Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

TL;DR

This paper explores the structure of crystal graphs in Fock space representations of affine type A, revealing explicit isomorphisms using combinatorial algorithms like Schensted's bumping, with implications for representation theory.

Contribution

It introduces a new crystal isomorphism acting on cylindric multipartitions, connecting classical combinatorics with crystal graph structures in affine type A.

Findings

Explicit description of a canonical crystal isomorphism

Connection between crystal graphs and combinatorial bumping algorithms

Characterization of vertices in Fock space crystal components

Abstract

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space.