math.RA — Rings and Algebras

Nilpotent-free spaces of 3 x 3 matrices over the field with three elements

Contributed by Deep Bhattacharjee

Wigderson conjectured that for every finite field $F$ and every $n\ge1$ there is a space of $n\times n$ matrices over $F$ of dimension $n^2-\binom n2$ whose only nilpotent element is zero. We show that every six-dimensional space of $3\times3$ matrices over $\F_3$ contains a nonzero nilpotent matrix, so the conjecture fails for $F=\F_3$ and $n=3$; the largest possible dimension is five.