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

Author(s)

Goncalves, Dimas Jose ^[1] ; Koshlukov, Plamen ^[2]

Affiliation

^[1] Univ Brasilia, Dept Matemat, BR-70910900 Brasilia, DF - Brazil

^[2] Univ Estadual Campinas, IMECC, Dept Math, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: Israel Journal of Mathematics; v. 186, n. 1, p. 407-426, NOV 2011.
• Web of Science Citations: 1

Abstract

Let K be a field of characteristic 0, and let UT (n) (K) be the algebra of n x n upper triangular matrices over K. We denote by P (n) the vector space of all multilinear polynomials of degree n in x (1), aEuro broken vertical bar, x (n) in the free associative algebra K(X). Then P (n) is a left S (n) -module where the symmetric group S (n) acts on P (n) by permuting the variables. The S (n) -modules P (n) and KS (n) are canonically isomorphic, a fact that lets us employ the representation theory in the study of algebras with polynomial identities. Denote by A (n) the alternative subgroup of S (n) . One may study KA (n) and its isomorphic copy in P (n) with the corresponding action of A (n) . 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 M (n) (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 a parts per thousand yen 6. Our main result consists in computing a lower bound for the minimal degree d(n) of an A-identity satisfied by the algebra UT (n) (K). It turns out that, given any positive integer k, there exists n (0) such that d(n) > 2n + k for all n > n (0). Moreover, we compute d(n) for n a parts per thousand currency sign 4 and for n = 6. It turns out that d(6) = 15 > 2 x 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 a parts per thousand currency sign x. (AU)

• FAPESP's process: 05/60337-2 - Lie and Jordan algebras, their representations and generalizations
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants