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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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Text of Sorting monoids on Coxeter groups - Bucknell linux. pm040/Slides/Schilling.pdf Sorting monoids on...

  • 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

  • 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

    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