A Fock space model for decomposition numbers for quantum groups at roots of unity

TL;DR

This paper introduces an abstract Fock space model for general Lie types, connecting it to affine Hecke algebras and Kazhdan-Lusztig polynomials, to aid in computing decomposition numbers for quantum groups at roots of unity.

Contribution

It constructs a new combinatorial Fock space framework for all Lie types, generalizing type A, and links it to affine Hecke algebras and Kazhdan-Lusztig theory.

Findings

Defines a new Fock space with bar involution and straightening relations.

Establishes the connection between the Fock space bases and Kazhdan-Lusztig polynomials.

Provides a combinatorial tool for calculating decomposition numbers of Weyl modules.

Abstract

In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge $q$-Fock space familiar in type $A$. Specifically, for each positive integer $\ell$, we define a $\mathbb{Z}[q,q^{-1}]$-module $\mathcal{F}_{\ell}$ with bar involution by specifying generators and "straightening relations" adapted from those appearing in the Kashiwara-Miwa-Stern formulation of the $q$-Fock space. By relating $\mathcal{F}_{\ell}$ to the corresponding affine Hecke algebra we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan-Lusztig polynomials. This property and the convenient combinatorial labeling of bases of $\mathcal{F}_{\ell}$ by dominant integral weights makes $\mathcal{F}_{\ell}$ a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.