Cartan determinant conjecture

Determine whether the Cartan determinant of a finite-dimensional algebra of finite global dimension can equal -1, or equivalently whether the Cartan determinant is necessarily 1.

Background

The paper recalls Eilenberg’s theorem that the Cartan matrix of a finite-dimensional algebra with finite global dimension has determinant either 1 or -1. It identifies as unresolved the possibility that the determinant equals -1, a question known as the Cartan determinant conjecture.

References

It is a major open problem, called the Cartan determinant conjecture, whether the Cartan determinant can be -1 or not, see [FZ] and [Z].

— Auslander regular algebras and Coxeter matrices  (2501.09447 - Klász et al., 16 Jan 2025) in Section 1, immediately before Corollary 1.3