Non-Elliptic Webs and Convex Sets in the Affine Building

Abstract

We describe the ${\mathfrak{sl}}_3$ non-elliptic webs in terms of convex sets in the affine building. Kuperberg defined the non-elliptic web basis in his work on rank-$2$ spider categories. B. Fontaine et al. [Compos. Math. 149, No. 11, 1871–1912 (2013; Zbl 1304.22016)] showed that the $\mathfrak{s}{l}_3$ non-elliptic webs are dual to CAT(0) triangulated diskoids in the affine building. We show that each such triangulated diskoid is the intersection of the min-convex and max-convex hulls of a generic polygon in the building. Choosing a generic polygon from each of the components of the Satake fiber produces (the duals of) the non-elliptic web basis. The convex hulls in the affine building were first introduced by G. Faltings [Prog. Math. 195, 157–184 (2001; Zbl 1028.14002)] and are related to tropical convexity, as discussed in work by M. Joswig et al. [Albanian J. Math. 1, No. 4, 187–211 (2007; Zbl 1133.52003)] and by L. Zhang ["Computing convex hulls in the affine building of ${\mathfrak{sl}}_d$", Preprint, arXiv:1811.08884].