Finiteness conditions for the number of isomorphism classes N_n(F)

Determine precise conditions on the ground field F under which the number N_n(F) of isomorphism classes of strictly lower triangular matrices (up to the isomorphism relation induced by the graded algebras A(T) defined in Definition 2.1) is finite for every positive integer n.

Background

The paper studies a category of graded commutative algebras A(T) attached to strictly lower triangular matrices T over a field F and the induced equivalence relation T ~ S when A(T) ≅ A(S). The authors denote by N_n(F) the number of ~-equivalence classes of n×n strictly lower triangular matrices over F.

They prove that N_n can be infinite depending on the field (e.g., over Q there are infinitely many classes for n=4), while N_n is finite over a finite field, and they give a general lower bound N_n ≥ C_{n−1}. Motivated by these findings, they ask for a general characterization of those fields F for which N_n(F) remains finite for all n.