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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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