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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ S$-operations in representation theory

Author: Evelyn Hutterer Boorman
Journal: Trans. Amer. Math. Soc. 205 (1975), 127-149
MSC: Primary 20C30
MathSciNet review: 0364424
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Abstract: For $ G$ a group and $ {\text{A} ^G}$ the category of $ G$-objects in a category A$ $, a collection of functors, called ``$ S$-operations,'' is introduced under mild restrictions on A$ $. With certain assumptions on A$ $ and with $ G$ the symmetric group $ {S_k}$, one obtains a unigeneration theorem for the Grothendieck ring formed from the isomorphism classes of objects in $ {\text{A} ^{{S_k}}}$. For A = finite-dimensional vector spaces over $ C$, the result says that the representation ring $ R({S_k})$ is generated, as a $ \lambda $-ring, by the canonical $ k$-dimensional permutation representation. When A = finite sets, the $ S$-operations are called ``$ \beta $-operations,'' and the result says that the Burnside ring $ B({S_k})$ is generated by the canonical $ {S_k}$-set if $ \beta $-operations are allowed along with addition and multiplication.

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Keywords: Representation ring, Grothendieck ring, $ \lambda $-ring, Burnside ring
Article copyright: © Copyright 1975 American Mathematical Society

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