Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes. Issue 11 (2nd November 2017)

• Record Type: Journal Article
• Authors: Herman, Allen; Hirasaka, Mitsugu; Oh, Semin
• Is Part Of: Communications in algebra. Volume 45:Issue 11(2017)
• Page Start: 4896
• Page End: 4905
• Publication Date: 2017-11-02
• Subjects: Adjacency algebras -- association schemes -- integral representation theory -- zeta functions; Primary: 16G30 -- Secondary: 05E30 -- 20C15; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1287268
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: The zeta function of an integral lattice Λ is the generating functionζ Λ ( s ) = ∑ n = 0 ∞ a n n − s, whose coefficients count the number of left ideals of Λ of index n . We derive a formula for the zeta function ofΛ 1 ⊗ Λ 2, where Λ 1 and Λ 2 are ℤ -orders contained in finite-dimensional semisimple ℚ -algebras that satisfy a "locally coprime" condition. We apply the formula obtained above to ℤS ⊗ ℤT and obtain the zeta function of the adjacency algebra of the direct product of two finite association schemes ( X, S ) and ( Y, T ) in several cases where the ℤ -orders ℤS and ℤT are locally coprime and their zeta functions are known.