Some Relations between Admissible Monomials for the Polynomial Algebra

Abstract

LetP(n)=F2[x1,…,xn]be the polynomial algebra innvariablesxi, of degree one, over the fieldF2of two elements. The mod-2 Steenrod algebraAacts onP(n)according to well known rules. A major problem in algebraic topology is of determiningA+P(n), the image of the action of the positively graded part ofA. We are interested in the related problem of determining a basis for the quotient vector spaceQ(n)=P(n)/A+P(n).Q(n)has been explicitly calculated forn=1,2,3,4but problems remain forn≥5. BothP(n)=⨁d≥0Pd(n)andQ(n)are graded, wherePd(n)denotes the set of homogeneous polynomials of degreed. In this paper, we show that ifu=x1m1⋯xn-1mn-1∈Pd′(n-1)is an admissible monomial (i.e.,umeets a criterion to be in a certain basis forQ(n-1)), then, for any pair of integers (j,λ),1≤j≤n, andλ≥0, the monomialhjλu=x1m1⋯xj-1mj-1xj2λ-1xj+1mj⋯xnmn-1∈Pd′+(2λ-1)(n)is admissible. As an application we consider a few cases whenn=5.