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Polynomial representations of \(\mathrm{GL }(n)\) and Schur–Weyl duality

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Abstract

Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur–Weyl duality is the key for an equivalence between both categories.

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Notes

  1. Given a basis \((v_i)_{i\in I}\) of \(V\), there are homogeneous polynomials \(f_{ij}\in k[X_{rs}]\) of degree \(d\) in \(n^2\) indeterminates such that \(\rho (\alpha )(v_i)=\sum _{j\in I}f_{ij}(\alpha _{rs}) v_j\) for each \(\alpha =(\alpha _{rs})\) in \(\mathrm{GL }_k(n)\).

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Acknowledgments

I am grateful to Steve Donkin, Steve Doty, and Andrew Hubery for helpful comments.

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  1. Fakultät für Mathematik, Universität Bielefeld, 33501, Bielefeld, Germany

    Henning Krause

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  1. Henning Krause
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Correspondence to Henning Krause.

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Krause, H. Polynomial representations of \(\mathrm{GL }(n)\) and Schur–Weyl duality. Beitr Algebra Geom 56, 769–773 (2015). https://doi.org/10.1007/s13366-015-0237-7

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  • DOI: https://doi.org/10.1007/s13366-015-0237-7

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