The 0-Rook Monoid Jo el Gay joint work with Florent Hivert e - - PowerPoint PPT Presentation

▶
the 0 rook monoid
SMART_READER_LITE
LIVE PREVIEW

The 0-Rook Monoid Jo el Gay joint work with Florent Hivert e - - PowerPoint PPT Presentation

Jun 01, 2023 385 likes •1.16k views

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress The 0-Rook Monoid Jo el Gay joint work with Florent Hivert e Paris-Sud, LRI & Universit Ecole Polytechnique, LIX S eminaire

slide-1
SLIDE 1

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

The 0-Rook Monoid

Jo¨ el Gay

joint work with Florent Hivert Universit´ e Paris-Sud, LRI & ´ Ecole Polytechnique, LIX

S´ eminaire Lotharingien de Combinatoire - March 28, 2017

Jo¨ el Gay The 0-Rook Monoid

slide-2
SLIDE 2

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

Contents

1 Symmetric Group and Rook Monoid 2 The 0-Rook Monoid 3 Representation theory 4 Work in Progress

Jo¨ el Gay The 0-Rook Monoid

slide-3
SLIDE 3

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

Sn

Symmetric Group

Jo¨ el Gay The 0-Rook Monoid

slide-4
SLIDE 4

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

Sn

q=1

← − Hn(q)

Symmetric Group Iwahori-Hecke algebra

Jo¨ el Gay The 0-Rook Monoid

slide-5
SLIDE 5

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0

Jo¨ el Gay The 0-Rook Monoid

slide-6
SLIDE 6

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0 s2

i = 1

si+1sisi+1 = sisi+1si sisj = sjsi T 2

i = q1 + (q − 1)Ti

Ti+1TiTi+1 = TiTi+1Ti TiTj = TjTi π2

i = πi

πi+1πiπi+1 = πiπi+1πi πiπj = πjπi

Jo¨ el Gay The 0-Rook Monoid

slide-7
SLIDE 7

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0 s2

i = 1

si+1sisi+1 = sisi+1si sisj = sjsi T 2

i = q1 + (q − 1)Ti

Ti+1TiTi+1 = TiTi+1Ti TiTj = TjTi π2

i = πi

πi+1πiπi+1 = πiπi+1πi πiπj = πjπi

Jo¨ el Gay The 0-Rook Monoid

slide-8
SLIDE 8

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0 s2

i = 1

si+1sisi+1 = sisi+1si sisj = sjsi T 2

i = q1 + (q − 1)Ti

Ti+1TiTi+1 = TiTi+1Ti TiTj = TjTi π2

i = πi

πi+1πiπi+1 = πiπi+1πi πiπj = πjπi

  • si = Ti

πi = Ti + 1

Jo¨ el Gay The 0-Rook Monoid

slide-9
SLIDE 9

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

Interesting properties of H0

n

|H0

n| = n!

Jo¨ el Gay The 0-Rook Monoid

slide-10
SLIDE 10

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

Interesting properties of H0

n

|H0

n| = n!

H0

n acts on Sn (bubble sort):

π5 · 3726145 = 3726415 π5 · 3726415 = 3726415

Jo¨ el Gay The 0-Rook Monoid

slide-11
SLIDE 11

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

Interesting properties of H0

n

|H0

n| = n!

H0

n acts on Sn (bubble sort):

π5 · 3726145 = 3726415 π5 · 3726415 = 3726415 An element of H0

n is characterized

by its action on the identity.

4312 4321 4213 4231 2341 2314 2143 2134 1432 1423 1342 1324 1243 1234 4132 4123 3421 3412 3241 3214 3142 3124 2431 2413

π1 π2 π3 Jo¨ el Gay The 0-Rook Monoid

slide-12
SLIDE 12

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

Interesting properties of H0

n

|H0

n| = n!

H0

n acts on Sn (bubble sort):

π5 · 3726145 = 3726415 π5 · 3726415 = 3726415 An element of H0

n is characterized

by its action on the identity.

4312 4321 4213 4231 2341 2314 2143 2134 1432 1423 1342 1324 1243 1234 4132 4123 3421 3412 3241 3214 3142 3124 2431 2413

π1 π2 π3 Jo¨ el Gay The 0-Rook Monoid

slide-13
SLIDE 13

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

Interesting properties of H0

n

|H0

n| = n!

H0

n acts on Sn (bubble sort):

π5 · 3726145 = 3726415 π5 · 3726415 = 3726415 An element of H0

n is characterized

by its action on the identity. Its simple and projective modules are well-known and combinatorial [Norton-Carter].

4312 4321 4213 4231 2341 2314 2143 2134 1432 1423 1342 1324 1243 1234 4132 4123 3421 3412 3241 3214 3142 3124 2431 2413

π1 π2 π3 Jo¨ el Gay The 0-Rook Monoid

slide-14
SLIDE 14

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

Interesting properties of H0

n

|H0

n| = n!

H0

n acts on Sn (bubble sort):

π5 · 3726145 = 3726415 π5 · 3726415 = 3726415 An element of H0

n is characterized

by its action on the identity. Its simple and projective modules are well-known and combinatorial [Norton-Carter]. The induction and restriction of modules gives us a structure of tower of monoids, linked to QSym and NCSF [Krob-Thibon].

4312 4321 4213 4231 2341 2314 2143 2134 1432 1423 1342 1324 1243 1234 4132 4123 3421 3412 3241 3214 3142 3124 2431 2413

π1 π2 π3 Jo¨ el Gay The 0-Rook Monoid

slide-15
SLIDE 15

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

The rook monoid

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

Jo¨ el Gay The 0-Rook Monoid

slide-16
SLIDE 16

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

The rook monoid

Rook matrix of size n = set of non attacking rooks on an n × n matrix. Rook Matrix                        

Jo¨ el Gay The 0-Rook Monoid

slide-17
SLIDE 17

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

The rook monoid

Rook matrix of size n = set of non attacking rooks on an n × n matrix. Rook Matrix                         Rook Vector 4 2 3 1 3 4 1

Jo¨ el Gay The 0-Rook Monoid

slide-18
SLIDE 18

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

The rook monoid

Rook matrix of size n = set of non attacking rooks on an n × n matrix. Rook Matrix                         Rook Vector 05 4 2 3 1 05 3 02 4 1

Jo¨ el Gay The 0-Rook Monoid

slide-19
SLIDE 19

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

The rook monoid

Rook matrix of size n = set of non attacking rooks on an n × n matrix. Rook Matrix                         Rook Vector 05 4 2 3 1 05 3 02 4 1 The product of two rook matrices is a rook matrix. Rook Monoid Rn = submonoid of the rook matrices Mn ⊃ Rn ⊃ Sn

Jo¨ el Gay The 0-Rook Monoid

slide-20
SLIDE 20

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

Sn

Jo¨ el Gay The 0-Rook Monoid

slide-21
SLIDE 21

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

Sn

q=1

← − Hn(q) Iwahori

Jo¨ el Gay The 0-Rook Monoid

slide-22
SLIDE 22

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

Iwahori

Jo¨ el Gay The 0-Rook Monoid

slide-23
SLIDE 23

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

֒ → Rn Iwahori

Jo¨ el Gay The 0-Rook Monoid

slide-24
SLIDE 24

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

֒ → ֒ → Rn

q=1

← − In(q) Iwahori , Solomon.

Jo¨ el Gay The 0-Rook Monoid

slide-25
SLIDE 25

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

֒ → ֒ → ֒ → Rn

q=1

← − In(q)

q=0

− → ?? Iwahori , Solomon.

Jo¨ el Gay The 0-Rook Monoid

slide-26
SLIDE 26

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

Sn

q=1

← − Hn(q)

q=0

− → H0

n

֒ → ֒ → ֒ → Rn

q=1

← − In(q)

q=0

− → R0

n

Iwahori , Solomon.

Jo¨ el Gay The 0-Rook Monoid

slide-27
SLIDE 27

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

Contents

1 Symmetric Group and Rook Monoid 2 The 0-Rook Monoid 3 Representation theory 4 Work in Progress

Jo¨ el Gay The 0-Rook Monoid

slide-28
SLIDE 28

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

Definition by right action on Rn

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

Jo¨ el Gay The 0-Rook Monoid

slide-29
SLIDE 29

Symmetric Group and Rook Monoid The 0-Rook Monoid Representation theory 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+1riri+2 . . . rn

if ri < ri+1, r1 . . . rn

  • therwise,

Deletion operator π0 :

(r1 . . . rn) · π0 = 0r2 . . . rn.

Jo¨ el Gay The 0-Rook Monoid

slide-30
SLIDE 30

Symmetric Group and Rook Monoid The 0-Rook Monoid Representation theory 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+1riri+2 . . . rn

if ri < ri+1, r1 . . . rn

  • therwise,

Deletion operator π0 :

(r1 . . . rn) · π0 = 0r2 . . . rn.

45321 · π1 = 54321 40321 · π1 = 40321 00321 · π2 = 03021 3027006 · π0 = 0027006

Jo¨ el Gay The 0-Rook Monoid

slide-31
SLIDE 31

Symmetric Group and Rook Monoid The 0-Rook Monoid Representation theory 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+1riri+2 . . . rn

if ri < ri+1, r1 . . . rn

  • therwise,

Deletion operator π0 :

(r1 . . . rn) · π0 = 0r2 . . . rn.

45321 · π1 = 54321 40321 · π1 = 40321 00321 · π2 = 03021 3027006 · π0 = 0027006

21 20 12 10 02 01 00

Jo¨ el Gay The 0-Rook Monoid

slide-32
SLIDE 32

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

Definition by presentation

Generators : π0, . . . , πn−1 Relations : π2

i = πi

0 ≤ i ≤ n − 1, πiπi+1πi = πi+1πiπi+1 1 ≤ i ≤ n − 2, πiπj = πjπi |i − j| > 1. π1π0π1π0 = π0π1π0 = π0π1π0π1

Jo¨ el Gay The 0-Rook Monoid

slide-33
SLIDE 33

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

Definition by presentation

Generators : π0, . . . , πn−1 Relations : π2

i = πi

0 ≤ i ≤ n − 1, πiπi+1πi = πi+1πiπi+1 1 ≤ i ≤ n − 2, πiπj = πjπi |i − j| > 1. π1π0π1π0 = π0π1π0 = π0π1π0π1

21 20 12 10 02 01 00

Jo¨ el Gay The 0-Rook Monoid

slide-34
SLIDE 34

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

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

Jo¨ el Gay The 0-Rook Monoid

slide-35
SLIDE 35

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

Theorem Both definitions (presentation and action on Rn) are equivalent. Key Fact : Theorem The map f :

  • R0

n

− → Rn r − → 1n · r is a bijection.

Jo¨ el Gay The 0-Rook Monoid

slide-36
SLIDE 36

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

Theorem Both definitions (presentation and action on Rn) are equivalent. Key Fact : Theorem The map f :

  • R0

n

− → Rn r − → 1n · r is a bijection. This also gives us canonical reduced expression of elements of R0

n.

Jo¨ el Gay The 0-Rook Monoid

slide-37
SLIDE 37

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145

Jo¨ el Gay The 0-Rook Monoid

slide-38
SLIDE 38

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145 ↓ π4 30154

Jo¨ el Gay The 0-Rook Monoid

slide-39
SLIDE 39

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145 ↓ π4 30154 ↓ π3 30514

Jo¨ el Gay The 0-Rook Monoid

slide-40
SLIDE 40

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145 ↓ π4 30154 ↓ π3 30514 ↓ π2 35014

Jo¨ el Gay The 0-Rook Monoid

slide-41
SLIDE 41

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145 ↓ π4 30154 ↓ π3 30514 ↓ π2 35014 ↓ π1 53014

Jo¨ el Gay The 0-Rook Monoid

slide-42
SLIDE 42

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145 ↓ π4 30154 ↓ π3 30514 ↓ π2 35014 ↓ π1 53014 ↓ π0 03014

Jo¨ el Gay The 0-Rook Monoid

slide-43
SLIDE 43

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145 ↓ π4 30154 ↓ π3 30514 ↓ π2 35014 ↓ π1 53014 ↓ π0 03014 ↓ π1 30014

Jo¨ el Gay The 0-Rook Monoid

slide-44
SLIDE 44

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

Canonical reduced expression

Example : using coset R0

5/R0 4

30145 ↓ π4 30154 ↓ π3 30514 ↓ π2 35014 ↓ π1 53014 ↓ π0 03014 ↓ π1 30014 π2

Jo¨ el Gay The 0-Rook Monoid

slide-45
SLIDE 45

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

Canonical reduced expression

Example : 30240

Jo¨ el Gay The 0-Rook Monoid

slide-46
SLIDE 46

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

Canonical reduced expression

Example : 30240 Index the zeros by the missing letters in decreasing order : 3052401 12345 15

Jo¨ el Gay The 0-Rook Monoid

slide-47
SLIDE 47

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

Canonical reduced expression

Example : 30240 Index the zeros by the missing letters in decreasing order : 3052401 12345 15 012345 · π0

Jo¨ el Gay The 0-Rook Monoid

slide-48
SLIDE 48

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

Canonical reduced expression

Example : 30240 Index the zeros by the missing letters in decreasing order : 3052401 12345 15 012345 · π0 201345 · π1

Jo¨ el Gay The 0-Rook Monoid

slide-49
SLIDE 49

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

Canonical reduced expression

Example : 30240 Index the zeros by the missing letters in decreasing order : 3052401 12345 15 012345 · π0 201345 · π1 320145 · π2π1

Jo¨ el Gay The 0-Rook Monoid

slide-50
SLIDE 50

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

Canonical reduced expression

Example : 30240 Index the zeros by the missing letters in decreasing order : 3052401 12345 15 012345 · π0 201345 · π1 320145 · π2π1 324015 · π3

Jo¨ el Gay The 0-Rook Monoid

slide-51
SLIDE 51

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

Canonical reduced expression

Example : 30240 Index the zeros by the missing letters in decreasing order : 3052401 12345 15 012345 · π0 201345 · π1 320145 · π2π1 324015 · π3 3052401 · π4π3π2π1π0π1

Jo¨ el Gay The 0-Rook Monoid

slide-52
SLIDE 52

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

Canonical reduced expression

Example : 30240 Index the zeros by the missing letters in decreasing order : 3052401 12345 15 012345 · π0 201345 · π1 320145 · π2π1 324015 · π3 3052401 · π4π3π2π1π0π1 Conclusion : 15 · [π0 · π1 · π2π1 · π3 · π4π3π2π1π0π1] = 30240.

Jo¨ el Gay The 0-Rook Monoid

slide-53
SLIDE 53

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

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 : H0

n

Jo¨ el Gay The 0-Rook Monoid

slide-54
SLIDE 54

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

J -triviality : H0

n right Cayley graph

4312 4321 4213 4231 2341 2314 2143 2134 1432 1423 1342 1324 1243 1234 4132 4123 3421 3412 3241 3214 3142 3124 2431 2413

Jo¨ el Gay The 0-Rook Monoid

slide-55
SLIDE 55

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

J -triviality : H0

n left Cayley graph

4312 4321 4213 4231 2341 2314 2143 2134 1432 1423 1342 1324 1243 1234 4132 4123 3421 3412 3241 3214 3142 3124 2431 2413

Jo¨ el Gay The 0-Rook Monoid

slide-56
SLIDE 56

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

J -triviality : H0

n bisided Cayley graph

4312 4321 4213 4231 2341 2314 2143 2134 1432 1423 1342 1324 1243 1234 4132 4123 3421 3412 3241 3214 3142 3124 2431 2413

Jo¨ el Gay The 0-Rook Monoid

slide-57
SLIDE 57

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

J -triviality : R0

n bisided Cayley graph

102 231 300 213 230 203 210 200 201 301 302 023 021 020 013 012 010 003 002 001 000 132 130 312 310 103 320 100 032 031 030 123 120 321

· π0 ✿ r✐❣❤t · π1 ✿ r✐❣❤t · π2 ✿ r✐❣❤t π0 · ✿ ❧❡❢t π1 · ✿ ❧❡❢t π2 · ✿ ❧❡❢t Jo¨ el Gay The 0-Rook Monoid

slide-58
SLIDE 58

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

Contents

1 Symmetric Group and Rook Monoid 2 The 0-Rook Monoid 3 Representation theory 4 Work in Progress

Jo¨ el Gay The 0-Rook Monoid

slide-59
SLIDE 59

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

Simple modules

Theorem R0

n is J -trivial.

Jo¨ el Gay The 0-Rook Monoid

slide-60
SLIDE 60

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

Simple modules

Theorem R0

n is J -trivial.

Corollary (Application of Denton-Hivert-Schilling-Thi´ ery) R0

n has 2n idempotents.

It has thus 2n simple modules of dimension 1.

Jo¨ el Gay The 0-Rook Monoid

slide-61
SLIDE 61

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

Descent set

Definition For π ∈ R0

n, we define its right R-descent set by

DR(π) = {0 ≤ i ≤ n − 1 | ππi = π}.

Jo¨ el Gay The 0-Rook Monoid

slide-62
SLIDE 62

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

Descent set

Definition For π ∈ R0

n, 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}

Jo¨ el Gay The 0-Rook Monoid

slide-63
SLIDE 63

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

Descent set

Definition For π ∈ R0

n, 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 4 2 3 0 7

Jo¨ el Gay The 0-Rook Monoid

slide-64
SLIDE 64

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

Descent set

Definition For π ∈ R0

n, 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 4 2 3 0 7 Warning : and not 0 0 .

Jo¨ el Gay The 0-Rook Monoid

slide-65
SLIDE 65

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

Descent class

List of the R-descent types for R0

4:

{} {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}

Jo¨ el Gay The 0-Rook Monoid

slide-66
SLIDE 66

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

Projective modules

Theorem (Application of Denton-Hivert-Schilling-Thi´ ery) The projective indecomposable R0

n-modules are indexed by the

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

0034 0024 0023 0014 0013 0012 0043 0042 0041 0040 0032 0031 0030 0021 0020 0010 0210 0300 0420 0410 0400 0321 0320 0310 0432 0431 0430 0100 0200 0421 0401 0301 0302 0423 0413 0412 0403 0402 0203 0324 0314 0312 0104 0214 0201 0103 0213 0304 0204 0102

Jo¨ el Gay The 0-Rook Monoid

slide-67
SLIDE 67

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

Projectivity over H0

n

Theorem The indecomposable projective R0

n-module splits as a H0 n-module

as the direct sum of all the indecomposable projective H0

n-modules

whose descent classes are explicit. Proof : explicit decomposition

= + 0 = + = + + + + + = + + + 2 + .

Jo¨ el Gay The 0-Rook Monoid

slide-68
SLIDE 68

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

= + 0 = + = + + + + + = + + + 2 + .

0324 0104 0401 0301 0423 0302 0413 0214 0403 0402 0203 0201 0314 0103 0304 0412 0312 0213 0102 0204

Jo¨ el Gay The 0-Rook Monoid

slide-69
SLIDE 69

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

= + 0 = + = + + + + + = + + + 2 + .

0324 0104 0401 0301 0423 0302 0413 0214 0403 0402 0203 0201 0314 0103 0304 0412 0312 0213 0102 0204

Jo¨ el Gay The 0-Rook Monoid

slide-70
SLIDE 70

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

= + 0 = + = + + + + + = + + + 2 + .

1324 3214 3241 4231 1423 4132 2413 3214 2143 3142 4123 4321 2314 4213 2134 3412 4312 4213 4312 3124

Jo¨ el Gay The 0-Rook Monoid

slide-71
SLIDE 71

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

Contents

1 Symmetric Group and Rook Monoid 2 The 0-Rook Monoid 3 Representation theory 4 Work in Progress

Jo¨ el Gay The 0-Rook Monoid

slide-72
SLIDE 72

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

R0

n is a lattice (analogous to permutohedron)

Jo¨ el Gay The 0-Rook Monoid

slide-73
SLIDE 73

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

R0

n is a lattice (analogous to permutohedron)

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

Jo¨ el Gay The 0-Rook Monoid

slide-74
SLIDE 74

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

R0

n is a lattice (analogous to permutohedron)

Tower of monoids : induction and restriction (linked to QSym and NCSF, work of Krob and Thibon) Renner Monoids (generalization for other Cartan types)

Jo¨ el Gay The 0-Rook Monoid

slide-75
SLIDE 75

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

THANK YOU FOR YOUR OUTSTANDING ATTENTION!!

Jo¨ el Gay The 0-Rook Monoid

slide-76
SLIDE 76

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

Descent classes are not intervals

4312 4301 4302 4203 4213 4103 4201 3002 3001 2104 2103 2004 2003 2001 1004 1003 1002 4102 4003 4002 4001 3214 3204 3201 3104 3102 3004

Jo¨ el Gay The 0-Rook Monoid