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

Abstract

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.

The central polynomials for the Grassmann algebra

“…Using this theorem D. Gonçalves and P. Koshlukov [5] gave an affirmative answer to the conjecture of Henke and Regev [7] about the description of the Aidentities of G. It was shown in [5] that the A-identities of G are determined by the polynomial (2).…”

Free assosymmetric algebras as modules of groups

“…Authors conjectured a finite generating set of the A n -identities for the Grassmann algebra. In [13] proved Henke-Regev conjecture.…”

A-identities for upper triangular matrices: A question of Henke and Regev

“…If R is any PI algebra with T-ideal T = T (R), then the A-identities of R are all polynomials in T ∩ P A n , n ≥ 1. Recently, A-identities have been the object of active research; see, for example, [6,7,5]. Thus in [7] it was shown that every PI algebra R satisfies some A-identity.…”

“…Furthermore, in [7, Conjecture 1.2] the authors conjectured that the above polynomial generates all A-identities of E. The affirmative answer to the above Conjecture was given in [5].…”