Florent Hivert ; Anne Schilling ; Nicolas M. Thiéry - The biHecke monoid of a finite Coxeter group

dmtcs:2851 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2851
The biHecke monoid of a finite Coxeter groupArticle

Authors: Florent Hivert ORCID1; Anne Schilling 2; Nicolas M. Thiéry ORCID3

  • 1 Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes
  • 2 Department of Mathematics [Univ California Davis]
  • 3 Laboratoire de Mathématiques d'Orsay

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 $H_0(W)$, 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 $w∈W$ a combinatorial module $T_w$ 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.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: Coxeter groups,Hecke algebras,representation theory,blocks of permutation matrices,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • FRG: Collaborative Research: Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects; Funder: National Science Foundation; Code: 0652652
  • FRG: Collaborative Research: Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects; Funder: National Science Foundation; Code: 0652641

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