Induced operators on the generalized symmetry classes of tensors
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.
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