A-identities for the Grassmann Algebra: the Conjecture of Henke and Regev

Let K be an algebraically closed field of characteristic 0, and let E be the infinite dimensional Grassmann (or exterior) algebra over K. Denote by Pn the vector space of the multilinear polynomials of degree n in x1, . . . , xn in the free associative algebra K(X). The symmetric group Sn acts on the left-hand side on Pn, thus turning it into an Sn-module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The Sn-modules Pn and KSn are canonically isomorphic. Letting An be the alternating group in Sn, one may study KAn and its isomorphic copy in Pn with the corresponding action of An. Henke and Regev described the An-codimensions of the Grassmann algebra E, and conjectured a finite generating set of the An-identities for E. Here we answer their conjecture in the affirmative.