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 ${\mathrm{SL}}_n$.