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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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When do the symmetric tensors of a commutative algebra form a Frobenius algebra?
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by Annetta G. Aramova and Luchezar L. Avramov PDF
Proc. Amer. Math. Soc. 85 (1982), 299-304 Request permission

Abstract:

For a commutative $k$-algebra $B$, consider the subalgebra ${({B^{ \otimes n}})^{{S_n}}}$ of the $n$th tensor power of $B$, formed by the tensors invariant under arbitrary permutations of the indices. Necessary and sufficient conditions are found for ${({B^{ \otimes n}})^{{S_n}}}$ to be Frobenius. When ${\dim _k}B \ne 2$, these say that $B$ is Frobenius and $n!$! is invertible in $k$, unless $B$ is separable. Some additional cases occur for two-dimensional algebras in positive characteristic, depending on the divisibility of $n + 1$.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 299-304
  • MSC: Primary 13E10
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0656088-5
  • MathSciNet review: 656088