Young tableaux and representations of Hecke algebras of type ADE

TL;DR

This paper introduces new affine Hecke algebras of type ADE, constructs their irreducible representations using generalized Young tableaux, and provides combinatorial models for finite Hecke algebra representations.

Contribution

It generalizes the combinatorial framework of Young tableaux to affine Hecke algebras of type ADE, offering explicit construction of irreducible representations.

Findings

Construction of irreducible calibrated representations

Description of the calibrated spectrum

Development of combinatorial models for finite Hecke algebra representations

Abstract

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.