Journal of Combinatorial Algebra

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Volume 1, Issue 4, 2017, pp. 371–423
DOI: 10.4171/JCA/1-4-2

Published online: 2017-10-09

Young tableaux and representations of Hecke algebras of type ADE

Loïc Poulain d'Andecy[1]

(1) Université de Reims Champagne-Ardenne, France

We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of type ADE. Representations are constructed by explicit formulas, in a seminormal form.

Keywords: Hecke algebras, Weyl groups, simply-laced root systems, affine Hecke algebras, Jucys–Murphy elements, skew partitions, Young tableaux, seminormal representations, calibrated representations

Poulain d'Andecy Loïc: Young tableaux and representations of Hecke algebras of type ADE. J. Comb. Algebra 1 (2017), 371-423. doi: 10.4171/JCA/1-4-2