Module categories for $A_n$ web categories from ${\tilde{A}}_{n-1}$-buildings

Abstract

We equip the category of vector bundles over the vertices of a locally finite ${\tilde{A}}_{n-1}$ building $\Delta$ with the structure of a module category over a category of type $A_n$ webs in positive characteristic. This module category is a $q$-analogue of the $Rep(S{L}_n)$ action on vector bundles over the $s{l}_n$ weight lattice. We show our module categories are equivariant with respect to symmetries of the building, and when a group $G$ acts simply transitively on the vertices of $\Delta$ this recovers the fiber functors constructed by Jones.

arXiv comment29 pages. Author contact: [email protected]