A graph-theoretic approach to a conjecture of Dixon and Pressman

Given $n\times n$ matrices, $A_1,\dots ,A_k$, consider the linear operator $L(A_1,\dots ,A_k)\text{\hspace{0.17em}⁣}:{\mathrm{M}}_n\to {\mathrm{M}}_n$ given by $$L(A_1,\dots ,A_k)(A_{k+1})=\sum _{\sigma \in S_{k+1}}\mathrm{sign}(\sigma )A_{\sigma (1)}{A}_{\sigma (2)}\cdots A_{\sigma (k+1)}\mathrm{.}$$ The Amitsur-Levitzki theorem asserts that $L(A_1,\dots ,A_k)$ is identically $0$ for every $k\ge 2n-1$. Dixon and Pressman conjectured that if $k$ is an even number between $2$ and $2n-2$, then the kernel of $L(A_1, \ldots, A_k)$isofdimension$k$for$A_1, \ldots, A_k\in \operatorname{M}_n($\mathbb{R}$)$ in general position. We prove this conjecture using graph-theoretic techniques.