# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.02169