Mathematics Colloquia and Seminars

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Description

(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 finite Coxeter groups. 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. While the focus of this talk will be on the combinatorial nature of the problem, we will show how our research was driven by this algebraic background together with computer exploration of examples by mean of the MuPAD-Combinat and Sage-Combinat software.