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

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