Let k be a field and let A be a finite-dimensional k-algebra of finite global dimension. Choose representatives P_1,…,P_n of the isomorphism classes of indecomposable projective A-modules and representatives S_1,…,S_n of the simple A-modules. Define the Cartan matrix C_A=(c_ij) by c_ij=[P_i:S_j], the composition multiplicity of S_j in P_i. Then det C_A=1.

Cartan determinant conjecture

The conjecture asks whether finite global dimension always forces Cartan determinant $1$ ; it was explicitly posed as an open problem by Dan Zacharia in $1983$ .

Known results

Eilenberg proved that finite global dimension implies $\det C_A\in\{1,-1\}$ .
The conjecture is known for algebras of global dimension $2$ .
It is also known for monomial and quasi-hereditary algebras.

August 2026 representation-finite result

On August 24, 2026, the preprint The Cartan determinant conjecture for representation-finite algebras proved $\det C(A)=1$ for finite-dimensional representation-finite algebras over algebraically closed fields with finite global dimension, plus a related endomorphism-algebra result. It explicitly leaves the unrestricted conjecture open. On August 25, 2026, its author said an AI produced the proof and paper, without identifying a model.

Current status (as of September 2026): The representation-finite case is claimed in an unrefereed preprint, while the general Cartan determinant conjecture remains open.

Cartan determinant conjecture

Primary source

Additional references

Progress summary

Solutions 0