On the Positivity of Numerical Cartan Determinants

In this talk, I will introduce a dg-categorical analogue of the Cartan determinant conjecture based on numerical Grothendieck groups. For a smooth proper dg category over an algebraically closed field, the Euler pairing induces a nondegenerate bilinear form on its numerical Grothendieck group, and one can therefore define the corresponding numerical Cartan determinant.

The classical Cartan determinant conjecture predicts that this determinant is equal to 1 for finite-dimensional smooth algebras. In the more general setting of smooth proper dg categories, however, the determinant need not be 1. This leads to a natural question: is the numerical Cartan determinant always positive?

This is joint work with Yeqin Liu and Ziyu Zhang.