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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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##### References:
  DOI: 10.1016/S0024-3795(73)80004-7 · Zbl 0283.15004 · doi:10.1016/S0024-3795(73)80004-7  Hungerford T. W., Algebra (1974) · Zbl 0293.12001  DOI: 10.1080/03081088608817710 · Zbl 0591.15020 · doi:10.1080/03081088608817710  DOI: 10.1080/03081087908817290 · Zbl 0418.15021 · doi:10.1080/03081087908817290  Serre J-P., Linear Representations of Finite Groups (1977)  Wang B. Y., to appear in Linear and Multilinear Algebra (1977)
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