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### Sorting monoids and algebras on Coxeter groups

**Algebra & Discrete Mathematics**

Speaker: | Nicolas M. Thiéry, Université Paris |

Location: | 2112 MSB |

Start time: | Fri, Jun 5 2009, 1:10PM |

This is joint work with Anne Schilling and Florent Hivert.

The usual combinatorial model for the 0-Hecke algebra H_n(0) (in type A) is to consider the algebra (or monoid) generated by the bubble sort operators pi_1,...,pi_{n-1}, where pi_i acts on words of length n and sorts the letters in positions i and i+1. This construction generalizes naturally to any Coxeter group.

By combining several variants of those operators (sorting, antisorting, affine) we construct several monoids and algebras. Astonishingly, they are endowed with very rich structures which relate to the combinatorics of descents and of several partial orders (such as Bruhat and left-right weak orders). These structures can be explained by numerous connections with representation theory, and in particular with affine Hecke algebras, and symmetric functions.

In the student run seminar on Thursday, we will leisurely explore one of the examples, namely the bi-Hecke monoid, and recall some prerequisites on Coxeter groups along the way. In this seminar we will present the overall picture.

While the focus will be on the combinatorics of the problem, we will show how our research was driven by the algebraic background together with computer exploration of examples by mean of the MuPAD-Combinat and Sage-Combinat software.