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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
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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

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