Zeta functions of generalized permutations with application to their factorization formulas
type: Item
instance of: scholarly article
title: Zeta functions of generalized permutations with application to their factorization formulas
zbMATH Open document ID: 1275.11126
author: Shin-ya Koyama
author: Sachiko Nakajima
published in: Proceedings of the Japan Academy. Series A
publication date: 2013-01-16T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1349355141
review text: 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.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 6126093
zbMATH Keywords: zeta functions
zbMATH Keywords: the field with one element
zbMATH Keywords: absolute mathematics
zbMATH Keywords: generalized permutation groups
reviewed by: Anton Deitmar
MaRDI profile type: Publication
OpenAlex ID: W2080603774
cites work: Representations of the generalized symmetric groups
cites work: The Schur Multiplier of the Generalized Symmetric Group
cites work: The Riemann hypothesis and functional equations for zeta functions over \(\mathbb F_1\)
DOI: 10.3792/PJAA.88.115
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description: scientific article; zbMATH DE number 6126093