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Transactions of the American Mathematical Society

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

 
 

 

Groups of linear operators defined by group characters


Authors: Marvin Marcus and James Holmes
Journal: Trans. Amer. Math. Soc. 172 (1972), 177-194
MSC: Primary 20G05
DOI: https://doi.org/10.1090/S0002-9947-1972-0310081-9
MathSciNet review: 0310081
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Abstract: Some of the recent work on invariance questions can be regarded as follows: Characterize those linear operators on $ \operatorname{Hom} (V,V)$ which preserve the character of a given representation of the full linear group. In this paper, for certain rational characters, necessary and sufficient conditions are described that ensure that the set of all such operators forms a group $ \mathfrak{L}$. The structure of $ \mathfrak{L}$ is also determined. The proofs depend on recent results concerning derivations on symmetry classes of tensors.


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

  • [1] A. C. Aitken, Determinants and matrices, Oliver and Boyd, Edinburgh; Interscience, New York, 1962, pp. 90-110.
  • [2] L. B. Beasley, Linear transformations on matrices: The invariance of the third elementary symmetric function, Canad. J. Math. 22 (1970), 746-752. MR 42 #3100. MR 0268201 (42:3100)
  • [3] G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen. I, S.-B. Preuss. Akad. Wiss. Berlin 1897, 994-1015.
  • [4] W. H. Greub, Multilinear algebra, Die Grundlehren der math. Wissenschaften, Band 136, Springer-Verlag, New York, 1967. MR 37 #222. MR 0224623 (37:222)
  • [5] M. Marcus, All linear operators leaving the unitary group invariant, Duke Math. J. 26 (1959), 155-163. MR 21 #54. MR 0101241 (21:54)
  • [6] -, Spectral properties of higher derivations on symmetry classes of tensors, Bull. Amer. Math. Soc. 75 (1969), 1303-1307. MR 41 #245. MR 0255584 (41:245)
  • [7] M. Marcus and W. R. Gordon, The structure of bases in tensor spaces, Amer. J. Math. 92 (1970), 623-640. MR 42 #7684. MR 0272803 (42:7684)
  • [8] M. Marcus and N. A. Kahn, A note on a group defined by a quadratic form, Canad. Math. Bull. 3 (1960), 143-148. MR 23 #A1653. MR 0124339 (23:A1653)
  • [9] M. Marcus and F. May, On a theorem of I. Schur concerning matrix transformations, Arch. Math. 11 (1960), 401-404. MR 24 #A134. MR 0130268 (24:A134)
  • [10] M. Marcus and R. Purves, Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Canad. J. Math. 11 (1959), 383-396. MR 21 #4167. MR 0105425 (21:4167)
  • [11] R. Merris, A generalization of the associated transformation, Linear Algebra and Appl. 4 (1971), 393-406. MR 0294380 (45:3450)
  • [12] I. Schur, Einige Bemerkungen zur Determinantentheorie, S.-B. Preuss. Akad. Wiss. Berlin 25 (1925), Satz II, 454-463.
  • [13] H. W. Turnbull, Theory of equations, Oliver and Boyd, Edinburgh; Interscience, New York, 1952, pp. 71-72.
  • [14] J. H. M. Wedderburn, Lectures on matrices, Amer. Math. Soc. Colloq. Publ., vol. 17, Amer. Math. Soc., Providence, R. I., 1934, 79pp.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0310081-9
Keywords: Representations, characters, linear transformations, elementary divisors, symmetry classes of tensors, derivations on symmetry classes
Article copyright: © Copyright 1972 American Mathematical Society

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