A note on Potter’s theorem for quasi-commutative matrices

We discuss the converse of a theorem of Potter stating that if the matrix equation AB=ωBA is satisfied with ω a primitive qth root of unity, then Aq+Bq=(A+B)q. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and q=3 and a conjecture for the general case. We also present some further partial results and conjectures.