A Note on Admissible Monomials of Degree 2^λ−1

Mbakiso Fix Mothebe^1,

¹Department of Mathematics, University of Botswana, Pvt Bag 00704, Gaborone, Botswana

Cite this paper:
Mbakiso Fix Mothebe. A Note on Admissible Monomials of Degree 2^λ−1. Journal of Mathematical Sciences and Applications. 2019; 7(1):10-14. doi: 10.12691/jmsa-7-1-2

Abstract

Let be the polynomial algebra in n variables x_i, 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 P^d(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, a_n 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.

Keywords:
Steenrod squares polynomial algebra Peterson hit problem

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References:

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