zbMATH — the first resource for mathematics

Induced operators on the generalized symmetry classes of tensors. (English) Zbl 07324052
Summary: Let \(V\) be a unitary space. Suppose \(G\) is a subgroup of the symmetric group of degree \(m\) and \(\Lambda\) is an irreducible unitary representation of \(G\) over a vector space \(U\). Consider the generalized symmetrizer on the tensor space \(U\otimes V^{\otimes m}\), \[ S_\Lambda (u\otimes v^\otimes)=\dfrac{1}{|G|} \sum_{\sigma\in G} \Lambda (\sigma)u \otimes v_{\sigma^{-1}(1)}\otimes \cdots \otimes v_{\sigma^{-1}(m)} \] defined by \(G\) and \(\Lambda\). The image of \(U\otimes V^{\otimes m}\) under the map \(S_\Lambda\) is called the generalized symmetry class of tensors associated with \(G\) and \(\Lambda\) and is denoted by \(V_\Lambda (G)\). The elements in \(V_\Lambda (G)\) of the form \(S_\Lambda (u\otimes v^\otimes)\) are called generalized decomposable tensors and are denoted by \(u{\circledast} v^{\circledast}\). For any linear operator \(T\) acting on \(V\), there is a unique induced operator \(K_\Lambda (T)\) acting on \(V_\Lambda (G)\) satisfying \[ K_\Lambda (T)(u \otimes v^\otimes)=u\circledast Tv_1 \circledast \cdots \circledast Tv_m. \] If \(\dim U=1\), then \(K_\Lambda (T)\) reduces to \(K_\lambda (T)\), induced operator on symmetry class of tensors \(V_\lambda (G)\). In this paper, the basic properties of the induced operator \(K_\Lambda (T)\) are studied. Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions.
20C30 Representations of finite symmetric groups
15A69 Multilinear algebra, tensor calculus
Full Text: DOI
[1] R. Bhatia and J. A. Dias da Silva, Variation of induced linear operators,Linear Algebra Appl.,341(2002) 391-402. · Zbl 1151.47301
[2] T. G. Lei, Generalized Schur functions and generalized decompossble symmetric tensors,Linear Algebra Appl.,263 (1997) 311-332. · Zbl 0894.15015
[3] C. K. Li and T. Y. Tam, Operator properties ofTandK(T),Linear Algebra Appl.,401(2005) 173-191. · Zbl 1076.15024
[4] C. K. Li and A. Zaharia, Induced operators on symmetry classes of tensors,Trans. Amer. Math. Soc.,342(2001) 807-836. · Zbl 0984.15025
[5] R. Merris,Multilinear Algebra, Gordon and Breach Science Publisher, Amsterdam, 1997.
[6] G. Rafatneshan and Y. Zamani, Generalized symmetry classes of tensors,Czechoslovak Math. J.,Published online July 8(2020) 1-13. · Zbl 07285970
[7] G. Rafatneshan and Y. Zamani,On the orthogonal basis of the generalized symmetry classes of tensors, to submitted. · Zbl 07285970
[8] M. Ranjbari and Y. Zamani, Induced operators on symmetry classes of polynomials,Int. J. Group Theory,6no. 2 (2017) 21-35. · Zbl 1445.20006
[9] Y. Zamani and S. Ahsani, On the decomposable numerical range of operators,Bull. Iranian. Math. Soc.,40no. 2 ( 2014) 387-396. · Zbl 1302.15029
[10] Y.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.