Discrete Mathematics & Theoretical Computer Science |

- 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

The Hecke group algebra $\operatorname{H} \mathring{W}$ of a finite Coxeter group $\mathring{W}$, as introduced by the first and last author, is obtained from $\mathring{W}$ by gluing appropriately its $0$-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $\mathring{W}$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $\operatorname{H} \mathring{W}$ is the natural quotient of the affine Hecke algebra $\operatorname{H}(W)(q)$ through its level $0$ representation. The proof relies on the following core combinatorial result: at level $0$ the $0$-Hecke algebra acts transitively on $\mathring{W}$. Equivalently, in type $A$, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level $0$ representation is a calibrated principal series representation $M(t)$ for a suitable choice of character $t$, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical $0$-Hecke algebra and that of the affine Hecke algebra at this specialization.

Source: HAL:hal-01185155v1

Volume: DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)

Section: Proceedings

Published on: January 1, 2008

Imported on: May 10, 2017

Keywords: Coxeter groups,(affine) Weyl groups,(affine) Hecke algebras,[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: 0652641*FRG: Collaborative Research: Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects*; Funder: National Science Foundation; Code: 0652652*Combinatorial Aspects of Representation Theory, Mathematical Physics and q-Series*; Funder: National Science Foundation; Code: 0501101

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