A Presentation for the Unipotent Group over Rings with Identity

✍ Scribed by Daniel K Biss; Samit Dasgupta

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 124 KB
• Volume: 237
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8604

✦ Synopsis

For a ring R with identity, define Unip R to be the group of upper-triangular n matrices over R all of whose diagonal entries are 1. For i s 1, 2, . . . , n y 1, let S i denote the matrix whose only nonzero off-diagonal entry is a 1 in the ith row and Ž . Ž . i q 1 st column. Then for any integer m including m s 0 , it is easy to see that Ž . the S generate Unip ZrmZ . Reiner gave relations among the S which he i n i Ž . conjectured gave a presentation for Unip Zr2Z . This conjecture was proven by n w Ž .

x Biss Comm. Algebra 26 1998 , 2971᎐2975 and an analogous conjecture was made Ž . for Unip ZrmZ in general. We prove this conjecture, as well as a generalization n of the conjecture to unipotent groups over arbitrary rings.