of H0n is characterized

by its action on the identity.

Its simple and projective modulesare well-known and combinatorial[Norton-Carter].

The induction and restriction ofmodules gives us a structure oftower of monoids, linked to QSymand NCSF [Krob-Thibon].

|H0n | = n!

π5 · 3726145 = 3726415π5 · 3726415 = 3726415

|H0n | = n!

π5 · 3726145 = 3726415π5 · 3726415 = 3726415

23142143

14231342

13241243

3412 3241

3214 3142

π1π2π3

|H0n | = n!

π5 · 3726145 = 3726415π5 · 3726415 = 3726415

23142143

14231342

13241243

3412 3241

3214 3142

π1π2π3

|H0n | = n!

π5 · 3726145 = 3726415π5 · 3726415 = 3726415

23142143

14231342

13241243

3412 3241

3214 3142

π1π2π3

|H0n | = n!

π5 · 3726145 = 3726415π5 · 3726415 = 3726415

23142143

14231342

13241243

3412 3241

3214 3142

π1π2π3

The rook monoid

Rook matrix of size n = set of non attacking rooks on an n × nmatrix.

The product of two rook matrices is a rook matrix.Rook Monoid Rn = submonoid of the rook matricesMn ⊃ Rn ⊃ Sn

The rook monoid

Rook Matrix

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0 0

0 0 0 00 0 0 0 00 0 0 00 0 0 00 0 0 0 0

Rook Vector 0 4 2 3 1 0 3 0 4 1

The rook monoid

Rook Matrix

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0 0

0 0 0 00 0 0 0 00 0 0 00 0 0 00 0 0 0 0

Rook Vector 0 4 2 3 1 0 3 0 4 1

The rook monoid

Rook Matrix

0 0 0 00 0 0 00 0 0 00 0 0 0

0 0 0 0

0 0 0 00 0 0 00 0 0 00 0 0 0

0 0 0 0

Rook Vector 05 4 2 3 1 05 3 02 4 1

The rook monoid

Rook Matrix

0 0 0 00 0 0 00 0 0 00 0 0 0

0 0 0 0

0 0 0 00 0 0 00 0 0 00 0 0 0

0 0 0 0

Rook Vector 05 4 2 3 1 05 3 02 4 1

q=1←− Hn(q)q=0−→ H0

↪→ ↪→ ↪→

Rnq=1←− In(q)

q=0−→Iwahori , Solomon.

Snq=1←− Hn(q)

q=0−→ H0n

↪→ ↪→ ↪→

Rnq=1←− In(q)

q=0−→

Iwahori

, Solomon.

Snq=1←− Hn(q)

q=0−→ H0n

↪→ ↪→ ↪→

Rnq=1←− In(q)

q=0−→

Iwahori

, Solomon.

Snq=1←− Hn(q)

q=0−→ H0n

↪→

↪→ ↪→

q=1←− In(q)q=0−→

Iwahori

, Solomon.

Snq=1←− Hn(q)

q=0−→ H0n

↪→ ↪→

↪→

Rnq=1←− In(q)

q=0−→

Iwahori , Solomon.

Snq=1←− Hn(q)

q=0−→ H0n

↪→ ↪→ ↪→

Rnq=1←− In(q)

q=0−→ ??

Iwahori , Solomon.

Snq=1←− Hn(q)

q=0−→ H0n

↪→ ↪→ ↪→

Rnq=1←− In(q)

q=0−→ R0n

Iwahori , Solomon.

Contents

2 The 0-Rook Monoid

4 Work in Progress

Definition by right action on Rn

Operators π0, π1, . . . πn−1 acting on rook vectors

Bubble sort operators π1, . . . , πn−1 :

(r1 . . . rn) · πi =

{r1 . . . ri−1ri+1ri ri+2 . . . rn if ri < ri+1,

r1 . . . rn otherwise,

Deletion operator π0 :(r1 . . . rn) · π0 = 0r2 . . . rn.

45321 · π1 = 5432140321 · π1 = 4032100321 · π2 = 03021

3027006 · π0 = 0027006

(r1 . . . rn) · πi =

45321 · π1 = 5432140321 · π1 = 4032100321 · π2 = 03021

3027006 · π0 = 0027006

(r1 . . . rn) · πi =

45321 · π1 = 5432140321 · π1 = 4032100321 · π2 = 03021

3027006 · π0 = 0027006

(r1 . . . rn) · πi =

45321 · π1 = 5432140321 · π1 = 4032100321 · π2 = 03021

3027006 · π0 = 0027006

Definition by presentation

Generators : π0, . . . , πn−1

Relations :

π2i = πi 0 ≤ i ≤ n − 1,

πiπi+1πi = πi+1πiπi+1 1 ≤ i ≤ n − 2,

πiπj = πjπi |i − j | > 1.

π1π0π1π0 = π0π1π0 = π0π1π0π1

Definition by presentation

Generators : π0, . . . , πn−1

Relations :

π2i = πi 0 ≤ i ≤ n − 1,

πiπi+1πi = πi+1πiπi+1 1 ≤ i ≤ n − 2,

πiπj = πjπi |i − j | > 1.

π1π0π1π0 = π0π1π0 = π0π1π0π1

Theorem

Both definitions (presentation and action on Rn) are equivalent.

Key Fact :

Theorem

The map f :

∣∣∣∣ R0n −→ Rn

r 7−→ 1n · r is a bijection.

This also gives us canonical reduced expression of elements of R0n .

Theorem

Key Fact :

Theorem

The map f :

∣∣∣∣ R0n −→ Rn

Theorem

Key Fact :

Theorem

The map f :

∣∣∣∣ R0n −→ Rn

Canonical reduced expression

Example : using coset R05/R

↓ π430154↓ π3

30514↓ π2

35014↓ π1

53014↓ π0

03014↓ π1

30014 π2

30145↓ π4

↓ π330514↓ π2

35014↓ π1

53014↓ π0

03014↓ π1

30014 π2

30145↓ π4

30154↓ π3

↓ π235014↓ π1

53014↓ π0

03014↓ π1

30014 π2

30145↓ π4

30154↓ π3

30514↓ π2

↓ π153014↓ π0

03014↓ π1

30014 π2

30145↓ π4

30154↓ π3

30514↓ π2

35014↓ π1

↓ π003014↓ π1

30014 π2

30145↓ π4

30154↓ π3

30514↓ π2

35014↓ π1

53014↓ π0

↓ π130014 π2

30145↓ π4

30154↓ π3

30514↓ π2

35014↓ π1

53014↓ π0

03014↓ π1

30145↓ π4

30154↓ π3

30514↓ π2

35014↓ π1

53014↓ π0

03014↓ π1

30014 π2

Example : 30240

Index the zeros by the missing letters in decreasing order : 3052401

12345 15012345 · π0201345 · π1320145 · π2π1324015 · π33052401 · π4π3π2π1π0π1

Conclusion : 15 · [π0 · π1 · π2π1 · π3 · π4π3π2π1π0π1] = 30240.

Example : 30240

12345 15

012345 · π0201345 · π1320145 · π2π1324015 · π33052401 · π4π3π2π1π0π1

Example : 30240

12345 15012345 · π0

201345 · π1320145 · π2π1324015 · π33052401 · π4π3π2π1π0π1

Example : 30240

12345 15012345 · π0201345 · π1

320145 · π2π1324015 · π33052401 · π4π3π2π1π0π1

Example : 30240

12345 15012345 · π0201345 · π1320145 · π2π1

324015 · π33052401 · π4π3π2π1π0π1

Example : 30240

12345 15012345 · π0201345 · π1320145 · π2π1324015 · π3

3052401 · π4π3π2π1π0π1

Example : 30240

12345 15012345 · π0201345 · π1320145 · π2π1324015 · π33052401 · π4π3π2π1π0π1

Example : 30240

12345 15012345 · π0201345 · π1320145 · π2π1324015 · π33052401 · π4π3π2π1π0π1

J -triviality

Definition (Green)

Let M a monoid, x , y ∈ M. We say that x ≤J y iff MxM ⊆ MyM.Equivalence relation : xJ y iff MxM = MyM.

Definition

A monoid is J -trivial if its J -classes are trivial.Equivalently, its bisided Cayley graph has no cycle except loops.

Example : H0n

J -triviality : H0n right Cayley graph

23142143

1423 1342

13241243

3412 3241

3214 3142

J -triviality : H0n left Cayley graph

23142143

1423 1342

13241243

3412 3241

3214 3142

J -triviality : H0n bisided Cayley graph

23142143

1423 1342

13241243

3412 3241

3214 3142

J -triviality : R0n bisided Cayley graph

· π0 : right· π1 : right· π2 : right

π0 · : leftπ1 · : leftπ2 · : left

Contents

2 The 0-Rook Monoid

4 Work in Progress

Simple modules

Theorem

R0n is J -trivial.

Corollary (Application of Denton-Hivert-Schilling-Thiery)

R0n has 2n idempotents.

It has thus 2n simple modules of dimension 1.

Simple modules

Theorem

R0n is J -trivial.

Corollary (Application of Denton-Hivert-Schilling-Thiery)

R0n has 2n idempotents.

It has thus 2n simple modules of dimension 1.

Descent set

Definition

For π ∈ R0n , we define its right R-descent set by

DR(π) = {0 ≤ i ≤ n − 1 |ππi = π}.

Example : positions

Let r = 0423007. 0 < 4 ≥ 2 < 3 ≥ 0 ≥ 0 < 7.

DR(r) = {0, 2, 4, 5} Notation : 0 42 3

Warning : 00

and not 0 0 .

Descent set

Definition

DR(π) = {0 ≤ i ≤ n − 1 |ππi = π}.

Example : positions

Let r = 0423007. 0 < 4 ≥ 2 < 3 ≥ 0 ≥ 0 < 7.

DR(r) = {0, 2, 4, 5}

Notation : 0 42 3

Warning : 00

and not 0 0 .

Descent set

Definition

DR(π) = {0 ≤ i ≤ n − 1 |ππi = π}.

Example : positions

Let r = 0423007. 0 < 4 ≥ 2 < 3 ≥ 0 ≥ 0 < 7.

DR(r) = {0, 2, 4, 5} Notation : 0 42 3

Warning : 00

and not 0 0 .

Descent set

Definition

DR(π) = {0 ≤ i ≤ n − 1 |ππi = π}.

Example : positions

Let r = 0423007. 0 < 4 ≥ 2 < 3 ≥ 0 ≥ 0 < 7.

DR(r) = {0, 2, 4, 5} Notation : 0 42 3

Warning : 00

and not 0 0 .

Descent class

List of the R-descent types for R04 :

{0} {1} {2} {3}

{0, 1}

{0, 2}

{0, 3}

{1, 2} {1, 3} {2, 3}

{0, 1, 2}

{0, 1, 3}

{0, 2, 3} {1, 2, 3}

{0, 1, 2, 3}

Projective modules

Theorem (Application of Denton-Hivert-Schilling-Thiery)

The projective indecomposable R0n -modules are indexed by the

R-descent type and isomorphic to the quotient of the associatedR-descent class by the finer R-descent class.

0023 0014

00300021

02100300

04120403

04020203

03120104

Projectivity over H0n

Theorem

The indecomposable projective R0n -module splits as a H0

n -moduleas the direct sum of all the indecomposable projective H0

n -moduleswhose descent classes are explicit.

Proof : explicit decomposition0 = 0 + 0

= + + + + + = + + + 2 + .

0 = 0 + 00 =

= + + + + + = + + + 2 + .

04130214

04020203

0 = 0 + 00 =

= + + + + + = + + + 2 + .

04130214

04020203

0 = 0 + 00 =

= + + + + + = + + + 2 + .

24133214

31424123

Contents

2 The 0-Rook Monoid

4 Work in Progress

R0n is a lattice (analogous to permutohedron)

Tower of monoids : induction and restriction (linked to QSymand NCSF, work of Krob and Thibon)

Renner Monoids (generalization for other Cartan types)

THANK YOU FOR YOUROUTSTANDINGATTENTION!!

Descent classes are not intervals

21032004

41024003

the 0-rook monoid€¦ · jo el gay joint work with florent hivert universit e paris-sud, lri &...

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