Conjectures on the ring of commuting matrices

Abstract

Let $X=(x_{ij})$ and $Y=(y_{ij})$ be generic $n$ by $n$ matrices and $Z=XY-YX$. Let $S=k[x_{11},...,x_{nn},y_{11},...,y_{nn}]$, where $k$ is a field, let $I$ be the ideal generated by the entries of $Z$ and let $R=S/I$. We give a conjecture on the first syzygies of $I$, show how these can be used to give a conjecture on the canonical module of $R$. Using this and the Hilbert series of $I$ we give a conjecture on the Betti numbers of $I$ in the $4 \times 4$ case. We also give some guesses on the structure of the resolution in general.