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On the orthogonal dimension of orbital sets. (English) Zbl 1077.15025
Let $$V$$ be a complex inner product vector space and let $$e_1,\dots, e_n$$ be an orthonormal basis of $$V$$. Let $$\lambda$$ be a complex irreducible representation of a subgroup $$G$$ of the full symmetric group $$S_m$$. Consider on $$V^{\otimes m}$$ the linear operator $$T(G,\lambda):=\frac{\lambda (\text{id})}{| G| }\sum_{\sigma\in G}\lambda(\sigma)P(\sigma)$$ where $$P(\sigma)$$ is defined by $$P(\sigma)(v_1\otimes\dots\otimes v_m)= v_{\sigma^{-1}(1)}\otimes\dots\otimes v_{\sigma^{-1}(m)}$$.
The authors give a combinatorial necessary and sufficient condition for orthogonality of critical decomposable symmetric tensors $$e^*_\alpha:=T(G,\lambda)(e_{\alpha(1)}\otimes\dots\otimes e_{\alpha(m)})$$ and $$e^*_\beta:=T(G,\lambda)(e_{\beta(1)}\otimes\dots\otimes e_{\beta(m)})$$. The notion of sign-uniform partition is introduced and the set of such partitions is described. The characterization of the sign-uniform partitions is used for obtaining more manageable conditions of orthogonality of $$e^*_\alpha$$ and $$e^*_\beta$$. The concept of orthogonal dimension of a finite set of nonzero vectors is introduced. The orthogonal dimension of critical orbital sets is computed for a class of irreducible characters $$\lambda$$.

MSC:
 15A72 Vector and tensor algebra, theory of invariants
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References:
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