Some Relations between Admissible Monomials for the Polynomial Algebra
Joint Authors
Mothebe, Mbakiso Fix
Uys, Lafras
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2015, Issue 2015 (31 Dec. 2015), pp.1-7, 7 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2015-07-14
Country of Publication
Egypt
No. of Pages
7
Main Subjects
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 .
American Psychological Association (APA)
Mothebe, Mbakiso Fix& Uys, Lafras. 2015. Some Relations between Admissible Monomials for the Polynomial Algebra. International Journal of Mathematics and Mathematical Sciences،Vol. 2015, no. 2015, pp.1-7.
https://search.emarefa.net/detail/BIM-1066189
Modern Language Association (MLA)
Mothebe, Mbakiso Fix& Uys, Lafras. Some Relations between Admissible Monomials for the Polynomial Algebra. International Journal of Mathematics and Mathematical Sciences No. 2015 (2015), pp.1-7.
https://search.emarefa.net/detail/BIM-1066189
American Medical Association (AMA)
Mothebe, Mbakiso Fix& Uys, Lafras. Some Relations between Admissible Monomials for the Polynomial Algebra. International Journal of Mathematics and Mathematical Sciences. 2015. Vol. 2015, no. 2015, pp.1-7.
https://search.emarefa.net/detail/BIM-1066189
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1066189