Linear preservers and representations with a 1-dimensional ring of invariants

Bermudez, H.; Garibaldi, S.; Larsen, V.

doi:10.1090/S0002-9947-2014-06081-9

Linear preservers and representations with a 1-dimensional ring of invariants
HTML articles powered by AMS MathViewer

by H. Bermudez, S. Garibaldi and V. Larsen PDF

Trans. Amer. Math. Soc. 366 (2014), 4755-4780 Request permission

Abstract:

We determine the group of linear transformations preserving a polynomial function $f$ on a vector space $V$ for several interesting pairs $(V,f)$ by introducing a subgroup $G$ of $\mathrm {GL}(V)$ and applying the theory of semisimple algebraic groups. Along the way, we give an explicit description of the normalizer $N_{\mathrm {GL}(V)}(G)$ and prove that, under a mild technical assumption, the normalizer agrees with the stabilizer in $\mathrm {GL}(V)$ of the orbit of the highest weight vector in $\mathbb {P}(V)$.

References

H. Azad, M. Barry, and G. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), no. 2, 551–562. MR 1047327, DOI 10.1080/00927879008823931
Michael Aschbacher, The $27$-dimensional module for $E_6$. I, Invent. Math. 89 (1987), no. 1, 159–195. MR 892190, DOI 10.1007/BF01404676
Michael Aschbacher, Chevalley groups of type $G_2$ as the group of a trilinear form, J. Algebra 109 (1987), no. 1, 193–259. MR 898346, DOI 10.1016/0021-8693(87)90173-6
L. Borsten, M. J. Duff, S. Ferrara, A. Marrani, and W. Rubens, Small orbits, Phys. Rev. D 85 (2012), no. 8, 086002.
LeRoy B. Beasley, Linear transformations on matrices: The invariance of the third elementary symmetric function, Canadian J. Math. 22 (1970), 746–752. MR 268201, DOI 10.4153/CJM-1970-084-x
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
Robert B. Brown, Groups of type $E_{7}$, J. Reine Angew. Math. 236 (1969), 79–102. MR 248185, DOI 10.1515/crll.1969.236.79
A. Cayley, On the theory of linear transformations, Cambridge Mathematical Journal IV (1845), 193–209, (= Coll. Math. Papers, vol. 1, pp. 80–94).
Skip Garibaldi and Michael Carr, Geometries, the principle of duality, and algebraic groups, Expo. Math. 24 (2006), no. 3, 195–234. MR 2250947, DOI 10.1016/j.exmath.2005.11.001
Arjeh M. Cohen and Aloysius G. Helminck, Trilinear alternating forms on a vector space of dimension $7$, Comm. Algebra 16 (1988), no. 1, 1–25. MR 921939, DOI 10.1080/00927878808823558
Claude C. Chevalley, The algebraic theory of spinors, Columbia University Press, New York, 1954. MR 0060497
G. H. Chan and M. H. Lim, Linear transformations on symmetric matrices. II, Linear and Multilinear Algebra 32 (1992), no. 3-4, 319–325. MR 1238013, DOI 10.1080/03081089208818172
Gin Hor Chan, Ming Huat Lim, and Kok-Keong Tan, Linear preservers on matrices, Linear Algebra Appl. 93 (1987), 67–80. MR 898543, DOI 10.1016/S0024-3795(87)90312-0
M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179–186. MR 435092, DOI 10.1007/BF01390108
M. Demazure and A. Grothendieck, SGA3: Schémas en groupes, Lecture Notes in Mathematics, vol. 151–153, Springer, 1970.
Jean Dieudonné, Sur une généralisation du groupe orthogonal à quatre variables, Arch. Math. 1 (1949), 282–287 (French). MR 29360, DOI 10.1007/BF02038756
Morris L. Eaton, On linear transformations which preserve the determinant, Illinois J. Math. 13 (1969), 722–727. MR 251050
F. Engel, Ein neues, dem linearen Komplexe analoges Gebilde, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Klasse 52 (1900), 63–76, 220–239.
Paul Eakin and James Silver, Rings which are almost polynomial rings, Trans. Amer. Math. Soc. 174 (1972), 425–449. MR 309924, DOI 10.1090/S0002-9947-1972-0309924-4
J. C. Ferrar, Strictly regular elements in Freudenthal triple systems, Trans. Amer. Math. Soc. 174 (1972), 313–331 (1973). MR 374223, DOI 10.1090/S0002-9947-1972-0374223-1
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Mathieu Florence, On higher trace forms of separable algebras, Arch. Math. (Basel) 97 (2011), no. 3, 247–249. MR 2836304, DOI 10.1007/s00013-011-0282-x
G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsberichte Deutsch. Akad. Wiss. Berlin (1897), 994–1015.
Skip Garibaldi, The characteristic polynomial and determinant are not ad hoc constructions, Amer. Math. Monthly 111 (2004), no. 9, 761–778. MR 2104048, DOI 10.2307/4145188
Skip Garibaldi, Cohomological invariants: exceptional groups and spin groups, Mem. Amer. Math. Soc. 200 (2009), no. 937, xii+81. With an appendix by Detlev W. Hoffmann. MR 2528487, DOI 10.1090/memo/0937
Wee Teck Gan, Benedict Gross, and Gordan Savin, Fourier coefficients of modular forms on $G_2$, Duke Math. J. 115 (2002), no. 1, 105–169. MR 1932327, DOI 10.1215/S0012-7094-02-11514-2
James A. Green, Polynomial representations of $\textrm {GL}_{n}$, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR 606556
G. B. Gurevich, Foundations of the theory of algebraic invariants, P. Noordhoff Ltd., Groningen, 1964. Translated by J. R. M. Radok and A. J. M. Spencer. MR 0183733
Robert M. Guralnick, Invertible preservers and algebraic groups. II. Preservers of similarity invariants and overgroups of $\textrm {PSL}_n(F)$, Linear and Multilinear Algebra 43 (1997), no. 1-3, 221–255. MR 1613065, DOI 10.1080/03081089708818527
Fred W. Helenius, Freudenthal triple systems by root system methods, J. Algebra 357 (2012), 116–137. MR 2905245, DOI 10.1016/j.jalgebra.2012.01.025
J. William Hoffman and Jorge Morales, Arithmetic of binary cubic forms, Enseign. Math. (2) 46 (2000), no. 1-2, 61–94. MR 1769537
Loo-Keng Hua, A theorem on matrices and its application to Grassmann space, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5 (1948), 150–181. MR 30928
J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, 1980, Third printing, revised.
—, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 21, Springer, 1981.
Jun-ichi Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997–1028. MR 277558, DOI 10.2307/2373406
N. Jacobson, Some groups of transformations defined by Jordan algebras. III, J. Reine Angew. Math. 207 (1961), 61–85. MR 159850, DOI 10.1515/crll.1961.207.61
Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190–213. MR 575790, DOI 10.1016/0021-8693(80)90141-6
T. Kambayashi and M. Miyanishi, On flat fibrations by the affine line, Illinois J. Math. 22 (1978), no. 4, 662–671. MR 503968
Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
M. H. Lim, Linear transformations on symmetric matrices, Linear and Multilinear Algebra 7 (1979), no. 1, 47–57. MR 523648, DOI 10.1080/03081087908817259
Chi-Kwong Li and Stephen Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), no. 7, 591–605. MR 1862098, DOI 10.2307/2695268
Chi-Kwong Li and Nam-Kiu Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162/164 (1992), 217–235. Directions in matrix theory (Auburn, AL, 1990). MR 1148401, DOI 10.1016/0024-3795(92)90377-M
Jacob Lurie, On simply laced Lie algebras and their minuscule representations, Comment. Math. Helv. 76 (2001), no. 3, 515–575. MR 1854697, DOI 10.1007/PL00013217
Marvin Marcus, Linear operations on matrices, Amer. Math. Monthly 69 (1962), 837–847. MR 147491, DOI 10.2307/2311230
Kurt Meyberg, Eine Theorie der Freudenthalschen Tripelsysteme. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 162–174, 175–190 (German). MR 0225838
Henryk Minc, Linear transformations on matrices: rank $1$ preservers and determinant preservers, Linear and Multilinear Algebra 4 (1976/77), no. 4, 265–272. MR 435098, DOI 10.1080/03081087708817161
Marvin Marcus and B. N. Moyls, Linear transformations on algebras of matrices, Canadian J. Math. 11 (1959), 61–66. MR 99996, DOI 10.4153/CJM-1959-008-0
Marvin Marcus and B. N. Moyls, Transformations on tensor product spaces, Pacific J. Math. 9 (1959), 1215–1221. MR 108503
Marvin Marcus and Roger Purves, Linear transformations on algebras of matrices: the invariance of the elementary symmetric functions, Canadian J. Math. 11 (1959), 383–396. MR 105425, DOI 10.4153/CJM-1959-039-4
Marvin Marcus and Roy Westwick, Linear maps on skew symmetric matrices: the invariance of elementary symmetric functions, Pacific J. Math. 10 (1960), 917–924. MR 114823
Akimasa Miyake and Miki Wadati, Multipartite entanglement and hyperdeterminants, Quantum Inf. Comput. 2 (2002), no. suppl., 540–555. ERATO Workshop on Quantum Information Science (Tokyo, 2002). MR 1975158
William C. Nemitz, Transformations preserving the Grassmannian, Trans. Amer. Math. Soc. 109 (1963), 400–410. MR 154882, DOI 10.1090/S0002-9947-1963-0154882-3
Vladimir P. Platonov and Dragomir Ž. Đoković, Linear preserver problems and algebraic groups, Math. Ann. 303 (1995), no. 1, 165–184. MR 1348361, DOI 10.1007/BF01460985
A survey of linear preserver problems, Gordon and Breach Science Publishers, Yverdon, 1992. Linear and Multilinear Algebra 33 (1992), no. 1-2. MR 1346777, DOI 10.1080/03081089208818176
V. L. Popov, Classification of spinors of dimension 14, Trans. Moscow Math. Soc. (1980), no. 1, 181–232.
V. L. Popov and E. B. Vinberg, Invariant theory, Encyclopedia of Mathematical Sciences, vol. 55, pp. 123–284, Springer-Verlag, 1994.
Philippe Revoy, Trivecteurs de rang $6$, Bull. Soc. Math. France Mém. 59 (1979), 141–155 (French). Colloque sur les Formes Quadratiques, 2 (Montpellier, 1977). MR 532012
Gerhard E. Röhrle, On extraspecial parabolic subgroups, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 143–155. MR 1247502, DOI 10.1090/conm/153/01310
M. Rost, On $14$-dimensional quadratic forms, their spinors, and the difference of two octonion algebras, unpublished note, March 1999.
—, On the Galois cohomology of ${\mathrm {Spin}(14)}$, unpublished note, March 1999.
Roger Richardson, Gerhard Röhrle, and Robert Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649–671. MR 1189494, DOI 10.1007/BF01231348
Hubert Rubenthaler, Algèbres de Lie et espaces préhomogènes, Travaux en Cours [Works in Progress], vol. 44, Hermann Éditeurs des Sciences et des Arts, Paris, 1992 (French). With a foreword by Jean-Michel Lemaire. MR 2412335
A. Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, Cambridge, 2000. With an appendix by Umberto Zannier. MR 1770638, DOI 10.1017/CBO9780511542916
Gerald W. Schwarz, Linear maps preserving fibers, J. Lie Theory 18 (2008), no. 2, 433–443. MR 2431126
Gary M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286. MR 888704, DOI 10.1090/memo/0365
C. S. Seshadri, Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), no. 3, 225–274. MR 466154, DOI 10.1016/0001-8708(77)90041-X
Ronald Shaw, Linear algebra and group representations. Vol. II, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1983. Multilinear algebra and group representations. MR 701854
Richard Sinkhorn, Linear adjugate preservers on the complex matrices, Linear and Multilinear Algebra 12 (1982/83), no. 3, 215–222. MR 678827, DOI 10.1080/03081088208817485
M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155. MR 430336
S. Solomon, Irreducible linear group-subgroup pairs with the same invariants, J. Lie Theory 15 (2005), no. 1, 105–123. MR 2115231
T. A. Springer, Some groups of type $E_7$, Nagoya Math. J. 182 (2006), 259–284. MR 2235344, DOI 10.1017/S0027763000026891
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1763974, DOI 10.1007/978-3-662-12622-6
William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117
William C. Waterhouse, Linear maps preserving reduced norms, Linear Algebra Appl. 43 (1982), 197–200. MR 656445, DOI 10.1016/0024-3795(82)90253-1
William C. Waterhouse, Invertibility of linear maps preserving matrix invariants, Linear and Multilinear Algebra 13 (1983), no. 2, 105–113. MR 697321, DOI 10.1080/03081088308817510
William C. Waterhouse, Automorphisms of $\textrm {det}(X_{ij})$: the group scheme approach, Adv. in Math. 65 (1987), no. 2, 171–203. MR 900267, DOI 10.1016/0001-8708(87)90021-1
William C. Waterhouse, Linear transformations preserving symmetric rank one matrices, J. Algebra 125 (1989), no. 2, 502–518. MR 1018960, DOI 10.1016/0021-8693(89)90179-8
William C. Waterhouse, Automorphism group schemes of basic matrix invariants, Trans. Amer. Math. Soc. 347 (1995), no. 10, 3859–3872. MR 1303128, DOI 10.1090/S0002-9947-1995-1303128-X
H. Weber, Lehrbuch der algebra, Friedrich Vieweg und Sohn, 1895.
R. Westwick, Linear transformations on Grassman spaces, Pacific J. Math. 14 (1964), 1123–1127. MR 167493
Roy Westwick, Transformations on tensor spaces, Pacific J. Math. 23 (1967), 613–620. MR 225805
Roy Westwick, Linear transformations on Grassmann spaces, Canadian J. Math. 21 (1969), 414–417. MR 241459, DOI 10.4153/CJM-1969-045-6
Xian Zhang, Linear/additive preservers of rank 2 on spaces of alternate matrices over fields, Linear Algebra Appl. 396 (2005), 91–102. MR 2112201, DOI 10.1016/j.laa.2004.08.028