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We study the derived tensor product of the representation rings of subgroups of a given compact Lie group G. That is, given two such subgroups $H_1$ and $H_2$, we study the tensor product of the associated representation rings $\mathrm{R}H_1$ and $\mathrm{R}H_2$ over the representation ring $\mathrm{R}G$, and prove a vanishing result for the associated higher Tor groups. This result can be viewed as a natural extension of the Theorem of Steinberg that asserts that the representation rings of maximal rank subgroups of G are free over $\mathrm{R}G$. It may also be viewed as an analogue of a result of Singhof on the cohomology of classifying spaces. We include an immediate application to the complex K-theory of biquotient manifolds.

We provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.

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An extension of Steinberg’s theorem to biquotient pairs of subgroups

The Representation Ring of the Twisted Quantum Double of a Finite Group

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2 results

An extension of Steinberg’s theorem to biquotient pairs of subgroups

The Representation Ring of the Twisted Quantum Double of a Finite Group