A Presentation for the Unipotent Group over Rings with Identity

Abstract

For a ring R with identity, define Unip_n(R) to be the group of upper-triangular matrices over R all of whose diagonal entries are 1. For i = 1, 2,…,n − 1, let S_i denote the matrix whose only nonzero off-diagonal entry is a 1 in the ith row and (i + 1)st column. Then for any integer m (including m = 0), it is easy to see that the S_i generate Unip_n(Z/mZ). Reiner gave relations among the S_i which he conjectured gave a presentation for Unip_n(Z/2Z). This conjecture was proven by Biss [Comm. Algebra26 (1998), 2971–2975] and an analogous conjecture was made for Unip_n(Z/mZ) in general. We prove this conjecture, as well as a generalization of the conjecture to unipotent groups over arbitrary rings.