Categorification of tensor powers of the vector representation of Uq(gl(1|1))

Abstract

We consider the monoidal subcategory of finite-dimensional representations of U_q(gl(1|1)) generated by the vector representation, and we provide a graphical calculus for the intertwining operators, which enables to compute explicitly the canonical basis, as well as the action of U_q(gl(1|1)). We construct a categorification using graded subquotient categories of the BGG category O(gl_n) and graded functors between them (translation, Zuckermann’s and coapproximation functors). We describe then the regular blocks of these categories as modules over explicit diagram algebras, which are defined using Soergel modules and combinatorics of symmetric polynomials. We construct diagrammatically standard and proper standard modules for the properly stratified structure of these algebras.