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⌊2m⌋+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.Comment: 22 pages, 7 figures, since version1 the statement of the lower bound
got extended and a conjecture has been adde