Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues

TL;DR

This paper proves that one-dimensional sums in classical affine types decompose into sums related to affine type A when the rank is large, linking them to parabolic Lusztig q-analogues.

Contribution

It establishes a decomposition of one-dimensional sums in classical affine types into type A sums for large rank, confirming a conjecture and connecting to Lusztig q-analogues.

Findings

Decomposition of sums in classical affine types into type A sums for large rank

Proof of a conjecture by Zabrocki and the third author

Connection between one-dimensional sums and Lusztig q-analogues

Abstract

This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of the third author and Zabrocki by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of the root system considered is sufficiently large. As a consequence, any one-dimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.