The Minimal Degree Standard Identity on $M_n{E}^2$ and $M_n{E}^3$

Abstract

We prove an Amitsur--Levitzki-type theorem for Grassmann algebras, stating that the minimal degree of a standard identity that is a polynomial identity of the ring of $n\times n$ matrices over the $m$-generated Grassmann algebra is at least $2\lfloor \frac{m}{2}\rfloor +4n-4$ for all $n,m\ge 2$ and this bound is sharp for $m=2,3$ and any $n\ge 2$. The arguments are purely combinatorial, based on computing sums of signs corresponding to Eulerian trails in directed graphs.

22 pages, 7 figures, since version1 the statement of the lower bound got extended and a conjecture has been added