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Transactions of the American Mathematical Society

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A class of representations of the full linear group. II


Author: Stephen Pierce
Journal: Trans. Amer. Math. Soc. 173 (1972), 251-262
MSC: Primary 20G05
DOI: https://doi.org/10.1090/S0002-9947-1972-0310082-0
MathSciNet review: 0310082
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Abstract: Let $ V$ be an $ n$-dimensional vector space over complex numbers $ C$. Let $ W$ be the $ m$th tensor product of $ V$. If $ T \in {\operatorname{Hom} _C}(V,V)$, let $ { \otimes ^m}T \in {\operatorname{Hom} _C}(W,W)$ be the $ m$th tensor product of $ T$. The homomorphism $ T \to { \otimes ^m}T$ is a representation of the full linear group $ {\text{G}}{{\text{L}}_n}(C)$. If $ H$ is a subgroup of the symmetric group $ {S_m}$, and $ \chi $ a linear character on $ H$, let $ V_\chi ^m(G)$ be the subspace of $ W$ consisting of all tensors symmetric with respect to $ H$ and $ \chi $. Then $ V_\chi ^m(H)$ is invariant under $ { \otimes ^m}T$. Let $ K(T)$ be the restriction of $ { \otimes ^m}T$ to $ V_\chi ^m(H)$. For $ n$ large compared with $ m$ and for $ H$ transitive, we determine all cases when the representation $ T \to K(T)$ is irreducible.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0310082-0
Keywords: Young tableaus, symmetry class of tensors, permutation groups
Article copyright: © Copyright 1972 American Mathematical Society

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