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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 |
Description
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.