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

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 Lotharingien de Combinatoire - March 28, 2017 Jo¨ el Gay The 0 -Rook Monoid

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

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress S n Symmetric Group Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 ← − H n ( q ) S n Symmetric Group Iwahori-Hecke algebra Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0 Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0 s 2 T 2 π 2 i = 1 i = q 1 + ( q − 1) T i i = π i s i +1 s i s i +1 = s i s i +1 s i T i +1 T i T i +1 = T i T i +1 T i π i +1 π i π i +1 = π i π i +1 π i s i s j = s j s i T i T j = T j T i π i π j = π j π i Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0 s 2 T 2 π 2 i = 1 i = q 1 + ( q − 1) T i i = π i s i +1 s i s i +1 = s i s i +1 s i T i +1 T i T i +1 = T i T i +1 T i π i +1 π i π i +1 = π i π i +1 π i s i s j = s j s i T i T j = T j T i π i π j = π j π i Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n Symmetric Group Iwahori-Hecke Hecke monoid at algebra q = 0 s 2 T 2 π 2 i = 1 i = q 1 + ( q − 1) T i i = π i s i +1 s i s i +1 = s i s i +1 s i T i +1 T i T i +1 = T i T i +1 T i π i +1 π i π i +1 = π i π i +1 π i s i s j = s j s i T i T j = T j T i π i π j = π j π i � � s i = T i π i = T i + 1 Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress Interesting properties of H 0 n | H 0 n | = n ! Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress Interesting properties of H 0 n | H 0 n | = n ! H 0 n acts on S n (bubble sort): π 5 · 3726145 = 3726415 π 5 · 3726415 = 3726415 Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress Interesting properties of H 0 n | H 0 n | = n ! 1234 H 0 n acts on S n (bubble sort): 2134 1243 1324 π 5 · 3726145 = 3726415 π 5 · 3726415 = 3726415 3124 2143 2314 1342 1423 An element of H 0 n is characterized 4123 3214 3142 2413 2341 1432 by its action on the identity. 4213 4132 3412 3241 2431 4312 4231 3421 π 1 π 2 π 3 4321 Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress Interesting properties of H 0 n | H 0 n | = n ! 1234 H 0 n acts on S n (bubble sort): 2134 1243 1324 π 5 · 3726145 = 3726415 π 5 · 3726415 = 3726415 3124 2143 2314 1342 1423 An element of H 0 n is characterized 4123 3214 3142 2413 2341 1432 by its action on the identity. 4213 4132 3412 3241 2431 4312 4231 3421 π 1 π 2 π 3 4321 Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress Interesting properties of H 0 n | H 0 n | = n ! 1234 H 0 n acts on S n (bubble sort): 2134 1243 1324 π 5 · 3726145 = 3726415 π 5 · 3726415 = 3726415 3124 2143 2314 1342 1423 An element of H 0 n is characterized 4123 3214 3142 2413 2341 1432 by its action on the identity. Its simple and projective modules 4213 4132 3412 3241 2431 are well-known and combinatorial 4312 4231 3421 [Norton-Carter]. π 1 π 2 π 3 4321 Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress Interesting properties of H 0 n | H 0 n | = n ! 1234 H 0 n acts on S n (bubble sort): 2134 1243 1324 π 5 · 3726145 = 3726415 π 5 · 3726415 = 3726415 3124 2143 2314 1342 1423 An element of H 0 n is characterized 4123 3214 3142 2413 2341 1432 by its action on the identity. Its simple and projective modules 4213 4132 3412 3241 2431 are well-known and combinatorial 4312 4231 3421 [Norton-Carter]. π 1 π 2 The induction and restriction of π 3 4321 modules gives us a structure of tower of monoids, linked to QSym and NCSF [Krob-Thibon]. Jo¨ el Gay The 0 -Rook Monoid

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

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.  0 0 0 0   0 0 0 0  0 0 0 0 0 0 0 0 0         Rook Matrix 0 0 0 0 0 0 0 0         0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 Jo¨ el Gay The 0 -Rook Monoid

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.  0 0 0 0   0 0 0 0  0 0 0 0 0 0 0 0 0         Rook Matrix 0 0 0 0 0 0 0 0         0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 Rook Vector 0 4 2 3 1 0 3 0 4 1 Jo¨ el Gay The 0 -Rook Monoid

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.  0 0 0 0   0 0 0 0  0 0 0 0 0 0 0 0         Rook Matrix 0 0 0 0 0 0 0 0         0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 Rook Vector 0 5 4 2 3 1 0 5 3 0 2 4 1 Jo¨ el Gay The 0 -Rook Monoid

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.  0 0 0 0   0 0 0 0  0 0 0 0 0 0 0 0         Rook Matrix 0 0 0 0 0 0 0 0         0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 Rook Vector 0 5 4 2 3 1 0 5 3 0 2 4 1 The product of two rook matrices is a rook matrix. Rook Monoid R n = submonoid of the rook matrices M n ⊃ R n ⊃ S n Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress S n Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 ← − H n ( q ) S n Iwahori Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n Iwahori Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n ֒ → R n Iwahori Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n ֒ ֒ → → q =1 ← − I n ( q ) R n Iwahori , Solomon. Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n ֒ ֒ ֒ → → → q =1 q =0 ← − I n ( q ) − → ?? R n Iwahori , Solomon. Jo¨ el Gay The 0 -Rook Monoid

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress q =1 q =0 → H 0 ← − H n ( q ) − S n n ֒ ֒ ֒ → → → q =1 q =0 → R 0 ← − I n ( q ) − R n n Iwahori , Solomon. Jo¨ el Gay The 0 -Rook Monoid

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

Symmetric Group and Rook Monoid The 0 -Rook Monoid Representation theory Work in Progress Definition by right action on R n Operators π 0 , π 1 , . . . π n − 1 acting on rook vectors Jo¨ el Gay The 0 -Rook Monoid