Instances of the Kaplansky-Lvov multilinear conjecture for polynomials of degree three

TL;DR

This paper proves the Kaplansky-Lvov conjecture for degree-three multilinear polynomials on matrix algebras when the dimension is even or less than 17, using Sylvester's matrices.

Contribution

It introduces a technique involving one-wiggle families of Sylvester's matrices to establish the conjecture for specific degrees and dimensions.

Findings

Conjecture holds for degree three polynomials when d is even.

Conjecture holds for degree three polynomials when d<17.

Technique effectively uses Sylvester's matrices to prove the conjecture.

Abstract

Given a positive integer d, the Kaplansky-Lvov conjecture states that the set of values of a multilinear noncommutative polynomial f on the matrix algebra M_d(C) is a vector subspace. In this article the technique of using one-wiggle families of Sylvester's clock-and-shift matrices is championed to establish the conjecture for polynomials f of degree three when d is even or d<17.