For any finite Coxeter group
,
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
. The
construction of the biHecke monoid relies on the usual combinatorial model for the
-Hecke
algebra
,
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
simple
and projective modules. In order to construct the simple modules, we introduce for each
a combinatorial
module
whose
support is the interval
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