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

TL;DR

This paper constructs a module category structure on vector bundles over vertices of a locally finite A_{n-1} building, linking it to A_{n} web categories and generalizing known fiber functors in positive characteristic.

Contribution

It introduces a new module category framework for vector bundles over A_{n-1} buildings, connecting geometric structures with quantum algebraic categories in positive characteristic.

Findings

Module categories are equivariant under building symmetries.

Recovers Jones' fiber functors when a group acts transitively.

Provides a q-analogue of Rep(SL_{n}) actions on vector bundles.

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 $Rep(SL_{n})$ action on vector bundles over the $sl_n$ 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.