Webs for Type Q Lie Superalgebras

Abstract

In the first part of this dissertation, we construct a monoidal supercategory whose morphism spaces are spanned by equivalence classes of diagrams called oriented type Q webs, modulo certain relations. We then prove a monoidal superequivalence between this category and the full subcategory of modules over the type Q Lie superalgebra q_n, tensor-generated by the symmetric powers of the natural module and their duals. This affords a diagrammatic presentation by generators and relations of the q_n-morphisms between these modules in terms of webs. The strategy behind the proof is an application of the method of Cautis-Kamnitzer-Morrison to the (q_m,q_n) Howe duality established by Cheng-Wang.

In the second part, we prove a similar result for the so-called spin permutation modules of the Sergeev superalgebra Ser_k, obtaining a diagrammatic description of the Ser_k-morphisms between them in terms of webs. We also develop the combinatorics of supertabloids, and use them to produce a diagrammatic basis for the space of Ser_k-morphisms between any two spin permutation modules in terms of webs.