# Sorting monoids on Coxeter groups - Bucknell linux. pm040/Slides/Schilling.pdf Sorting monoids on Coxeter groups Florent Hivert1 Anne Schilling2 Nicolas M. Thi ery2;3 1LITIS/LIFAR,

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• 1 / 19

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. Thiéry2,3

1LITIS/LIFAR, Université Rouen, France

2University of California at Davis, USA

3Laboratoire de Mathématiques d’Orsay, Université 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

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

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

1243

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

1423

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4123

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4123

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4132

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4312

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4312

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4321

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4321

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4321

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4321

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4321

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: s2i = 1, (s1s2) 3 = 1, (s2s3)

3 = 1, (s1s3) 2 = 1

• 2 / 19

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

Bubble (anti) sort algorithm

4321

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: s2i = 1, (s1s2)

3 = 1, (s2s3) 3 = 1, (s1s3)

2 = 1

• 3 / 19

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

Coxeter groups

Definition (Coxeter group W )

Generators

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