Equations of some wonderful compactifications

Pascal Hivert[1]

  • [1] Université de Versailles–Saint-Quentin en Yvelines Département de Mathématiques Bâtiment Fermat 45, avenue des État-Unis 78035 VERSAILLES cedex (France)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 2121-2138
  • ISSN: 0373-0956

Abstract

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De Concini and Procesi have defined the wonderful compactification X ¯ of a symmetric space X = G / G σ where G is a complex semisimple adjoint group and G σ the subgroup of fixed points of G by an involution σ . It is a closed subvariety of a Grassmannian of the Lie algebra 𝔤 of G . In this paper we prove that, when the rank of X is equal to the rank of G , the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form w on 𝔤 vanishes on the ( - 1 ) -eigenspace of σ .

How to cite

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Hivert, Pascal. "Equations of some wonderful compactifications." Annales de l’institut Fourier 61.5 (2011): 2121-2138. <http://eudml.org/doc/219840>.

@article{Hivert2011,
abstract = {De Concini and Procesi have defined the wonderful compactification $\bar\{X\}$ of a symmetric space $X=G/G^\sigma $ where $G$ is a complex semisimple adjoint group and $G^\sigma $ the subgroup of fixed points of $G$ by an involution $\sigma $. It is a closed subvariety of a Grassmannian of the Lie algebra $\mathfrak\{g\}$ of $G$. In this paper we prove that, when the rank of $X$ is equal to the rank of $G$, the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form $w$ on $\mathfrak\{g\}$ vanishes on the $(-1)$-eigenspace of $\sigma $.},
affiliation = {Université de Versailles–Saint-Quentin en Yvelines Département de Mathématiques Bâtiment Fermat 45, avenue des État-Unis 78035 VERSAILLES cedex (France)},
author = {Hivert, Pascal},
journal = {Annales de l’institut Fourier},
keywords = {Wonderful compactification; symmetric space; Lie algebra; adjoint group; scheme; wonderful compactification},
language = {eng},
number = {5},
pages = {2121-2138},
publisher = {Association des Annales de l’institut Fourier},
title = {Equations of some wonderful compactifications},
url = {http://eudml.org/doc/219840},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Hivert, Pascal
TI - Equations of some wonderful compactifications
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 2121
EP - 2138
AB - De Concini and Procesi have defined the wonderful compactification $\bar{X}$ of a symmetric space $X=G/G^\sigma $ where $G$ is a complex semisimple adjoint group and $G^\sigma $ the subgroup of fixed points of $G$ by an involution $\sigma $. It is a closed subvariety of a Grassmannian of the Lie algebra $\mathfrak{g}$ of $G$. In this paper we prove that, when the rank of $X$ is equal to the rank of $G$, the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form $w$ on $\mathfrak{g}$ vanishes on the $(-1)$-eigenspace of $\sigma $.
LA - eng
KW - Wonderful compactification; symmetric space; Lie algebra; adjoint group; scheme; wonderful compactification
UR - http://eudml.org/doc/219840
ER -

References

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  1. Nicolas Bourbaki, Groupes et algèbres de Lie. Chapitres 7–8, (1990), Masson, Paris Zbl0483.22001
  2. C. De Concini, C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982) 996 (1983), 1-44, Springer, Berlin Zbl0581.14041
  3. Jan Draisma, Hanspeter Kraft, Jochen Kuttler, Nilpotent subspaces of maximal dimension in semi-simple Lie algebras, Compos. Math. 142 (2006), 464-476 Zbl1163.17011MR2218906
  4. William Fulton, Joe Harris, Representation theory, 129 (1991), Springer-Verlag, New York Zbl0744.22001MR1153249
  5. D. Garfinkle, A new construction of the Joseph ideal, (1982) 
  6. Michael Thaddeus, Complete collineations revisited, Math. Ann. 315 (1999), 469-495 Zbl0986.14030MR1725990
  7. M. A. A. van Leeuwen, A. M. Cohen, B. Lisser, LiE, A Package for Lie Group Computations, (1992) 
  8. È. B. Vinberg, V. V. Gorbatsevich, A. L. Onishchik, Structure of Lie groups and Lie algebras, Current problems in mathematics. Fundamental directions, Vol. 41 (Russian) (1990), 5-259, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow Zbl0733.22003MR1056486
  9. Jerzy Weyman, Cohomology of vector bundles and syzygies, 149 (2003), Cambridge University Press, Cambridge Zbl1075.13007MR1988690

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