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2013 The biHecke monoid of a finite Coxeter group and its representations
Florent Hivert, Anne Schilling, Nicolas Thiéry
Algebra Number Theory 7(3): 595-671 (2013). DOI: 10.2140/ant.2013.7.595


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 H0(W), 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 wW a combinatorial module Tw whose support is the interval [1,w]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.


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Florent Hivert. Anne Schilling. Nicolas Thiéry. "The biHecke monoid of a finite Coxeter group and its representations." Algebra Number Theory 7 (3) 595 - 671, 2013.


Received: 8 June 2011; Revised: 20 February 2012; Accepted: 4 April 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1305.20071
MathSciNet: MR3095222
Digital Object Identifier: 10.2140/ant.2013.7.595

Primary: 20F55 , 20M30
Secondary: 06D75 , 16G99 , 20C08

Keywords: blocks of permutation matrices , Coxeter groups , Hecke algebras , representation theory

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.7 • No. 3 • 2013
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