Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

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


Authors: H. Bermudez, S. Garibaldi and V. Larsen
Journal: Trans. Amer. Math. Soc. 366 (2014), 4755-4780
MSC (2010): Primary 47B49; Secondary 15A04, 15A72, 20G15
DOI: https://doi.org/10.1090/S0002-9947-2014-06081-9
Published electronically: April 16, 2014
MathSciNet review: 3217699
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [ABS90] H. Azad, M. Barry, and G. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), no. 2, 551-562. MR 1047327 (91d:20048), https://doi.org/10.1080/00927879008823931
  • [Asc87a] Michael Aschbacher, The $ 27$-dimensional module for $ E_6$. I, Invent. Math. 89 (1987), no. 1, 159-195. MR 892190 (88h:20045), https://doi.org/10.1007/BF01404676
  • [Asc87b] 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 (88g:20089), https://doi.org/10.1016/0021-8693(87)90173-6
  • [BDF$^+$12] L. Borsten, M. J. Duff, S. Ferrara, A. Marrani, and W. Rubens, Small orbits, Phys. Rev. D 85 (2012), no. 8, 086002.
  • [Bea70] LeRoy B. Beasley, Linear transformations on matrices: The invariance of the third elementary symmetric function, Canad. J. Math. 22 (1970), 746-752. MR 0268201 (42 #3100)
  • [Bor91] Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
  • [Bou02] 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 (2003a:17001)
  • [Bou05] Nicolas Bourbaki, Lie groups and Lie algebras: Chapters 7-9, Springer-Verlag, Berlin, 2005.MR 2109105
  • [Bro69] Robert B. Brown, Groups of type $ E_{7}$, J. Reine Angew. Math. 236 (1969), 79-102. MR 0248185 (40 #1439)
  • [Cay45] A. Cayley, On the theory of linear transformations, Cambridge Mathematical Journal IV (1845), 193-209, (= Coll. Math. Papers, vol. 1, pp. 80-94).
  • [CG06] Skip Garibaldi and Michael Carr, Geometries, the principle of duality, and algebraic groups, Expo. Math. 24 (2006), no. 3, 195-234. MR 2250947 (2007f:20080), https://doi.org/10.1016/j.exmath.2005.11.001
  • [CH88] 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 (89h:20060), https://doi.org/10.1080/00927878808823558
  • [Che97] Claude C. Chevalley, The algebraic theory of spinors, Columbia University Press, New York, 1954. MR 0060497 (15,678d)
  • [CL92] 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 (94i:15025), https://doi.org/10.1080/03081089208818172
  • [CLT87] Gin Hor Chan, Ming Huat Lim, and Kok-Keong Tan, Linear preservers on matrices, Linear Algebra Appl. 93 (1987), 67-80. MR 898543 (88h:15004), https://doi.org/10.1016/S0024-3795(87)90312-0
  • [Dem77] M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179-186. MR 0435092 (55 #8054)
  • [DG70] M. Demazure and A. Grothendieck, SGA3: Schémas en groupes, Lecture Notes in Mathematics, vol. 151-153, Springer, 1970.
  • [Die49] Jean Dieudonné, Sur une généralisation du groupe orthogonal à quatre variables, Arch. Math. 1 (1949), 282-287 (French). MR 0029360 (10,586l)
  • [Eat69] Morris L. Eaton, On linear transformations which preserve the determinant, Illinois J. Math. 13 (1969), 722-727. MR 0251050 (40 #4281)
  • [Eng00] 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.
  • [ES72] Paul Eakin and James Silver, Rings which are almost polynomial rings, Trans. Amer. Math. Soc. 174 (1972), 425-449. MR 0309924 (46 #9028)
  • [Fer72] J. C. Ferrar, Strictly regular elements in Freudenthal triple systems, Trans. Amer. Math. Soc. 174 (1972), 313-331 (1973). MR 0374223 (51 #10423)
  • [FH91] 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 (93a:20069)
  • [Flo11] Mathieu Florence, On higher trace forms of separable algebras, Arch. Math. (Basel) 97 (2011), no. 3, 247-249. MR 2836304 (2012g:16034), https://doi.org/10.1007/s00013-011-0282-x
  • [Fro97] G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsberichte Deutsch. Akad. Wiss. Berlin (1897), 994-1015.
  • [Gar04] Skip Garibaldi, The characteristic polynomial and determinant are not ad hoc constructions, Amer. Math. Monthly 111 (2004), no. 9, 761-778. MR 2104048 (2006d:15012), https://doi.org/10.2307/4145188
  • [Gar09] 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 (2010g:20079)
  • [GGS02] 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 (2004a:11036), https://doi.org/10.1215/S0012-7094-02-11514-2
  • [Gre80] James A. Green, Polynomial representations of $ {\rm GL}_{n}$, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin, 1980. MR 606556 (83j:20003)
  • [Gur64] G. B. Gurevich, Foundations of the theory of algebraic invariants, Translated by J. R. M. Radok and A. J. M. Spencer, P. Noordhoff Ltd., Groningen, 1964. MR 0183733 (32 #1211)
  • [Gur97] Robert M. Guralnick, Invertible preservers and algebraic groups. II. Preservers of similarity invariants and overgroups of $ {\rm PSL}_n(F)$, Linear and Multilinear Algebra 43 (1997), no. 1-3, 221-255. MR 1613065 (99m:20108), https://doi.org/10.1080/03081089708818527
  • [Hel12] Fred W. Helenius, Freudenthal triple systems by root system methods, J. Algebra 357 (2012), 116-137. MR 2905245, https://doi.org/10.1016/j.jalgebra.2012.01.025
  • [HM00] J. William Hoffman and Jorge Morales, Arithmetic of binary cubic forms, Enseign. Math. (2) 46 (2000), no. 1-2, 61-94. MR 1769537 (2001h:11048)
  • [Hua48] 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 0030928 (11,75a)
  • [Hum80] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, 1980, Third printing, revised.
  • [Hum81] -, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 21, Springer, 1981.
  • [Igu70] Jun-ichi Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997-1028. MR 0277558 (43 #3291)
  • [Jac61] N. Jacobson, Some groups of transformations defined by Jordan algebras. III, J. Reine Angew. Math. 207 (1961), 61-85. MR 0159850 (28 #3066)
  • [Jac68] N. Jacobson, Structure and representations of Jordan algebras, Coll. Pub., vol. 39, Amer. Math. Soc., Providence, RI, 1968.MR 0251099
  • [Jan03] Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057 (2004h:20061)
  • [Kac80] V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190-213. MR 575790 (81i:17005), https://doi.org/10.1016/0021-8693(80)90141-6
  • [KM78] T. Kambayashi and M. Miyanishi, On flat fibrations by the affine line, Illinois J. Math. 22 (1978), no. 4, 662-671. MR 503968 (80f:14028)
  • [KMRT98] 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 (2000a:16031)
  • [Lim79] M. H. Lim, Linear transformations on symmetric matrices, Linear and Multilinear Algebra 7 (1979), no. 1, 47-57. MR 523648 (80f:15027), https://doi.org/10.1080/03081087908817259
  • [LP01] Chi-Kwong Li and Stephen Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), no. 7, 591-605. MR 1862098 (2002g:15005), https://doi.org/10.2307/2695268
  • [LT92] 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 (93b:15003), https://doi.org/10.1016/0024-3795(92)90377-M
  • [Lur01] Jacob Lurie, On simply laced Lie algebras and their minuscule representations, Comment. Math. Helv. 76 (2001), no. 3, 515-575. MR 1854697 (2002g:17015), https://doi.org/10.1007/PL00013217
  • [Mar62] Marvin Marcus, Linear operations on matrices, Amer. Math. Monthly 69 (1962), 837-847. MR 0147491 (26 #5007)
  • [Mey68] 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 (37 #1429)
  • [Min77] Henryk Minc, Linear transformations on matrices: rank $ 1$ preservers and determinant preservers, Linear and Multilinear Algebra 4 (1976/77), no. 4, 265-272. MR 0435098 (55 #8060)
  • [MM59a] Marvin Marcus and B. N. Moyls, Linear transformations on algebras of matrices, Canad. J. Math. 11 (1959), 61-66. MR 0099996 (20 #6432)
  • [MM59b] Marvin Marcus and B. N. Moyls, Transformations on tensor product spaces, Pacific J. Math. 9 (1959), 1215-1221. MR 0108503 (21 #7219)
  • [MP59] Marvin Marcus and Roger Purves, Linear transformations on algebras of matrices: the invariance of the elementary symmetric functions, Canad. J. Math. 11 (1959), 383-396. MR 0105425 (21 #4167)
  • [MW60] 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 0114823 (22 #5641)
  • [MW02] 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 (2004e:81028)
  • [Nem63] William C. Nemitz, Transformations preserving the Grassmannian, Trans. Amer. Math. Soc. 109 (1963), 400-410. MR 0154882 (27 #4826)
  • [PD95] Vladimir P. Platonov and Dragomir Ž. oković, Linear preserver problems and algebraic groups, Math. Ann. 303 (1995), no. 1, 165-184. MR 1348361 (96m:20072), https://doi.org/10.1007/BF01460985
  • [PLL$^+$92] S. Pierce, M. H. Lim, R. Loewy, C.K. Li, N.K. Tsing, B. McDonald, and L. Beasley, A survey of linear preserver problems, Gordon and Breach Science Publishers, Yverdon, 1992, Linear and Multilinear Algebra 33, no. 1-2.MR 1346777 (96c:15043)
  • [Pop80] V. L. Popov, Classification of spinors of dimension 14, Trans. Moscow Math. Soc. (1980), no. 1, 181-232.
  • [PV94] V. L. Popov and E. B. Vinberg, Invariant theory, Encyclopedia of Mathematical Sciences, vol. 55, pp. 123-284, Springer-Verlag, 1994.
  • [Rev79] 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 (80k:15041)
  • [Röh93] 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 (94k:20082), https://doi.org/10.1090/conm/153/01310
  • [Ros99a] M. Rost, On $ 14$-dimensional quadratic forms, their spinors, and the difference of two octonion algebras, unpublished note, March 1999.
  • [Ros99b] -, On the Galois cohomology of $ {\mathrm {Spin}(14)}$, unpublished note, March 1999.
  • [RRS92] Roger Richardson, Gerhard Röhrle, and Robert Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649-671. MR 1189494 (93j:20092), https://doi.org/10.1007/BF01231348
  • [Rub92] 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 (2009e:20101)
  • [Sch00] 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 (2001h:11135)
  • [Sch08] Gerald W. Schwarz, Linear maps preserving fibers, J. Lie Theory 18 (2008), no. 2, 433-443. MR 2431126 (2009d:20111)
  • [Sei87] Gary M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286. MR 888704 (88g:20092)
  • [Ses77] C. S. Seshadri, Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), no. 3, 225-274. MR 0466154 (57 #6035)
  • [Sha83] Ronald Shaw, Linear algebra and group representations. Vol. II, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1983. Multilinear algebra and group representations. MR 701854 (84m:15003)
  • [Sin83] Richard Sinkhorn, Linear adjugate preservers on the complex matrices, Linear and Multilinear Algebra 12 (1982/83), no. 3, 215-222. MR 678827 (84a:15002), https://doi.org/10.1080/03081088208817485
  • [SK77] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1-155. MR 0430336 (55 #3341)
  • [Sol05] S. Solomon, Irreducible linear group-subgroup pairs with the same invariants, J. Lie Theory 15 (2005), no. 1, 105-123. MR 2115231 (2005j:13008)
  • [Spr06] T. A. Springer, Some groups of type $ E_7$, Nagoya Math. J. 182 (2006), 259-284. MR 2235344 (2007b:20096)
  • [SV00] Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1763974 (2001f:17006)
  • [Wat79] William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York, 1979. MR 547117 (82e:14003)
  • [Wat82] William C. Waterhouse, Linear maps preserving reduced norms, Linear Algebra Appl. 43 (1982), 197-200. MR 656445 (83i:16025), https://doi.org/10.1016/0024-3795(82)90253-1
  • [Wat83] William C. Waterhouse, Invertibility of linear maps preserving matrix invariants, Linear and Multilinear Algebra 13 (1983), no. 2, 105-113. MR 697321 (84i:15020), https://doi.org/10.1080/03081088308817510
  • [Wat87] William C. Waterhouse, Automorphisms of $ {\rm det}(X_{ij})$: the group scheme approach, Adv. in Math. 65 (1987), no. 2, 171-203. MR 900267 (88k:14025), https://doi.org/10.1016/0001-8708(87)90021-1
  • [Wat89] William C. Waterhouse, Linear transformations preserving symmetric rank one matrices, J. Algebra 125 (1989), no. 2, 502-518. MR 1018960 (91d:16059), https://doi.org/10.1016/0021-8693(89)90179-8
  • [Wat95] William C. Waterhouse, Automorphism group schemes of basic matrix invariants, Trans. Amer. Math. Soc. 347 (1995), no. 10, 3859-3872. MR 1303128 (96c:14036), https://doi.org/10.2307/2155207
  • [Web95] H. Weber, Lehrbuch der algebra, Friedrich Vieweg und Sohn, 1895.
  • [Wes64] R. Westwick, Linear transformations on Grassman spaces, Pacific J. Math. 14 (1964), 1123-1127. MR 0167493 (29 #4766)
  • [Wes67] Roy Westwick, Transformations on tensor spaces, Pacific J. Math. 23 (1967), 613-620. MR 0225805 (37 #1397)
  • [Wes69] Roy Westwick, Linear transformations on Grassmann spaces, Canad. J. Math. 21 (1969), 414-417. MR 0241459 (39 #2799)
  • [Zha05] Xian Zhang, Linear/additive preservers of rank 2 on spaces of alternate matrices over fields, Linear Algebra Appl. 396 (2005), 91-102. MR 2112201 (2006a:15011), https://doi.org/10.1016/j.laa.2004.08.028

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47B49, 15A04, 15A72, 20G15

Retrieve articles in all journals with MSC (2010): 47B49, 15A04, 15A72, 20G15


Additional Information

H. Bermudez
Affiliation: Department of Mathematics and Computer Science, MSC W401, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322

S. Garibaldi
Affiliation: Department of Mathematics and Computer Science, MSC W401, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322

V. Larsen
Affiliation: Department of Mathematics and Computer Science, MSC W401, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322

DOI: https://doi.org/10.1090/S0002-9947-2014-06081-9
Received by editor(s): January 24, 2012
Received by editor(s) in revised form: October 20, 2012
Published electronically: April 16, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society