Triple crystal action in Fock spaces

TL;DR

This paper explicitly constructs a triple crystal structure on higher level Fock spaces, revealing new combinatorial and algebraic insights into the representation theory of cyclotomic rational Cherednik algebras.

Contribution

It introduces a new combinatorial indexing of basis elements and defines a Heisenberg crystal that commutes with two affine quantum group crystals, advancing understanding of related algebraic structures.

Findings

New basis indexing makes quantum group crystals commute

Defined a Heisenberg crystal commuting with quantum group crystals

Explicit labeling of finite-dimensional simple modules in Cherednik algebras

Abstract

We make explicit a triple crystal structure on higher level Fock spaces, by investigating at the combinatorial level the actions of two affine quantum groups and of a Heisenberg algebra. To this end, we first determine a new indexation of the basis elements that makes the two quantum group crystals commute. Then, we define a so-called Heisenberg crystal, commuting with the other two. This gives new information about the representation theory of cyclotomic rational Cherednik algebras, relying on some recent results of Shan and Vasserot and of Losev. In particular, we give an explicit labelling of their finite-dimensional simple modules.