Web bases for the Lie superalgebra $\mathfrak{gl}(1|1)$

Abstract

Queffelec--Sartori and Tubbenhauer--Vaz--Wedrich give a diagrammatic web category via generators and relations to describe the subcategory of Rep($\mathfrak{gl}(m|n)$) generated by tensor products of exterior powers of the defining representation. We construct the first explicit bases for homomorphism spaces in the web category for $\mathfrak{gl}(1|1)$, giving an alternative diagrammatic presentation of this web category, and explore connections to the combinatorics of planar networks. Our basis comes equipped with an algorithm to decompose any web into a linear combination of basis webs, giving an effective way to compute in the $\mathfrak{gl}(1|1)$-skein theory.