Induced operators on the generalized symmetry classes of tensors

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

• irreducible representation
• generalized schur function
• generalized symmetrizer
• generalized symmetry class of tensors
• induced operator