Witt vectors and Tambara functors
In ``The Burnside ring of profinite groups and the Witt vector construction'' [Adv. Math. 70, 87--132 (1988; Zbl 0691.13026)] \textit{A. Dress} and \textit{C. Siebeneicher} constructed for each profinite group \(G\) an endofunctor \(\mathcal {W}_G\) on the category of commutative rings, which for \(G = \mathbb Z_p\) assigns to a commutative ring \(A\) the ring of \(p\)-typical Witt vectors. For \(A=\mathbb Z\) the construction yields a completed Burnside ring for the pro-finite group \(G\). \smallskip The author gives a combinatorial description of \(\mathcal {W}_G(A)\) for a polynomial algebra \(A\) over \(\mathbb Z\) with trivial \(G\)-action. This result is then used for finite groups \(G\) to describe the relation between \(\mathcal {W}_G\) and so-called Tambara functors introduced by \textit{D. Tambara} [``On multiplicative transfer'', Commun. Algebra 21, 1393--1420 (1993; Zbl 0797.19001)]. The functor \(\mathcal {W}_G\) is shown to be related to the left adjoint of the algebraic functor from the category of \(G\)-Tambara functors to the category of commutative rings with an action of \(G\) via ring automorphisms.
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