THE CYCLOTOMIC IDENTITY AND THE CYCLIC GROUP

A. Dress and Ch. Siebeneicher

Universität Bielefeld

Metropolis and Rota [1] discuss the cyclotomic identity

\frac{1}{1 − αt} = ∏_{n∈ℕ} (\frac{1}{1 − t^{n}} )^{M(α,n)} .

Thereby they define the necklace-algebra Nr(A) for a commutative ring A with unit element 1. We show that Nr(A) is the Burnside-ring Ω̂(ℤ) of almost finite ℤ-sets, where an almost finite ℤ-set is a set with an operation of the infinite cyclic group ℤ, such that every element lies in a finite orbit and every orbit type ℤ/nℤ occurs only with finite multiplicity. For every n ∈ ℕ we have a homomorphism

φ_{n} : Ω̂(ℤ) ⟶ ℤ,

which assigns to a ℤ-set the number of elements invariant under the subgroup nℤ. The family φ_{n} with n > 0 defines an injective homomorphism

φ : Ω̂(ℤ) ⟶ ℤ^{J}

of Ω̂(ℤ) into the product ring ℤ^{J}, where J denotes the set of positive integers. By defining symmetric powers of almost finite ℤ-sets, we get a homomorphism

s_{t} : Ω̂(ℤ) ⟶ 1 + tΩ(ℤ)[[t]]

of the additive group Ω̂(ℤ) into the multiplicative group of formal power-series with constant term 1 and coefficients in the ring Ω̂(ℤ). Combining this homomorphism with φ_{1}, we get the isomorphism

φ_{1} ∘ s_{t} : Ω̂(ℤ) ⟶ 1 + tℤ[[t]].

Finally we show that Ω̂(ℤ) is isomorphic to the (generalized) ring of Witt-vectors W (ℤ).

The Burnside-ring construction applies to every profinite group providing thereby a generalized cyclotomic(?) identity. The construction allows also to define for an arbitrary profinite group and given commutative ring A a generalized ring of Wittvectors over A.

References

1. N. Metropolis and C.-C. Rota, Witt vectors and the Algebra of Necklaces, Adv. in Math. 50 (1983), 95–125.