Soit un groupe complexe réductif connexe où est un espace vectoriel complexe de dimension finie. Soient et . D’aprés Raïs nous disons que l’orbite est caractéristique pour si la composante connexe de l’identité de est . Si est semi-simple, nous disons que est semi-caractéristique pour si la composante connexe de l’identité de est une extension de par un tore. Nous classifions les orbites de qui ne sont pas (semi)-caractéristiques dans plusieurs cas.
Let be a connected complex reductive group where is a finite-dimensional complex vector space. Let and let . Following Raïs we say that the orbit is characteristic for if the identity component of is . If is semisimple, we say that is semi-characteristic for if the identity component of is an extension of by a torus. We classify the -orbits which are not (semi)-characteristic in many cases.
Keywords: Characteristic orbits, linear preserver problems
Mot clés : Orbites caractéristiques, problèmes de préservation linéaires
Schwarz, Gerald W. 1
@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}, mrnumber = {2985513}, zbl = {1255.14040}, language = {en}, url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.org/articles/10.5802/aif.2691/ DO - 10.5802/aif.2691 LA - en ID - AIF_2012__62_2_667_0 ER -
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2691/
[1] 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] 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] Maximal subgroups of the classical groups, Trudy Moskov. Mat. Obšč., Volume 1 (1952), pp. 39-166 | MR | Zbl
[4] 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] 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] Invertible preservers and algebraic groups. II. Preservers of similarity invariants and overgroups of , Linear and Multilinear Algebra, Volume 43 (1997) no. 1-3, pp. 221-255 | DOI | MR | Zbl
[7] 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] 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] The structure of Lie groups, Holden-Day Inc., San Francisco, 1965 | MR | Zbl
[10] A note on automorphisms of Lie algebras, Pacific J. Math., Volume 12 (1962), pp. 303-315 | MR | Zbl
[11] 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] Lie algebras, Dover Publications Inc., New York, 1979 (Republication of the 1962 original) | MR
[13] Linear preserver problems, Amer. Math. Monthly, Volume 108 (2001) no. 7, pp. 591-605 | DOI | MR
[14] 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] Adhérences d’orbite et invariants, Invent. Math., Volume 29 (1975) no. 3, pp. 231-238 | DOI | MR | Zbl
[16] Inclusion relations between transitive compact transformation groups, Trudy Moskov. Mat. Obšč., Volume 11 (1962), pp. 199-242 | MR | Zbl
[17] Decompositions of reductive Lie groups, Mat. Sb. (N.S.), Volume 80 (122) (1969), pp. 553-599 | MR | Zbl
[18] Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994 | MR | Zbl
[19] Linear preserver problems and algebraic groups, Math. Ann., Volume 303 (1995) no. 1, pp. 165-184 | DOI | MR | Zbl
[20] Notes sur la notion d’invariant caractéristique, 2007 (arxiv.org/abs/0707.0782v1)
[21] Algebraic quotients of compact group actions, J. Algebra, Volume 244 (2001) no. 2, pp. 365-378 | DOI | MR
[22] 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] What is a Wilf-Zeilberger pair?, Notices Amer. Math. Soc., Volume 57 (2010) no. 4, pp. 508-509 | MR
[24] LiE, a software package for Lie group computations, Euromath Bull., Volume 1 (1994) no. 2, pp. 83-94 | MR | Zbl
[25] A package for Lie group computations, Computer Algebra Nederland, Amsterdam, 1992
[26] A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat., Volume 36 (1972), pp. 749-764 | MR | Zbl
[27] The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939 | MR | Zbl
[28] Rational functions certify combinatorial identities, J. Amer. Math. Soc., Volume 3 (1990) no. 1, pp. 147-158 | DOI | MR | Zbl
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