Geodesic
Non-Elliptic Webs and Convex Sets in the Affine Building
Documenta mathematica, Tome 25 (2020), pp. 2413-2443
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We describe the sl3​ 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 sl3​ 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 sld​", Preprint, arXiv:1811.08884].
DOI : 10.4171/dm/802
Classification : 05E10, 14M15, 51E24, 52B55
Mots-clés : convexity, affine building, affine Grassmannian, Kuperberg webs 
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     author = {Tair Akhmejanov},
     title = {Non-Elliptic {Webs} and {Convex} {Sets} in the {Affine} {Building}},
     journal = {Documenta mathematica},
     pages = {2413--2443},
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Tair Akhmejanov. Non-Elliptic Webs and Convex Sets in the Affine Building. Documenta mathematica, Tome 25 (2020), pp. 2413-2443. doi: 10.4171/dm/802

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