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A tropical view on Bruhat-Tits buildings and their compactifications. (English) Zbl 1227.20031

Author’s summary: We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If \(G\) is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of \(G\) and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of \(G\). We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character of the representation.

MSC:

20E42 Groups with a \(BN\)-pair; buildings
14T05 Tropical geometry (MSC2010)
20G25 Linear algebraic groups over local fields and their integers
51E24 Buildings and the geometry of diagrams
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[1] Akian M., Bapat R., Gaubert S., Max-plus algebra, In: Handbook of Linear Algebra, Discrete Math. Appl. (Boca Raton), Chapman and Hall, Boca Raton, 2007, #25; · Zbl 0922.15001
[2] Bruhat F., Tits J., Groupes réductifs sur un corps local: I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math., 1972, 41, 5-251 http://dx.doi.org/10.1007/BF02715544; · Zbl 0254.14017
[3] Bruhat F., Tits J., Groupes réductifs sur un corps local: II. Schémas en groups. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math., 1984, 60, 197-376 http://dx.doi.org/10.1007/BF02700560; · Zbl 0597.14041
[4] Bruhat F., Tits J., Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France, 1984, 112(2), 259-301; · Zbl 0565.14028
[5] Einsiedler M., Kapranov M., Lind D., Non-archimedean amoebas and tropical varieties, J. Reine Angew. Math., 2006, 601, 139-157; · Zbl 1115.14051
[6] Goldman O., Iwahori N., Thespace of p-adic norms, Acta Math., 1963, 109(1), 137-177 http://dx.doi.org/10.1007/BF02391811; · Zbl 0133.29402
[7] Green J.A., Polynomial Representations of GL n, Lecture Notes in Math., 830, Springer, Berlin-New York, 1980;
[8] Joswig M., Tropical convex hull computations, In: Tropical and Idempotent Mathematics, Contemp. Math., 495, AMS, Providence, 2009, 193-212; · Zbl 1202.52004
[9] Joswig M., Sturmfels B., Yu J., Affine buildings and tropical convexity, Albanian J. Math., 2007, 1(4), 187-211; · Zbl 1133.52003
[10] Landvogt E., A Compactification of the Bruhat-Tits Building, Lecture Notes in Math., 1619, Springer, Berlin, 1996; · Zbl 0935.20034
[11] Landvogt E., Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math., 2000, 518, 213-241; · Zbl 0937.20026
[12] Rémy B., Thuillier A., Werner A., Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings, Ann. Sci. Éc. Norm. Sup., 2010, 43(3), 461-554; · Zbl 1198.51006
[13] Rémy B., Thuillier A., Werner A., Bruhat-Tits theory from Berkovich’s point of view. II. Satake compactifications of buildings, J. Inst. Math. Jussieu (in press); · Zbl 1241.51003
[14] Werner A., Compactification of the Bruhat-Tits building of PGL by lattices of smaller rank, Doc. Math., 2001, 6, 315-342; · Zbl 1048.20014
[15] Werner A., Compactifications of Bruhat-Tits buildings associated to linear representations, Proc. Lond. Math. Soc., 2007, 95(2), 497-518 http://dx.doi.org/10.1112/plms/pdm019; · Zbl 1131.20019
[16] Ziegler G. M., Lectures on Polytopes, Grad. Texts in Math., 152, Springer, New York, 2007; · Zbl 0823.52002
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