$A$-identities for the Grassmann algebra: The conjecture of Henke and Regev

Abstract

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 P n the vector space of the multilinear polynomials of degree n in x 1 , ..., x n in the free associative algebra K(X). The symmetric group S n acts on the left-hand side on P n , thus turning it into an S n -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 S n -modules P n and KS n are canonically isomorphic. Letting An be the alternating group in S n , one may study KA n and its isomorphic copy in P n with the corresponding action of An. Henke and Regev described the A n -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.