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On the orthogonal bases of symmetry classes. (English) Zbl 0935.15024
Let $$V$$ be an $$n$$-dimensional complex inner product space and let $$\{e_1,\dots,e_n\}$$ be an orthonormal basis of $$V$$. Suppose $$G$$ is a permutation group of degree $$m$$ and $$X$$ is an irreducible character of $$G$$. The symmetry class of tensors associated with $$G$$ and $$X$$ is denoted by $$V_X(G)$$. This note contains one major theorem. The author discusses the problem of existing orthogonal bases for $$V_X(G)$$ consisting of symmetrized decomposable tensors $$e_\alpha^*$$.
Reviewer: Y.Kuo (Knoxville)

##### MSC:
 15A72 Vector and tensor algebra, theory of invariants
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##### References:
 [1] Freese, R., Inequalities for generalized matrix functions based on arbitrary characters, Linear algebra appl., 7, 337-345, (1973) · Zbl 0283.15004 [2] Holmes, R.R., Orthogonal bases of symmetrized tensor spaces, Linear and multilinear algebra, 39, 241-243, (1995) · Zbl 0831.15018 [3] Holmes, R.R.; Tam, T.Y., Symmetry classes of tensors associated with certain groups, Linear and multilinear algebra, 32, 21-31, (1992) · Zbl 0762.15015 [4] Marcus, M., Finite dimensional multilinear algebra, part I, (1973), Dekker New York · Zbl 0284.15024 [5] Merris, R., Recent advances in symmetry classes of tensors, Linear and multilinear algebra, 7, 317-328, (1979) · Zbl 0418.15021
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