Super $q$-Howe duality and web categories

TL;DR

This paper employs super $q$-Howe duality to develop diagrammatic presentations of certain algebraic categories related to $rak{gl}_N$ and $rak{gl}_{N|M}$, linking representation theory with knot invariants.

Contribution

It introduces a diagrammatic approach to categories of $rak{gl}_N$-modules using super $q$-Howe duality, providing new insights into algebraic structures and knot polynomial symmetries.

Findings

Diagrammatic presentations of Hecke algebra and $rak{gl}_N$-categories.

Representation-theoretic explanation of HOMFLY--PT polynomial symmetry.

Connection between super $q$-Howe duality and knot invariants.

Abstract

We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY--PT polynomials.