# Presentations of categories of modules using the Cautis–Kamnitzer–Morrison principle

### Giulian Wiggins

University of Sydney, Australia

## Abstract

We study commuting actions of a complex reductive Lie algebra, $g$, and an algebra $A$ on a finite dimensional vector space $V$. We outline how such commuting actions can be used to give a presentation of the category of $A$-modules isomorphic to $g$-weight spaces of $V$. This generalizes a result of Cautis–Kamnitzer–Morrison (2014) in the non-quantum setting. As examples, we apply this to classical Schur–Weyl duality and the Brauer Schur–Weyl dualities. In particular we obtain a diagrammatic presentation, in terms of generators and relations, of the category of permutation modules of the symmetric group, $S_{d}$. This is used to give a new description of the Kronecker product of the $S_{d}$-equivariant morphisms between permutation modules.

## Cite this article

Giulian Wiggins, Presentations of categories of modules using the Cautis–Kamnitzer–Morrison principle. J. Comb. Algebra 3 (2019), no. 1, pp. 71–112

DOI 10.4171/JCA/27