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On the special basis of a certain full symmetry class of tensors. (English) Zbl 1212.15053
Summary: The problem of the existence of a special basis for the symmetry classes of tensors has been studied by several authors. In [Linear Multilinear Algebra 39, No. 3, 241–243 (1995; Zbl 0831.15018)], R. R. Holmes proved that if \(V\) is an \(n\)-dimensional inner product space over \(\mathbb{C}\), \(n\geq 3\), and \(\lambda\) is an irreducible character of \(S_m\) of the form \((m-1,1)\), \(m\geq 3\), then the full symmetry class of tensors associated with \(\lambda\), i.e., \(V_\lambda(S_m)\) is non-zero and does not have a special basis. The nonexistence of a special basis for the full symmetry class of tensors associated with irreducible character \((l_1,l_2)\), \(l_1\geq 3\) of \(S_m\) is concluded by J. A. Dias da Silva and M. M. Torres [Linear Algebra Appl. 401, 77–107 (2005; Zbl 1077.15025)], by computation of the orthogonal dimension of critical orbital sets.
In this paper we prove this result by a new method. Indeed, we show that if \(V\) is an \(n\)-dimensional inner product space over \(\mathbb{C}\), \(n\geq 2\) and \(\lambda\) is an irreducible character of \(S_m\) of the form \((m-l,l)\), \(m\geq 3l\), then \(V_\lambda(S_m)\) is non-zero and has no special basis.
MSC:
15A72 Vector and tensor algebra, theory of invariants
15A69 Multilinear algebra, tensor calculus
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
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