Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes

ABSTRACT

The zeta function of an integral lattice Λ is the generating function ${\zeta }_{\Lambda }\left(s\right)={\sum }_{n=0}^{\infty }{a}_n{n}^{-s}$, whose coefficients count the number of left ideals of Λ of index n. We derive a formula for the zeta function of ${\Lambda }_1\otimes {\Lambda }_2$, where Λ₁ and Λ₂ 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.

KEYWORDS:

• Adjacency algebras
• association schemes
• integral representation theory
• zeta functions

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16G30
• Secondary: 05E30
• 20C15

Acknowledgment

We would like to take the opportunity to thank the referee for their observations that helped improve the quality of this article.