# A counterexample to a conjecture of Kiyota, Murai and Wada

## TL;DR

This paper presents a counterexample disproving a 2002 conjecture that the largest eigenvalue of a block's Cartan matrix is rational if and only if all eigenvalues are rational, challenging previous assumptions in group theory.

## Contribution

The authors provide the first known counterexample to the conjecture, showing that the largest eigenvalue can be irrational even when all eigenvalues are rational.

## Key findings

- Counterexample with irrational largest eigenvalue
- Disproves the if-and-only-if condition of the conjecture
- Discusses implications for the theory of Cartan matrices

## Abstract

Kiyota, Murai and Wada conjectured in 2002 that the largest eigenvalue of the Cartan matrix C of a block of a finite group is rational if and only if all eigenvalues of C are rational. We provide a counterexample to this conjecture and discuss related questions.