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 $Uotimes V^{otimes m}$‎, ‎$$ S_{Lambda}(uotimes v^{otimes})=dfrac{1}{|G|}sum_{sigmain G}Lambda(sigma)uotimes v_{sigma^{-1}(1)}otimescdotsotimes v_{sigma^{-1}(m)} $$ defined by $G$ and $Lambda$‎. ‎The image of $Uotimes V^{otimes m}$ 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 $S_{Lambda}(uotimes v^{otimes})$ are called generalized decomposable tensors and are denoted by $ucircledast v^{circledast}$‎. ‎For any linear operator $T$ acting on $V$‎, ‎there is a unique induced operator $K_{Lambda}(T)$ acting on $V_{Lambda}(G)$ satisfying $$ K_{Lambda}(T)(uotimes v^{otimes})=ucircledast Tv_{1}circledast cdots circledast Tv_{m}‎. ‎$$ 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.