The existence of two commuting nilpotent matrices of specified Jordan types depends on the characteristic *and* size of the field. This was first shown in my joint paper with John Britnell, *Types and classes of commuting matrices*, J. Lond. Math. Soc. **83** (2011) 470–492.

Specifically, Proposition 4.7 states that there are matrices
of Jordan types $(n,n)$ and $(n+1,n-1)$ that commute over $\mathbb{F}_{p^r}$ if and only if $n$ is not divisible by $p(p^{2r}-1)/e$, where

$$e = \begin{cases} 1 & \text{if $p=2$} \\ 2 & \text{otherwise.} \end{cases}$$

The smallest example of this type is that there are matrices of Jordan types $(6,6)$ and $(7,5)$ that commute with entries in $\mathbb{F}_{4}$, $\mathbb{F}_8$, and so on, but no such matrices with entries in $\mathbb{F}_2$.

Section 4.4 of the paper gives some further field-dependent results of this type, including a classification of all commuting Jordan types labelled by partitions with at most two parts.

11more comments