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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.
##### MSC:
 20C30 Representations of finite symmetric groups 15A69 Multilinear algebra, tensor calculus
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##### References:
 [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.
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