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

Abstract

Let K be a field of characteristic 0, and let UTn(K) be the algebra of n × n upper triangular matrices over K. We denote by Pn the vector space of all multilinear polynomials of degree n in x1, …, xn in the free associative algebra K(X). Then Pn is a left Sn-module where the symmetric group Sn acts on Pn by permuting the variables. The Sn-modules Pn and KSn are canonically isomorphic, a fact that lets us employ the representation theory in the study of algebras with polynomial identities. Denote by An the alternative subgroup of Sn. One may study KAn and its isomorphic copy in Pn with the corresponding action of An. Henke and Regev studied A-identities. They described the A-codimensions of the Grassmann algebra and conjectured a finite generating set of the A-identities for E. In an earlier paper we answered in the affirmative their conjecture. Another problem posed by Henke and Regev concerned the minimal degree of an A-identity satisfied by the full matrix algebra Mn(K). They asked whether this minimal degree equals 2n+2. In this paper we show that this is not the case as long as n ≥ 6. Our main result consists in computing a lower bound for the minimal degree d(n) of an A-identity satisfied by the algebra UTn(K). It turns out that, given any positive integer k, there exists n0 such that d(n) > 2n + k for all n > n0. Moreover, we compute d(n) for n ≤ 4 and for n = 6. It turns out that d(6) = 15 > 2 × 6 + 2. We have reasons to believe that for every n, d(n) = [(5n + 1)/2] holds, where as usual [x] stands for the integer part of the real number x, that is the largest integer ≤ x.