paper

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

arXiv:2010.04679

Abstract

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

32 pages