The minimal degree standard identity on MnE2 and MnE3

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 × n matrices over the m-generated Grassmann algebra is at least \(2\lfloor\frac{m}{2}\rfloor+4n-4\) for all n, m ≥ 2 and this bound is sharp for m = 2,3 and any n ≥ 2. The arguments are purely combinatorial, based on computing sums of signs corresponding to Eulerian trails in directed graphs.