A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$

TL;DR

This paper develops a categorification of the representation category of quantum superalgebra U_q(gl(m|n)) using foam theory, connecting it to known link homology theories and introducing non-local relations.

Contribution

It introduces a new categorification of Rep(gl(m|n)) via foam diagrams, extending existing theories and relating to non-local relations in knot homology.

Findings

Categorifies Rep(gl(m|n)) using foam diagrams.

Recovers sl(m) foams for n=0 case.

Defines a categorification of symmetric powers of the standard representation.

Abstract

Using quantum skew-Howe duality, we study the category $\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powers of the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a categorification of $\operatorname{Rep}(\mathfrak{gl}(m|n))$ using the theory of foams. In the case of $n=0$, we show that we can recover $\mathfrak{sl}(m)$ foams introduced by Queffelec and Rose to define Khovanov-Rozansky $\mathfrak{sl}(m)$ link homology. We also define a categorification of the monoidal category of symmetric powers of the standard representation of $U_q(\mathfrak{gl}(n))$, since this category can be identified with $\operatorname{Rep}(\mathfrak{gl}(0|n))$. The relations on our foams are non-local, since the number of dots that can appear on a facet depends on the position of the dot in the foam, rather than just on its colouring. This may be related to Gilmore's non-local relations in Heegaard Floer knot homology.