$A$-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 $A_n$ be the alternating group in $S_n$, one may study $KA_n$ and its isomorphic copy in $P_n$ with the corresponding action of $A_n$. Henke and Regev described the $A_n$-codimensions of the Grassmann algebra $E$, and conjectured a finite generating set of the $A_n$-identities for $E$. Here we answer their conjecture in the affirmative.