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Induced operators on the generalized symmetry classes of tensors. (English) Zbl 07324052
Summary: 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.
MSC:
20C30 Representations of finite symmetric groups
15A69 Multilinear algebra, tensor calculus
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