Parallelotope tilings and $q$-decomposition numbers

TL;DR

This paper derives explicit formulas for certain canonical basis vectors in the Fock space representation of quantum affine algebras, using geometric structures called parallelotope tilings, and relates these to quiver subgraphs in representation theory.

Contribution

It introduces a novel geometric approach using parallelotope tilings to compute canonical basis vectors and connects these to quiver subgraphs in the context of $v$-Schur algebras.

Findings

Closed formulas for a large subset of canonical basis vectors.

Polytopal complexes formed by parallelotope tilings.

Descriptions of Ext^1-quivers via polytopal complexes.

Abstract

We provide closed formulas for a large subset of the canonical basis vectors of the Fock space representation of $U_q(\widehat{\mathfrak{sl}}_e)$. These formulas arise from parallelotopes which assemble to form polytopal complexes. The subgraphs of the $\mathrm{Ext}^1$-quivers of $v$-Schur algebras at complex $e$-th roots of unity generated by simple modules corresponding to these canonical basis vectors are given by the $1$-skeletons of the polytopal complexes.