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Symmetry classes of tensors associated with certain groups. (English) Zbl 0762.15015
Let \(V\) be a complex inner product space of dimension \(m\). To each permutation \(\alpha\) in the full symmetric group \(S_ n\) there corresponds a linear operator \(P(\alpha)\) on \(\otimes^ nV\) determined by \(P(\alpha)v_ 1\otimes\cdots\otimes v_ n=v_{\alpha^{- 1}(1)}\otimes\cdots\otimes v_{\alpha^{-1}(n)}\).
Let \(G\) be a subgroup of \(S_ n\) and \(\chi\) an irreducible character of \(G\). Define \(T(G,\chi)=(\chi(e)/| G|)\) \(\sum_{\sigma\in G}\chi(\sigma)P(\sigma)\). The image of \(\otimes^ nV\) under the map \(T(G,\chi)\) is called the symmetry class of tensors associated with \(G\) and \(\chi\) and is denoted \(V_ \chi(G)\). The image of \(v_ 1\otimes\cdots\otimes v_ n\) under \(T(G,\chi)\) is denoted \(v_ 1*\cdots*v_ n\).
Let \(\{e_ 1,\cdots,e_ m\}\) be a basis of \(V\). Denote by \(e^*\) the tensor \(e_{\gamma(1)}*\cdots*e_{\gamma(n)}\) for any sequence \(\gamma=(\gamma_ 1,\dots,\gamma_ n)\) with \(i\leq\gamma_ i\leq m\).
The first goal of this paper is the investigation of the existence of a set \(S\) of sequences \(\gamma\) such that \(\{e^*_ \gamma|\gamma\in S\}\) is an orthogonal basis of \(V_ \chi(G)\). The second goal is to figure out the dimension of the corresponding symmetry classes of tensors. Actually the cases when \(G\) is a cyclic group, a dihedral group, \(A_ 4\) and \(S_ 4\), are considered.
Reviewer: V.L.Popov (Moskva)

MSC:
15A72 Vector and tensor algebra, theory of invariants
20C30 Representations of finite symmetric groups
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