Some Relations between Admissible Monomials for the Polynomial Algebra

Abstract EN

Let P ( n ) = F 2 [ x 1 , … , x n ] be the polynomial algebra in n variables x i , of degree one, over the field F 2 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 ≥ 0 P d ( n ) and Q ( n ) are graded, where P d ( n ) denotes the set of homogeneous polynomials of degree d .

In this paper, we show that if u = x 1 m 1 ⋯ x n - 1 m n - 1 ∈ P d ′ ( n - 1 ) is an admissible monomial (i.e., u meets a criterion to be in a certain basis for Q ( n - 1 ) ), then, for any pair of integers ( j , λ ), 1 ≤ j ≤ n , and λ ≥ 0 , the monomial h j λ u = x 1 m 1 ⋯ x j - 1 m j - 1 x j 2 λ - 1 x j + 1 m j ⋯ x n m n - 1 ∈ P d ′ + ( 2 λ - 1 ) ( n ) is admissible.

As an application we consider a few cases when n = 5 .