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

摘要

In this paper we construct an "abstract Fock space" for general Lie typesthat serves as a generalisation of the infinite wedge $q$-Fock space familiarin type $A$. Specifically, for each positive integer $ell$, we define a$mathbb{Z}[q,q^{-1}]$-module $mathcal{F}_{ell}$ with bar involution byspecifying generators and "straightening relations" adapted from thoseappearing in the Kashiwara-Miwa-Stern formulation of the $q$-Fock space. Byrelating $mathcal{F}_{ell}$ to the corresponding affine Hecke algebra we showthat the abstract Fock space has standard and canonical bases for which thetransition matrix produces parabolic affine Kazhdan-Lusztig polynomials. Thisproperty and the convenient combinatorial labeling of bases of$mathcal{F}_{ell}$ by dominant integral weights makes $mathcal{F}_{ell}$ auseful combinatorial tool for determining decomposition numbers of Weyl modulesfor quantum groups at roots of unity.