In 2011, Bonnafe-Kemper proved that the invariant ring of one vector and one convector for the upper triangular matrices along with diagonals 1 over a finite field, is a complete intersection, as well as they raised a conjecture to the invariant ring of one vector and one covector for the general linear groups. In 2017, David L. Wehlau and the speaker confirmed the conjecture of BonnafeKemper, i.e., we exhibited a minimal generating set for the invariant ring and proved it is not a complete intersection but Gorenstein.

In this talk, we present our conjecture on relations among these generators of the invariant ring for the general linear groups.