Unimodular Matrices and Parsons Numbers

✍ Scribed by Aiden Bruen; David Wehlau; Zhang Zhaoji

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 185 KB
• Volume: 74
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.0054

✦ Synopsis

Let [A 1 , ..., A m ] be a set of m matrices of size n_n over the field F such that A i # SL(n, F) for 1 i m and such that A i &A j # SL(n, F) for 1 i< j m. The largest integer m for which such a set exists is called the Parsons number for n and F, denoted m(n, F). We will call such a set of m(n, F) matrices a Parsons set: such a set arises in a combinatorial setting (see [Z]). Parsons asserted (see [Z]) that m(n, F q ) q n if F q is the Galois field of order q. Here we will consider the case n=2. Our result is the following.

Theorem. Let F be any field. Let p be the characteristic of F ( possibly zero). Then 2 m(2, F) 5. Moreover, m(2, F)=5 if and only if p=5 and F contains a primitive cube root of unity. If p=3 then 3 m(2, F) 4, and m(2, F)=4 if and only if F contains a square root of &1. Finally if p{2, 3 and F contains a square root of &3, then m(2, F)=4 or m(2, F)=3 according as F does or does not contain a square root of 33.