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2011 | 9 | 2 | 390-402
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A tropical view on Bruhat-Tits buildings and their compactifications

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EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
9
Numer
2
Strony
390-402
Opis fizyczny
Daty
wydano
2011-04-01
online
2011-02-18
Twórcy
Bibliografia
  • [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
  • [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
  • [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
  • [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
  • [5] Einsiedler M., Kapranov M., Lind D., Non-archimedean amoebas and tropical varieties, J. Reine Angew. Math., 2006, 601, 139–157
  • [6] Goldman O., Iwahori N., Thespace of p-adic norms, Acta Math., 1963, 109(1), 137–177 http://dx.doi.org/10.1007/BF02391811
  • [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
  • [9] Joswig M., Sturmfels B., Yu J., Affine buildings and tropical convexity, Albanian J. Math., 2007, 1(4), 187–211
  • [10] Landvogt E., A Compactification of the Bruhat-Tits Building, Lecture Notes in Math., 1619, Springer, Berlin, 1996
  • [11] Landvogt E., Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math., 2000, 518, 213–241
  • [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
  • [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)
  • [14] Werner A., Compactification of the Bruhat-Tits building of PGL by lattices of smaller rank, Doc. Math., 2001, 6, 315–342
  • [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
  • [16] Ziegler G. M., Lectures on Polytopes, Grad. Texts in Math., 152, Springer, New York, 2007
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0005-3
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