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

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{A}_{n-1},\ (n \ge 3) . From a type \tilde{A}_{n-1} triangle presentation on a geometry of order q , we construct a fiber functor on the diagrammatic monoidal category \mathrm{Web}(\mathrm{SL}^{-}_{n}) over any field \mathbb{k} with characteristic p\ge n − 1 such that q \equiv 1\ \mathrm{mod}\ p . When \mathbb{k} is algebraically closed and n odd, this gives new fiber functors on the category of tilting modules for \mathrm{SL}_{n} .