Subregular J-Rings of the Finite Irreducible Coxeter Systems

TL;DR

This paper demonstrates how to classify the algebraic structure of subregular J-rings for finite irreducible Coxeter systems by establishing isomorphisms to matrix rings over fields, completing the classification in all cases.

Contribution

It extends previous work by applying isomorphism theorems to all remaining cases, showing that their path algebras are isomorphic to sums of matrix rings over rational fields or their extensions.

Findings

Path algebras are isomorphic to matrix rings over fields.

Complete classification of subregular J-rings for all finite irreducible Coxeter systems.

Identification of algebraic structures as sums of matrix rings over rational fields or extensions.

Abstract

In previous papers, the author showed that in many cases of interest there exists an isomorphism between certain path algebras related to the structure of the subregular J-rings of Coxeter systems and matrix rings over a free product of rings. For the finite irreducible Coxeter systems, the application of the isomorphism theorems was straightforward in all but a few cases. In this paper we show how the isomorphism theorems can be applied to show that the path algebras in the remaining cases are isomorphic to a direct sum of matrix rings over the rational numbers or over a direct product of field extensions of the rational numbers.