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

Let K be a field of characteristic 0, and let UT _n (K) be the algebra of n × n upper triangular matrices over K. We denote by P _n the vector space of all multilinear polynomials of degree n in x ₁, …, 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 ≥ 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 ≤ 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.     

Hidden in plain sight: unlocking the full potential of cyclic voltammetry with the thin-film rotating (ring) disk electrode studies for the investigation of oxygen reduction reaction electrocatalysts

Cyclic voltammetry conducted with a thin-film rotating disk electrode, or ring-disk electrode (CV-TF-R(R)DE) is a very popular ‘ex situ’ tool for the rapid screening of electrocatalysts for their activity in oxygen reduction reaction. Despite its popularity and broad use, in most instances only a small part of the information that could be accessed by CV-TF-R(R)DE is actually used by scientists in their research. This work outlines both innovative and more traditional (but half-forgotten) ways of using CV-TF-R(R)DE to its optimal or full potential in the ongoing quest to study the most relevant features of oxygen reduction reaction electrocatalysts and quickly identify the most promising candidates for their applications in fuel cells or other electrochemical devices.