Counting cosets of unimodular groups over Dedekind domains

TL;DR

This paper develops a formula to count right cosets within double cosets of unimodular groups over Dedekind domains, with applications to index calculations and Hecke algebra computations.

Contribution

It introduces a new formula for counting cosets and applies it to index theory and Hecke algebra calculations in the context of Dedekind domains.

Findings

Derived a formula for coset counting in unimodular groups over Dedekind domains

Provided an index formula for congruence subgroups

Developed an algorithm for explicit Hecke algebra product calculations

Abstract

In this paper, a formula for the calculation of the number of right cosets contained in a double coset with respect to a unimodular group over a Dedekind domain is developed, and applications of this formula in the theory of congruence subgroups -- an index formula -- and the theory of abstract Hecke algebras -- a reduction theorem and an algorithm for the explicit calculation of products -- are given.