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- W1182051282 abstract "Let G be a subgroup of the symmetric group Sn for n ∈ N and let V be an inner product space. Orthogonality properties of the set of standard (decomposable) symmetrized tensors in V ⊗n corresponding to G have been studied for more than two decades [WG91, HT92, Hol95, DP99, BPR03, Hol04, TS12]. The determination of such properties would be facilitated by an understanding of the more general geometric properties of this set. Building on fairly isolated insights throughout the relevant literature, we obtain some new results, and in doing so we begin to weave a coherent narrative for the future exploration of this geometry. The space V ⊗n is an orthogonal direct sum of orbital subspaces, so it suffices to study the sets of standard symmetrized tensors in these subspaces. In order to do so, we investigate for each irreducible character χ of G and each subgroup H of G the set Σ of standard vectors in what we call the coset space C H . This coset space is itself a vector space and a CG-module, and for each (H,χ)-pairing it corresponds to one and only one orbital subspace of V ⊗n. Hence the coset space serves as a proxy tool for our inquiries into the geometry of a given orbital subspace. The structure of C H as a vector space is easily understood, but for the sake of parallelism with orbital subspaces, we need also to understand its structure as a CG-module. The first result furnishes an isomorphism that renders this structure in the well-known terms of the associated group algebra. The isomorphism may prove useful in the future, as groundwork for a more module-theoretic approach to these matters." @default.
- W1182051282 created "2016-06-24" @default.
- W1182051282 creator A5087304222 @default.
- W1182051282 date "2014-05-01" @default.
- W1182051282 modified "2023-09-27" @default.
- W1182051282 title "Some Geometry Of Symmetrized Tensor Spaces" @default.
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