Triangle presentations and tilting modules for $\text{SL}_{2k+1}$

TL;DR

This paper constructs new fiber functors on categories related to SL_{n} using combinatorial triangle presentations on finite projective geometries, linking geometric structures to representation theory.

Contribution

It introduces a method to derive fiber functors on Web categories for SL_{n} from triangle presentations, expanding the understanding of tilting modules in characteristic p.

Findings

New fiber functors for SL_{n} categories constructed

Connection between triangle presentations and tilting modules established

Results applicable over fields with characteristic p ≥ n-1

Abstract

Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of a locally finite building of type $\tilde{\text{A}}_{n-1}$ ($n\ge3$). From a type $\tilde{\text{A}}_{n-1}$ triangle presentation on a geometry of order $q$, we construct a fiber functor on the diagrammatic monoidal category $\text{Web}(\text{SL}^{-}_{n})$ over any field $\mathbb{k}$ with characteristic $p\ge n-1$ such that $q \equiv 1$ mod $p$. When $\mathbb{k}$ is algebraically closed and $n$ odd, this gives new fiber functors on the category of tilting modules for $\text{SL}_{n}$.