Symmetrized tensors and spherical functions. (English) Zbl 1400.15026
Bebiano, Natália (ed.), Applied and computational matrix analysis. MAT-TRIAD, Coimbra, Portugal, September 7–11, 2015. Selected, revised contributions. Cham: Springer (ISBN 978-3-319-49982-6/hbk; 978-3-319-49984-0/ebook). Springer Proceedings in Mathematics & Statistics 192, 253-262 (2017).
Summary: Let \(G\) be a subgroup of the symmetric group and \(\varphi \) a complex function on \(G\). A longstanding question in Multilinear Algebra is to find conditions for the vanishing of the decomposable symmetrized tensor associated with \(G\) and \(\varphi \) (we recall the definition below). When \(\varphi \) is an irreducible complex character of \(G\), the problem has been studied by several authors, see for example [J. A. Dias da Silva, Linear Multilinear Algebra 27, No. 1, 25–32 (1990; Zbl 0739.05020); J. A. Dias da Silva and A. Fonseca, Linear Multilinear Algebra 27, No. 1, 49–55 (1990; Zbl 0701.15026); C. Gamas, Linear Algebra Appl. 108, 83–119 (1988; Zbl 0652.15023); T. H. Pate, Linear Multilinear Algebra 28, No. 3, 175–184 (1990; Zbl 0722.15032)]. In the present paper we study and solve the vanishing problem for the case when \(G\) is the full symmetric group and \(\varphi \) is a certain type of spherical function.
For the entire collection see [Zbl 1369.65002].
For the entire collection see [Zbl 1369.65002].
MSC:
| 15A72 | Vector and tensor algebra, theory of invariants |
| 15A69 | Multilinear algebra, tensor calculus |
| 43A90 | Harmonic analysis and spherical functions |
| 20B30 | Symmetric groups |
References:
| [1] | Dias da Silva, J.A.: On \(\(μ -\)\)colorings of a matroid. Linear Multilinear Algebra 27, 25-32 (1990) · Zbl 0739.05020 · doi:10.1080/03081089008817990 |
| [2] | Dias da Silva, J.A., Fonseca, A.: Nonzero star products. Linear Multilinear Algebra 27, 49-55 (1990) · Zbl 0701.15026 · doi:10.1080/03081089008817992 |
| [3] | Gamas, C.: Conditions for a symmetrized decomposable tensor to be zero. Linear Algebra Appl. 108, 83-119 (1988) · Zbl 0652.15023 · doi:10.1016/0024-3795(88)90180-2 |
| [4] | Gamas, C.: Symmetrized Tensors and spherical functions. Submitted · Zbl 0962.15004 |
| [5] | Pate, T.H.: Immanants and decomposable tensors that symmetrized to zero. L. M. A. 28, 175-184 (1990) · Zbl 0722.15032 |
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