A tropical view on Bruhat-Tits buildings and their compactifications

Annette Werner

Open Mathematics (2011)

  • Volume: 9, Issue: 2, page 390-402
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Annette Werner. "A tropical view on Bruhat-Tits buildings and their compactifications." Open Mathematics 9.2 (2011): 390-402. <http://eudml.org/doc/269659>.

@article{AnnetteWerner2011,
abstract = {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.},
author = {Annette Werner},
journal = {Open Mathematics},
keywords = {Bruhat-Tits buildings; Tropical geometry; Weight polytopes; tropical geometry; weight polytopes; connected reductive groups; maximal split tori; tropical stabiliser subgroups; compactifications},
language = {eng},
number = {2},
pages = {390-402},
title = {A tropical view on Bruhat-Tits buildings and their compactifications},
url = {http://eudml.org/doc/269659},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Annette Werner
TI - A tropical view on Bruhat-Tits buildings and their compactifications
JO - Open Mathematics
PY - 2011
VL - 9
IS - 2
SP - 390
EP - 402
AB - 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.
LA - eng
KW - Bruhat-Tits buildings; Tropical geometry; Weight polytopes; tropical geometry; weight polytopes; connected reductive groups; maximal split tori; tropical stabiliser subgroups; compactifications
UR - http://eudml.org/doc/269659
ER -

References

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  1. [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 Zbl0922.15001
  2. [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 
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  4. [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 Zbl0565.14028
  5. [5] Einsiedler M., Kapranov M., Lind D., Non-archimedean amoebas and tropical varieties, J. Reine Angew. Math., 2006, 601, 139–157 Zbl1115.14051
  6. [6] Goldman O., Iwahori N., Thespace of p-adic norms, Acta Math., 1963, 109(1), 137–177 http://dx.doi.org/10.1007/BF02391811 Zbl0133.29402
  7. [7] Green J.A., Polynomial Representations of GL n, Lecture Notes in Math., 830, Springer, Berlin-New York, 1980 
  8. [8] Joswig M., Tropical convex hull computations, In: Tropical and Idempotent Mathematics, Contemp. Math., 495, AMS, Providence, 2009, 193–212 Zbl1202.52004
  9. [9] Joswig M., Sturmfels B., Yu J., Affine buildings and tropical convexity, Albanian J. Math., 2007, 1(4), 187–211 Zbl1133.52003
  10. [10] Landvogt E., A Compactification of the Bruhat-Tits Building, Lecture Notes in Math., 1619, Springer, Berlin, 1996 Zbl0935.20034
  11. [11] Landvogt E., Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math., 2000, 518, 213–241 Zbl0937.20026
  12. [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 Zbl1198.51006
  13. [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) Zbl1241.51003
  14. [14] Werner A., Compactification of the Bruhat-Tits building of PGL by lattices of smaller rank, Doc. Math., 2001, 6, 315–342 Zbl1048.20014
  15. [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 Zbl1131.20019
  16. [16] Ziegler G. M., Lectures on Polytopes, Grad. Texts in Math., 152, Springer, New York, 2007 Zbl0823.52002

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