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

TL;DR

This paper proves a conjecture relating to the kernel dimension of a specific multilinear operator on matrices, using graph theory, extending classical results like the Amitsur-Levitzki theorem.

Contribution

The paper introduces a graph-theoretic method to prove Dixon and Pressman's conjecture on kernel dimensions of a multilinear matrix operator.

Findings

Confirmed the conjecture for even k between 2 and 2n-2

Established the kernel dimension as k in general position

Extended classical matrix polynomial identities

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.