Vol. 7, No. 3, 2013

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The biHecke monoid of a finite Coxeter group and its representations

Florent Hivert, Anne Schilling and Nicolas Thiéry

Vol. 7 (2013), No. 3, 595–671
Abstract

For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0$-Hecke algebra ${H}_{0}\left(W\right)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits $|W|$ simple and projective modules. In order to construct the simple modules, we introduce for each $w\in W$ a combinatorial module ${T}_{w}$ whose support is the interval ${\left[1,w\right]}_{R}$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.

Keywords
Coxeter groups, Hecke algebras, representation theory, blocks of permutation matrices
Mathematical Subject Classification 2010
Primary: 20M30, 20F55
Secondary: 06D75, 16G99, 20C08
Milestones
Received: 8 June 2011
Revised: 20 February 2012
Accepted: 4 April 2012
Published: 23 August 2013
Authors
 Florent Hivert Laboratoire de Recherche en Informatique (UMR CNRS 8623) Université Paris-Sud 11 91405 Orsay cedex France Anne Schilling Department of Mathematics University of California One Shields Avenue Davis, CA 95616-8633 United States Nicolas Thiéry Laboratoire de Mathématiques d’Orsay Université Paris-Sud 11 91405 Orsay cedex France Laboratoire de Recherche en Informatique (UMR CNRS 8623) Université Paris-Sud 11 91405 Orsay cedex France