Affine sl_p controls the representation theory of the symmetric group and related Hecke algebras

TL;DR

This paper establishes a deep connection between the representation theory of affine Hecke algebras and integrable highest weight modules of affine Lie algebras, introducing a p-canonical basis with remarkable properties.

Contribution

It provides a new parametrization of representations via crystal graphs and introduces the p-canonical basis with positivity properties, extending classical bases to positive characteristic.

Findings

Parameterization of algebra representations by crystal graph nodes

Introduction of the p-canonical basis with positivity properties

Equivalence of the p-canonical basis to the usual basis when p=0

Abstract

In this paper we prove theorems that describe how the representation theory of the affine Hecke algebra of type A and of related algebras such as the group algebra of the symmetric group are controlled by integrable highest weight representations of the characteristic zero affine Lie algebra \hat{sl}_l. In particular we parameterise the representations of these algebras by the nodes of the crystal graph, and give various Hecke theoretic descriptions of the edges. As a consequence we find for each prime p a basis of the integrable representations of \hat{sl}_l which shares many of the remarkable properties, such as positivity, of the global crystal basis/canonical basis of Lusztig and Kashiwara. This {\it $p$-canonical basis} is the usual one when p = 0, and the crystal of the p-canonical basis is always the usual one. The paper is self-contained, and our techniques are elementary (no perverse sheaves or algebraic geometry is invoked).