Some relations between admissible monomials for the polynomial algebra

Abstract

Let P(𝑛) = F2[𝑥1, . . . , 𝑥𝑛] be the polynomial algebra in 𝑛 variables 𝑥𝑖, of degree one, over the field F2 of two elements. The mod-2 Steenrod algebra A acts on P(𝑛) according to well known rules. A major problem in algebraic topology is of determining A+P(𝑛), the image of the action of the positively graded part of A. We are interested in the related problem of determining a basis for the quotient vector space Q(𝑛) = P(𝑛)/A+P(𝑛). Q(𝑛) has been explicitly calculated for 𝑛 = 1, 2, 3, 4 but problems remain for 𝑛 ≥ 5. Both P(𝑛) = ⨁𝑑≥0P𝑑(𝑛) and Q(𝑛) are graded, where P𝑑(𝑛) denotes the set of homogeneous polynomials of degree 𝑑. In this paper, we show that if𝑢 = 𝑥𝑚1 1 ⋅ ⋅ ⋅ 𝑥𝑚𝑛−1 𝑛−1 ∈ P𝑑 𸀠 (𝑛−1) is an admissible monomial (i.e., 𝑢 meets a criterion to be in a certain basis forQ(𝑛−1)), then, for any pair of integers (𝑗, 𝜆), 1 ≤ 𝑗 ≤ 𝑛, and 𝜆 ≥ 0, the monomial ℎ𝜆𝑗 (𝑢) = 𝑥𝑚1 1 ⋅ ⋅ ⋅ 𝑥 𝑚𝑗−1 𝑗−1 𝑥2 𝜆−1 𝑗 𝑥𝑚𝑗 𝑗+1 ⋅ ⋅ ⋅ 𝑥𝑚𝑛−1 𝑛 ∈ P𝑑 𸀠 +(2𝜆 −1)(𝑛) is admissible. As an application we consider a few cases when 𝑛 = 5.