The $\mathfrak{sl}_\infty$-crystal combinatorics of higher level Fock spaces

TL;DR

This paper develops a combinatorial rule for the $rak{sl}_ty$-crystal structure on higher level Fock spaces, enabling precise analysis of charged partitions and applications to Cherednik algebra representations.

Contribution

It introduces a new combinatorial method to compute crystal arrows in higher level Fock spaces, linking crystal theory with Cherednik algebra representations.

Findings

Computed the support of spherical Cherednik algebra representations.

Characterized parameters for finite-dimensional Cherednik algebra modules.

Provided an abacus-based description of finite-dimensional type B Cherednik modules.

Abstract

For integers $e,\ell\geq 2$, the level $\ell$ Fock space has an $\mathfrak{sl}_\infty$-crystal structure arising from the action of a Heisenberg algebra, intertwining the $\widehat{\mathfrak{sl}_e}$-crystal. The vertices of these crystals are charged $\ell$-partitions. We give the combinatorial rule for computing the arrows anywhere in the $\mathfrak{sl}_\infty$-crystal. This allows us to pinpoint the location of any charged $\ell$-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of parameters such that it is finite-dimensional. We also give an easy abacus characterization of all finite-dimensional representations of type $B$ Cherednik algebras.