Combinatorial Fock spaces and quantum symmetric pairs

Abstract

A way to construct the natural representation of the quantized affine algebra $U_v({\widehat{\mathfrak{s}\mathfrak{l}}}_{\mathcal{l}})$ is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for $U_{\mathrm{q}}(\mathfrak{s}{\mathfrak{l}}_N)$ were $\mathrm{q}$ is a root of unity to the action of $U_v({\widehat{\mathfrak{s}\mathfrak{l}}}_{\mathcal{l}})$ as N tends toward 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 $U_{\mathrm{q}}(\mathfrak{g})$-modules 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.

Keywords:

• Fock spaces
• quantum groups at roots of unity
• quantum symmetric pairs

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B37
• 20C20

Acknowledgments

We would like to thank Catharina Stroppel and Daniel Tubbenhauer for comments and remarks. We would also like to thank Weiqiang Wang and the referee for corrections and suggestion to improve the content of the paper.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. 12050410261 and the Beijing Natural Science Foundation under Grant No. 1232017.