Linear maps preserving orbits
[Applications linéaires qui préservent des orbites]
Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 667-706.

Soit HGL(V) un groupe complexe réductif connexe où V est un espace vectoriel complexe de dimension finie. Soient vV et G={g GL (V)gHv=Hv}. D’aprés Raïs nous disons que l’orbite Hv est caractéristique pour H si la composante connexe de l’identité de G est H. Si H est semi-simple, nous disons que Hv est semi-caractéristique pour H si la composante connexe de l’identité de G est une extension de H par un tore. Nous classifions les orbites de H qui ne sont pas (semi)-caractéristiques dans plusieurs cas.

Let HGL(V) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let vV and let G={g GL (V)gHv=Hv}. Following Raïs we say that the orbit Hv is characteristic for H if the identity component of G is H. If H is semisimple, we say that Hv is semi-characteristic for H if the identity component of G is an extension of H by a torus. We classify the H-orbits which are not (semi)-characteristic in many cases.

DOI : 10.5802/aif.2691
Classification : 20G20, 22E46
Keywords: Characteristic orbits, linear preserver problems
Mot clés : Orbites caractéristiques, problèmes de préservation linéaires
Schwarz, Gerald W. 1

1 Brandeis University Department of Mathematics Waltham, MA 02454-9110 (USA)
@article{AIF_2012__62_2_667_0,
     author = {Schwarz, Gerald W.},
     title = {Linear maps preserving orbits},
     journal = {Annales de l'Institut Fourier},
     pages = {667--706},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     doi = {10.5802/aif.2691},
     zbl = {1255.14040},
     mrnumber = {2985513},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2691/}
}
TY  - JOUR
AU  - Schwarz, Gerald W.
TI  - Linear maps preserving orbits
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 667
EP  - 706
VL  - 62
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2691/
DO  - 10.5802/aif.2691
LA  - en
ID  - AIF_2012__62_2_667_0
ER  - 
%0 Journal Article
%A Schwarz, Gerald W.
%T Linear maps preserving orbits
%J Annales de l'Institut Fourier
%D 2012
%P 667-706
%V 62
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2691/
%R 10.5802/aif.2691
%G en
%F AIF_2012__62_2_667_0
Schwarz, Gerald W. Linear maps preserving orbits. Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 667-706. doi : 10.5802/aif.2691. http://www.numdam.org/articles/10.5802/aif.2691/

[1] Doković, Dragomir Ž.; Li, Chi-Kwong Overgroups of some classical linear groups with applications to linear preserver problems, Linear Algebra Appl., Volume 197/198 (1994), pp. 31-61 Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992) | DOI | MR | Zbl

[2] Doković, Dragomir Ž.; Platonov, Vladimir P. Algebraic groups and linear preserver problems, C. R. Acad. Sci. Paris Sér. I Math., Volume 317 (1993) no. 10, pp. 925-930 | MR | Zbl

[3] Dynkin, E. B. Maximal subgroups of the classical groups, Trudy Moskov. Mat. Obšč., Volume 1 (1952), pp. 39-166 | MR | Zbl

[4] Gorbatsevich, V. V.; Onishchik, A. L. Lie transformation groups [see MR0950861 (89m:22010)], Lie groups and Lie algebras, I (Encyclopaedia Math. Sci.), Volume 20, Springer, Berlin, 1993, pp. 95-235 | MR | Zbl

[5] Guralnick, Robert M. Invertible preservers and algebraic groups, Proceedings of the 3rd ILAS Conference (Pensacola, FL, 1993), Volume 212/213 (1994), pp. 249-257 | DOI | MR | Zbl

[6] Guralnick, Robert M. Invertible preservers and algebraic groups. II. Preservers of similarity invariants and overgroups of PSL n (F), Linear and Multilinear Algebra, Volume 43 (1997) no. 1-3, pp. 221-255 | DOI | MR | Zbl

[7] Guralnick, Robert M.; Li, Chi-Kwong Invertible preservers and algebraic groups. III. Preservers of unitary similarity (congruence) invariants and overgroups of some unitary subgroups, Linear and Multilinear Algebra, Volume 43 (1997) no. 1-3, pp. 257-282 | DOI | MR | Zbl

[8] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978 | MR | Zbl

[9] Hochschild, G. The structure of Lie groups, Holden-Day Inc., San Francisco, 1965 | MR | Zbl

[10] Jacobson, N. A note on automorphisms of Lie algebras, Pacific J. Math., Volume 12 (1962), pp. 303-315 | MR | Zbl

[11] Jacobson, Nathan Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962 | MR | Zbl

[12] Jacobson, Nathan Lie algebras, Dover Publications Inc., New York, 1979 (Republication of the 1962 original) | MR

[13] Li, Chi-Kwong; Pierce, Stephen Linear preserver problems, Amer. Math. Monthly, Volume 108 (2001) no. 7, pp. 591-605 | DOI | MR

[14] Luna, Domingo Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, p. 81-105. Bull. Soc. Math. France, Paris, Mémoire 33 | Numdam | MR | Zbl

[15] Luna, Domingo Adhérences d’orbite et invariants, Invent. Math., Volume 29 (1975) no. 3, pp. 231-238 | DOI | MR | Zbl

[16] Oniščik, Arkadi L. Inclusion relations between transitive compact transformation groups, Trudy Moskov. Mat. Obšč., Volume 11 (1962), pp. 199-242 | MR | Zbl

[17] Oniščik, Arkadi L. Decompositions of reductive Lie groups, Mat. Sb. (N.S.), Volume 80 (122) (1969), pp. 553-599 | MR | Zbl

[18] Oniščik, Arkadi L. Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994 | MR | Zbl

[19] Platonov, Vladimir P.; Doković, Dragomir Ž. Linear preserver problems and algebraic groups, Math. Ann., Volume 303 (1995) no. 1, pp. 165-184 | DOI | MR | Zbl

[20] Raïs, Mustapha Notes sur la notion d’invariant caractéristique, 2007 (arxiv.org/abs/0707.0782v1)

[21] Schwarz, Gerald W. Algebraic quotients of compact group actions, J. Algebra, Volume 244 (2001) no. 2, pp. 365-378 | DOI | MR

[22] Stanley, Richard P. Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.), Volume 1 (1979) no. 3, pp. 475-511 | DOI | MR | Zbl

[23] Tefera, Akalu What is a Wilf-Zeilberger pair?, Notices Amer. Math. Soc., Volume 57 (2010) no. 4, pp. 508-509 | MR

[24] van Leeuwen, M. A. A. LiE, a software package for Lie group computations, Euromath Bull., Volume 1 (1994) no. 2, pp. 83-94 | MR | Zbl

[25] van Leeuwen, M. A. A.; Cohen, A. M.; Lisser, B. A package for Lie group computations, Computer Algebra Nederland, Amsterdam, 1992

[26] Vinberg, È. B.; Popov, V. L. A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat., Volume 36 (1972), pp. 749-764 | MR | Zbl

[27] Weyl, Hermann The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939 | MR | Zbl

[28] Wilf, Herbert S.; Zeilberger, Doron Rational functions certify combinatorial identities, J. Amer. Math. Soc., Volume 3 (1990) no. 1, pp. 147-158 | DOI | MR | Zbl

Cité par Sources :