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.

A few extra words

We discuss a machine that “takes dualities and produces diagrammatic presentations of the related representation theoretical categories”. Specifically, we feed the machine with super q-Howe duality between the superalgebra \(\textbf{U}_q(\mathfrak{gl}(m|n))\) and \(\textbf{U}_q(\mathfrak{gl}_N)\).
We construct diagrammatic presentations of an idempotented form of the Iwahori-Hecke algebra as well as of categories of \(\textbf{U}_q(\mathfrak{gl}_N)\)-modules by using the “green-red” web categories \(\infty\text{-}\!\textbf{Web}_{\mathrm{gr}}\) and \(N\text{-}\!\textbf{Web}_{\mathrm{gr}}\). Morphisms in these \(\mathbb{C}_q\)-linear categories are combinations of planar, upwards oriented, trivalent graphs with edges labeled by positive integers and colored black, green or red modulo local relations.
An example of a green-red web is:

A very similar approach works for the corresponding categories of \(\mathfrak{gl}_{N|M}\)-modules as well as we show in an extra section.