Zeta Functions of Lattices of the Symmetric Group. Issue 5 (3rd May 2016)

• Record Type: Journal Article
• Authors: Hofmann, Tommy
• Is Part Of: Communications in algebra. Volume 44:Issue 5(2016)
• Page Start: 2243
• Page End: 2255
• Publication Date: 2016-05-03
• Subjects: Integral representation -- Symmetric group -- Zeta function; Primary: 20C10 -- Secondary: 11S45; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2015.1044102
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract : The symmetric group 𝔖 n +1 of degree n + 1 admits an n -dimensional irreducible Q𝔖 n -module V corresponding to the hook partition (2, 1 n −1 ). By the work of Craig and Plesken, we know that there are σ( n + 1) many isomorphism classes of Z𝔖 n +1 -lattices which are rationally equivalent to V, where σ denotes the divisor counting function. In the present article, we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the Z𝔖 n +1 -lattice defined by the Specht basis.