%0 Journal Article
%T Induced operators on the generalized symmetry classes of tensors
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Rafatneshan, Gholamreza
%A Zamani, Yousef
%D 2021
%\ 12/01/2021
%V 10
%N 4
%P 197-211
%! Induced operators on the generalized symmetry classes of tensors
%K irreducible representation
%K generalized Schur function
%K generalized symmetrizer
%K generalized symmetry class of tensors
%K induced operator
%R 10.22108/ijgt.2020.122990.1622
%X 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.
%U https://ijgt.ui.ac.ir/article_25097_cf1ef37a223382337243a31231e11412.pdf