Representations of A-type Hecke algebras

TL;DR

This paper reviews the representation theory of A-type Hecke algebras, adapting methods from symmetric groups, and provides explicit constructions of idempotents and their q-dimensions.

Contribution

It introduces an explicit construction of idempotents in Hecke algebras using Jucys-Murphy elements and discusses Ocneanu's traces related to quantum groups.

Findings

Explicit idempotent constructions in Hecke algebras

Connection between traces and quantum group dimensions

Adaptation of symmetric group methods to Hecke algebras

Abstract

We review some facts about the representation theory of the Hecke algebra. We adapt for the Hecke algebra case the approach of Okounkov and Vershik which was developed for the representation theory of symmetric groups. We justify an explicit construction of the idempotents in the Hecke algebra in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented.