Combinatorial Fock spaces and quantum symmetric pairs

TL;DR

This paper extends the concept of deformed Fock spaces and quantum symmetric pairs beyond type A, exploring their structure and representation theory for other classical types at roots of unity.

Contribution

It constructs embeddings of Grothendieck groups into Fock spaces for classical types and defines an affine quantum symmetric pair action, generalizing type A results.

Findings

Embeddings of Grothendieck groups into Fock spaces for classical types.

Definition of an affine quantum symmetric pair action.

Relation of the action to linkage principles and tensor product multiplicities.

Abstract

The natural representation of the quantized affine algebra of type A can be defined via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for a type A quantum group at a root of unity to the action of the quantized affine algebra as the rank tends towards infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional modules for the corresponding quatum group at a root of unity into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra. We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities.