Zeta functions of generalized permutations with application to their factorization formulas (Q1931917)

In the field of geometry over the field \({\mathbb F}_1\) of one element, although there is no agreement on the right formal framework yet, most authors agree that there exists algebraic extensions \({\mathbb F}_{1^m}\) coming about by attaching \(m\)-th roots of unity. Further, in the framework of monoid schemes the group \(\mathrm{GL}_n({\mathbb F}_1)\) is the permutation group \(\mathrm{Per}(n)\) in \(n\) letters, so \(\mathrm{GL}_n({\mathbb F}_{1^m})\) is the so called generalized permutation group, that is, the group of all monomial matrices in \(\mathrm{GL}_n\) with entries in the group \(\mu_n\) of \(n\)-th roots of unity. In the previous paper [Proc. Japan Acad., Ser. A 85, No. 6, 75--80 (2009; Zbl 1275.11125)],\textit{S. Kim, S.-y. Koyama} and \textit{N. Kurokawa} established the functional equation of the dynamical Lefschetz-type \(L\)-function attached to an element \(\sigma\) of \(\mathrm{GL}_n({\mathbb F}_1)\). The clearly-written current paper contains the generalization of these results to \(\mathrm{GL}_n({\mathbb F}_{1^m})\). It is shown that the corresponding \(L\)-function has a natural determinant expression which yields meromorphic continuation, that is satisfies a natural functional equation and that it admits a factorization according to the factorization of a permutation into cycles.