On ℤ_p-graded identities and cocharacters of the Grassmann algebra

A note on $\mathbb{Z}$-gradings on the Grassmann algebra and Elementary Number Theory

“…In [4], the authors started the study of gradings and graded identities on E by a cyclic group of prime order (different from two). They reduced the study of graded identities to only two cases, which are as follows.…”

“…As mentioned in the Introduction, the methods we develop here suggest that an adequate usage of Elementary Number Theory could contribute to the study of gradings and graded identities not only on the Grassmann algebra E, but also on other important algebras. Recall that if d is a prime, one can apply our methods to the problems studied in the papers [3,4]. In other words, the description of gradings on E whose support is a subgroup of Z, and their graded identities, is "reduced" to: 1) the problem of the description of graded identities on E by a cyclic group, and 2) the usage of techniques derived from number theory.…”

“…A technique that has been used recently is based on Elementary Number Theory. We recall that such an approach appeared implicitly in [4], and more explicitly in [6,11]. Although elementary as a method, the results obtained suggest that a good understanding of such a relation between Number theory and PI theory could give us some insights for the nature of the gradings and graded identities not only on E, but also in the case of other important algebras.…”

“…(Here and in what follows Z 2 stands for the cyclic group of order 2.) Furthermore, in [2,4,12], gradings and graded identities on E by finite abelian groups of order larger than 2 were also investigated. Gradings on E by infinite groups were studied for the first time in [13].…”

“…The description of such identities employs various techniques. One may consider multilinear polynomial identities since they determine all identities of a given algebra in characteristic 0, see [3,4]. One of the main tools in this case is the representation theory of the symmetric and of the general linear groups, and refinements, see for a detailed account the monographs [5,8].…”