Computing zeta functions of table algebra orders using local zeta integrals

TL;DR

This paper explores an efficient method to compute Solomon's zeta functions for orders in integral table algebras, including adjacency algebras of association schemes, using local zeta integrals.

Contribution

It introduces a local zeta integral approach to overcome computational difficulties in calculating zeta functions for higher-rank orders.

Findings

Successfully computed explicit zeta functions for several examples.

Demonstrated the effectiveness of local zeta integrals over elementary methods.

Provided a practical framework for future computations in algebraic structures.

Abstract

We investigate Solomon's zeta function for orders in the special case of orders generated by the standard basis of an integral table algebra, a special case of which is the integral adjacency algebra of an association scheme. As Solomon's elementary method for computing this zeta function runs into computational difficulties for ranks $3$ or more, a more efficient method is desired. We give several examples to illustrate how the local zeta integral approach proposed by Bushnell and Reiner can be applied to compute explicit zeta functions for these orders.