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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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