In this talk, we will explain a (mysterious?) connection between the combinatorics of affine buildings and representation theory in type A. If a group acts simply and transitively on the vertices of an affine building in type A_n, it gives rise to a certain combinatorial structure called a triangle presentation. We will describe how triangle presentations can be used to build exotic fiber functors on the category of tilting modules for SL_n (for n odd) in positive characteristic.

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