Standard multipartitions and a combinatorial affine Schur-Weyl duality

TL;DR

This paper introduces standard multipartitions and establishes a combinatorial affine Schur-Weyl duality, linking multipartitions to irreducible representations of affine Hecke algebras and quantum loop algebras.

Contribution

It defines standard multipartitions and extends the affine Schur-Weyl duality to a combinatorial framework connecting multipartitions with algebraic representations.

Findings

One-to-one correspondence between standard multipartitions and irreducible representations of affine Hecke algebra.

Extension of the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras.

Combinatorial description of affine Schur-Weyl duality via multipartition residues.

Abstract

We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur--Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations completely determine all irreducible polynomial representations for the quantum loop algebra. We will see, in particular, the notion of standard multipartitions gives rise to a combinatorial description of the affine Schur--Weyl duality in terms of a column-reading vs. row reading of residues of a multipartition.