Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

Damon, James; Pike, Brian

doi:10.5802/aif.2956

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors
[Groupes solvables, diviseurs libres et singularités des matrices nonisolées I : Tours des diviseurs libres]

Damon, James ¹ ; Pike, Brian ²

¹ Department of Mathematics University of North Carolina Chapel Hill, NC 27599-3250 (USA)
² Dept. of Computer and Mathematical Sciences University of Toronto Scarborough 1265 Military Trail Toronto, ON M1C 1A4 (Canada)

Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1251-1300.

Résumé
Abstract

Nous introduisons une méthode pour obtenir des nouvelles classes de diviseurs libres à partir de représentations $V$ de groupes algébriques linéaires connexes $G$ pour lesquelles $dim G = dim V$ et $V$ a une orbite ouverte. Nous donnons des conditions suffisantes pour lesquelles le complémentaire de cette orbite ouverte, la « variété des orbites exceptionelles », est une diviseur libre (ou un diviseur libre* plus faible) pour des « représentations par blocs » à la fois des groupes solvables et des extensions des groupes réductifs par ces groupes. Ce sont des représentations pour lesquelles la matrice définie à partir d’une base des « champs des vecteurs associés » de la représentation $V$ , a une forme triangulaire bloc et les blocs satisfont certaines conditions de non-singularité.

Pour les tours de groupes de Lie et leurs représentations ce résultat donne une tour de diviseurs libres obtenue en avoisinant successivement des variétés de matrices singulières. Il s’applique aux groupes solvables qui donnent la factorisation classique du type Cholesky et une forme modifiée de celle ci, sur les espaces des matrices $m \times m$ symétriques, antisymétriques, ou générales. Pour les matrices antisymétriques, il s’étend aussi aux représentations des algèbres de Lie solvables et non-linéaires de dimension infinie.

We introduce a method for obtaining new classes of free divisors from representations $V$ of connected linear algebraic groups $G$ where $dim G = dim V$ , with $V$ having an open orbit. We give sufficient conditions that the complement of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor) for “block representations” of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated “representation vector fields” on $V$ has block triangular form, with blocks satisfying certain nonsingularity conditions.

For towers of Lie groups and representations this yields a tower of free divisors, successively obtained by adjoining varieties of singular matrices. This applies to solvable groups which give classical Cholesky-type factorization, and a modified form of it, on spaces of $m \times m$ symmetric, skew-symmetric or general matrices. For skew-symmetric matrices, it further extends to representations of nonlinear infinite dimensional solvable Lie algebras.

MR Zbl

DOI : 10.5802/aif.2956

Classification : 17B66, 22E27, 11S90
Keywords: prehomogeneous vector spaces, free divisors, linear free divisors, determinantal varieties, Pfaffian varieties, solvable algebraic groups, Cholesky-type factorizations, block representations, exceptional orbit varieties, infinite-dimensional solvable Lie algebras
Mot clés : espaces vectoriels préhomogènes, diviseurs libres, diviseurs libres linéaires, variétés déterminantales, variétés de Pfaff, groupes algébriques solvables, factorisations du type Cholesky, représentations par blocs, variété des orbites exceptionelles, algèbres de Lie solvables de dimension infinie

Affiliations des auteurs :

Damon, James ¹ ; Pike, Brian ²

@article{AIF_2015__65_3_1251_0,
     author = {Damon, James and Pike, Brian},
     title = {Solvable {Groups,} {Free} {Divisors} and {Nonisolated} {Matrix} {Singularities} {I:} {Towers} of {Free} {Divisors}},
     journal = {Annales de l'Institut Fourier},
     pages = {1251--1300},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     doi = {10.5802/aif.2956},
     zbl = {06497263},
     mrnumber = {3449179},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2956/}
}

TY  - JOUR
AU  - Damon, James
AU  - Pike, Brian
TI  - Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 1251
EP  - 1300
VL  - 65
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2956/
DO  - 10.5802/aif.2956
LA  - en
ID  - AIF_2015__65_3_1251_0
ER  -

%0 Journal Article
%A Damon, James
%A Pike, Brian
%T Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors
%J Annales de l'Institut Fourier
%D 2015
%P 1251-1300
%V 65
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2956/
%R 10.5802/aif.2956
%G en
%F AIF_2015__65_3_1251_0

Damon, James; Pike, Brian. Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1251-1300. doi : 10.5802/aif.2956. https://aif.centre-mersenne.org/articles/10.5802/aif.2956/

Bibliographie
Cité par

[1] Benner, Peter; Byers, Ralph; Fassbender, Heike; Mehrmann, Volker; Watkins, David Cholesky-like factorizations of skew-symmetric matrices, Electron. Trans. Numer. Anal., Volume 11 (2000), p. 85-93 (electronic) | MR | Zbl

[2] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991, pp. xii+288 | DOI | MR | Zbl

[3] Bruce, J. W. On families of symmetric matrices, Mosc. Math. J., Volume 3 (2003) no. 2, pp. 335-360 | MR | Zbl

[4] Bruce, J. W.; Tari, F. On families of square matrices, Proc. London Math. Soc. (3), Volume 89 (2004) no. 3, pp. 738-762 | DOI | MR | Zbl

[5] Buchsbaum, David A.; Eisenbud, David Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$ , Amer. J. Math., Volume 99 (1977) no. 3, pp. 447-485 | DOI | MR | Zbl

[6] Buchweitz, Ragnar-Olaf; Mond, David Linear free divisors and quiver representations, Singularities and computer algebra (London Math. Soc. Lecture Note Ser.), Volume 324, Cambridge Univ. Press, Cambridge, 2006, pp. 41-77 | DOI | MR | Zbl

[7] Burch, Lindsay On ideals of finite homological dimension in local rings, Proc. Cambridge Philos. Soc., Volume 64 (1968), pp. 941-948 | DOI | MR | Zbl

[8] Damon, James Higher multiplicities and almost free divisors and complete intersections, Mem. Amer. Math. Soc., Volume 123 (1996) no. 589, pp. x+113 | DOI | MR | Zbl

[9] Damon, James On the legacy of free divisors: discriminants and Morse-type singularities, Amer. J. Math., Volume 120 (1998) no. 3, pp. 453-492 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.3damon.pdf | DOI | MR | Zbl

[10] Damon, James On the legacy of free divisors. II. Free $^{*}$ divisors and complete intersections, Mosc. Math. J., Volume 3 (2003) no. 2, pp. 361-395 | MR | Zbl

[11] Damon, James; Mond, David $𝒜$ -codimension and the vanishing topology of discriminants, Invent. Math., Volume 106 (1991) no. 2, pp. 217-242 | DOI | MR | Zbl

[12] Damon, James; Pike, Brian Solvable group representations and free divisors whose complements are $K (π, 1)$ ’s, Topology Appl., Volume 159 (2012) no. 2, pp. 437-449 | DOI | MR | Zbl

[13] Damon, James; Pike, Brian Solvable groups, free divisors and nonisolated matrix singularities II: vanishing topology, Geom. Topol., Volume 18 (2014) no. 2, pp. 911-962 | DOI | MR | Zbl

[14] Demmel, James W. Applied numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997, pp. xii+419 | DOI | MR | Zbl

[15] Dress, Andreas W. M.; Wenzel, Walter A simple proof of an identity concerning Pfaffians of skew symmetric matrices, Adv. Math., Volume 112 (1995) no. 1, pp. 120-134 | DOI | MR | Zbl

[16] Eagon, J. A.; Northcott, D. G. Ideals defined by matrices and a certain complex associated with them., Proc. Roy. Soc. Ser. A, Volume 269 (1962), pp. 188-204 | DOI | MR | Zbl

[17] Frühbis-Krüger, Anne Classification of simple space curve singularities, Comm. Algebra, Volume 27 (1999) no. 8, pp. 3993-4013 | DOI | MR | Zbl

[18] Frühbis-Krüger, Anne; Neumer, Alexander Simple Cohen-Macaulay codimension 2 singularities, Comm. Algebra, Volume 38 (2010) no. 2, pp. 454-495 | DOI | MR | Zbl

[19] Goryunov, V. V.; Zakalyukin, V. M. Simple symmetric matrix singularities and the subgroups of Weyl groups $A_{μ}$ , $D_{μ}$ , $E_{μ}$ , Mosc. Math. J., Volume 3 (2003) no. 2, pp. 507-530 | MR | Zbl

[20] Granger, Michel; Mond, David; Nieto-Reyes, Alicia; Schulze, Mathias Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 2, pp. 811-850 | DOI | Numdam | MR | Zbl

[21] Granger, Michel; Mond, David; Schulze, Mathias Free divisors in prehomogeneous vector spaces, Proc. Lond. Math. Soc. (3), Volume 102 (2011) no. 5, pp. 923-950 | DOI | MR | Zbl

[22] Greuel, Gert-Martin; Lê, Dung Tráng Spitzen, Doppelpunkte und vertikale Tangenten in der Diskriminante verseller Deformationen von vollständigen Durchschnitten, Math. Ann., Volume 222 (1976) no. 1, pp. 71-88 | DOI | MR | Zbl

[23] Haslinger, G. Families of Skew-Symmetric Matrices, University of Liverpool (2001) (Ph. D. Thesis)

[24] Hilbert, David Ueber die Theorie der algebraischen Formen, Math. Ann., Volume 36 (1890) no. 4, pp. 473-534 | DOI | MR

[25] Kimura, Tatsuo Introduction to prehomogeneous vector spaces, Translations of Mathematical Monographs, 215, American Mathematical Society, Providence, RI, 2003, pp. xxii+288 (Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author) | MR | Zbl

[26] Macaulay, F. S. The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994, pp. xxxii+112 (Revised reprint of the 1916 original, With an introduction by Paul Roberts) | MR | Zbl

[27] Muir, Thomas A treatise on the theory of determinants, Revised and enlarged by William H. Metzler, Dover Publications, Inc., New York, 1960, pp. vii+766 | MR

[28] Pike, B. Singular Milnor numbers of non-isolated matrix singularities, Dept. of Mathematics, University of North Carolina (2010) (Ph. D. Thesis) | MR

[29] Saito, Kyoji Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 27 (1980) no. 2, pp. 265-291 | MR | Zbl

[30] Sato, M.; Kimura, T. A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., Volume 65 (1977), pp. 1-155 | MR | Zbl

[31] Sato, Mikio Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J., Volume 120 (1990), pp. 1-34 http://projecteuclid.org/euclid.nmj/1118782193 (Notes by Takuro Shintani, Translated from the Japanese by Masakazu Muro) | MR | Zbl

[32] Schaps, Mary Deformations of Cohen-Macaulay schemes of codimension $2$ and non-singular deformations of space curves, Amer. J. Math., Volume 99 (1977) no. 4, pp. 669-685 | DOI | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT