Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T14:59:15.751Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational Tensor Representations of Hom(V, V) and an Extension of an Inequality of I. Schur.

Published online by Cambridge University Press:  20 November 2018

Marvin Marcus  and
William Robert Gordon
Marvin Marcus
Affiliation:
University of California, Santa Barbara, California
William Robert Gordon
Affiliation:
University of Victoria, Victoria, British Columbia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let V be an n-dimensional vector space over the complex numbers equipped with an inner product (x, y), and let (P, μ) be a symmetry class in the mth tensor product of V associated with a permutation group G and a character χ (see below). Then for each T ∊ Hom (V, V) the function φ which sends each m-tuple (v1, … , vm) of elements of V to the tensor μ(TV1, … , Tvm) is symmetric with respect to G and x, and so there is a unique linear map K(T) from P to P such that φ = K(T)μ.

It is easily checked that K: Hom(V, V) → Hom(P, P) is a rational representation of the multiplicative semi-group in Hom(V, V): for any two linear operators S and T on V

K(ST) = K(S)K(T).

Moreover, if T is normal then, with respect to the inner product induced on P by the inner product on V (see below), K(T) is normal.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 686 - 695
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Rainer, Kress, Hans Ludwig, de Vries, and Rudolf, Wegmann, On non-normal matrices (to appear in Linear Algebra and Appl.).Google Scholar
2. Marvin, Marcus and Henryk, Minc, Introduction to linear algebra (Macmillan Company, New York, 1965).Google Scholar
3. Marvin, Marcus and Henryk, Minc, Generalized matrix functions, Trans. Amer. Math. Soc. 116 (1965), 316–329.Google Scholar
4. Marvin, Marcus and Henryk, Minc, Permutations on symmetry classes, J. Algebra 5 (1967), 59–71.Google Scholar
5. Marvin, Marcus, On two classical results of I. Schur, Bull. Amer. Math. Soc. 70 (1964), 685–688.Google Scholar
6. Murnaghan, F. D., On the unitary invariants of a square matrix, An. Acad. Brasil. Ci. 26 (1954), 1–7.Google Scholar