A Note on Admissible Monomials of Degree 2 λ −1

Abstract

Let be the polynomial algebra in n variables xi, of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on according to well known rules. A major problem in algebraic topology is that of determining 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 Both and Q(n) are graded, where Pd(n) denotes the set of homogeneous polynomials of degree d. In this note we show that the monomial is the only one among all its permutation representatives that is admissible, (that is, an meets a criterion to be in a certain basis for Q(n)). We show further that if with m ≥ n, then there are exactly permutation representatives of the product monomial that are admissible.