Rational Tensor Representations of Hom(V, V) and an Extension of an Inequality of I. Schur
| Autor: | Marvin Marcus, William R. Gordon |
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| Rok vydání: | 1972 |
| Předmět: | |
| Zdroj: | Canadian Journal of Mathematics. 24:686-695 |
| ISSN: | 1496-4279 0008-414X |
| DOI: | 10.4153/cjm-1972-064-8 |
| Popis: | 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 VK(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. |
| Databáze: | OpenAIRE |
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