# Induced operators on the generalized symmetry classes of tensors

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Basic Sciences, Sahand University of Technology, P.O. Box 51335/1996, Tabriz, Iran

Abstract

‎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.

Keywords

Main Subjects

#### References

[1] R. Bhatia and J. A. Dias da Silva, Variation of induced linear operators, Linear Algebra Appl., 341 (2002) 391–402.
[2] T. G. Lei, Generalized Schur functions and generalized decompossble symmetric tensors, Linear Algebra Appl., 263
(1997) 311–332.
[3] C. K. Li and T. Y. Tam, Operator properties of T and K(T), Linear Algebra Appl., 401 (2005) 173–191.
[4] C. K. Li and A. Zaharia, Induced operators on symmetry classes of tensors, Trans. Amer. Math. Soc., 342 (2001)
807–836.
[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.
[7] G. Rafatneshan and Y. Zamani, On the orthogonal basis of the generalized symmetry classes of tensors, to submitted.
[8] M. Ranjbari and Y. Zamani, Induced operators on symmetry classes of polynomials, Int. J. Group Theory, 6 no. 2
(2017) 21–35.
[9] Y. Zamani and S. Ahsani, On the decomposable numerical range of operators, Bull. Iranian. Math. Soc., 40 no. 2
( 2014) 387–396.
[10] Y. Zamani and M. Ranjbari, Representations of the general linear group over symmetry classes of polynomials,
Czechoslovak Math. J., 68 no. 143 (2018) 267–276.