The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, \(\lambda\)-rings, and the universal ring of Witt vectors
Let \(a(t)=1+\sum_{n=1}a_ nt^ n \in {\mathbb{Z}}t\) be a formal power series with integral coefficients. The authors establish isomorphisms between: the Burnside ring \({\hat \Omega}(C)\) of the infinite cyclic group C, the Grothendieck ring \(\Lambda(Z)\) [\textit{A. Grothendieck}, Bull. Soc. Math. France 86 (1958), 137-154 (1959; Zbl 0091.331)] of formal power series with constant term 1 and the ring of Witt vectors \(W(Z)\) [\textit{E. Witt}, J. Reine Angew. Math. 176, 126-140 (1937; Zbl 0016.05101)]. Then the well-known product decomposition \[ a(t)=\prod^{\infty}_{n=1}(\frac{1}{1-t^ n})^{b_ n} =\prod^{\infty}_{n=1}\frac{1}{1-q_ nt^ n} =\exp (\int \prod^{\infty}_{n=1}d_ nt^{n-1}dt ) \] and a surprisingly simple combinatorial interpretation of the relations between the systems \((a_ 1,a_ 2,...)\), \((b_ 1,b_ 2,...)\), \((q_ 1,q_ 2,...)\), and \((d_ 1,d_ 2,...)\) are obtained.
product decomposition of formal power series
scientific article; zbMATH DE number 4130521
scientific article