Geodesic
Non-Elliptic Webs and Convex Sets in the Affine Building
Documenta mathematica, Tome 25 (2020), pp. 2413-2443.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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. \textit{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 \textit{G. Faltings} [Prog. Math. 195, 157--184 (2001; Zbl 1028.14002)] and are related to tropical convexity, as discussed in work by \textit{M. Joswig} et al. [Albanian J. Math. 1, No. 4, 187--211 (2007; Zbl 1133.52003)] and by \textit{L. Zhang} [``Computing convex hulls in the affine building of $\mathfrak{sl}_d$'', Preprint, \url{arXiv:1811.08884}].
Classification : 05E10, 51E24, 52B55, 14M15
Keywords: Kuperberg webs, affine building, affine Grassmannian, convexity 
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Akhmejanov, Tair. Non-Elliptic Webs and Convex Sets in the Affine Building. Documenta mathematica, Tome 25 (2020), pp. 2413-2443. http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a5/