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

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 ${\tilde{\text{A}}}_{n-1}$ ($n\ge 3$). 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 ${\text{SL}}_n$.

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