A note on graded polynomial identities for tensor products by the Grassmann algebra in positive characteristic

Let G be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the TG-ideal of G-graded identities of a G-graded algebra in characteristic 0 and the generators of the TG× &2-ideal of G × ;2-graded identities of its tensor product by the infinite-dimensional Grassmann algebra E endowed with the canonical grading have pairly the same degree. In this paper, we deal with &2 × ;2-graded identities of Ek E over an infinite field of characteristic p > 2, where Ek is E endowed with a specific &2-grading. We find identities of degree p + 1 and p + 2 while the maximal degree of a generator of the ;2-graded identities of Ek is p if p > k. Moreover, we find a basis of the &2 × ;2-graded identities of Ek E and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the &2 × ;2-graded Gelfand-Kirillov (GK) dimension of Ek E