Quantum Polynomial Functors

TL;DR

This paper develops a quantum analogue of strict polynomial functors, establishing their fundamental properties and applying them to derive quantum duality and invariant theory results.

Contribution

It introduces a category of quantum polynomial functors, extending classical theory, and constructs quantum Schur/Weyl functors to derive duality and invariant results.

Findings

Established structural properties of quantum polynomial functors

Constructed quantum Schur/Weyl functors

Derived quantum $(GL_m,GL_n)$ duality

Abstract

We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this category (duality, projective generators, braiding etc.) in analogy with classical strict polynomial functors. We then apply the work of Hashimoto and Hayashi in this context to construct quantum Schur/Weyl functors, and use this to provide new and easy derivations of quantum $(GL_m,GL_n)$ duality, along with other results in quantum invariant theory.