The images of multilinear polynomials evaluated on $3\times 3$ matrices

What is it about?

If $p$ is a multilinear, non-commutative polynomial, then Kaplansky asked what are the possible images of $p$ evaluated on $M_n(K)$, the $n\times n$ matrices over a field $K$. If $p$ is a polynomial identity for $M_n(K)$ then the image is $\{0\}$ and if $p$ is central (but not a polynomial identity) then the image is $K$, identified with scalar matrices. Two other obvious possibilities are $\germ{sl}_n(K)$ and all of $M_n(K)$. Kaplansky conjectured that these are the only possibilities. In a previous paper [Proc. Amer. Math. Soc. 140 (2012), no. 2, 465–478; MR2846315] the authors considered the case of $n=2$, and in the current work they consider $n=3$, in both cases for an algebraically closed field $K$ of any characteristic. Although they do not settle Kaplansky's conjecture, they show that if it is not true for $n=3$ then the image of $p$ will be one of the following: (1) a dense subset of $M_3(K)$, (2) a dense subset of $\germ{sl}_3(K)$, (3) the set of what they call 3-scalar matrices, defined to be matrices whose eigenvalues are of the form $\{\beta,\beta\epsilon,\beta\epsilon^2\}$, where $\epsilon$ is a cube root of 1, or (4) $K$ plus the set of 3-scalar matrices. Reviewed by Allan Berele

Featured Image