Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

Abstract

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ 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 that 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 ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. In this paper we give explicit formulae for the dimension of ${\Q}(n)$ in degrees less than or equal to $12.$

On Peterson's open problem and representations of the general linear groups

“…We end this section by establishing a formula below on the dimension of QP 0 d in degree n, which will be used in the next section. Note that this formula is similar to the one of [29].…”

“…From this event, one of the applications of the hit problem of Peterson is to study the representations of the general linear groups over Z/2. Therefrom the hit problem has attracted great interest of many algebraic topologists (see Crabb and Hubbuck [12], Crossley [13], Kameko [21], Mothebe and his collaborators [27,28,29], Nam [30], Pengelley and Williams [33,35], Priddy [52], Silverman and Singer [54], Singer [56], Peterson [38], the present author [39,40,41,42,43,44,50,51], Sum [59,60,61,62,63,64,65], Walker and Wood [69,70,71], Wood [74,75] and others).…”

“…We remark that for t ∈ {1, 2}, since ( Sq 0 * ) (5,13.2 t −5) : (QP 5 ) 13.2 t −5 → (QP 5 ) 13.2 t−1 −5 is an epimorphism, we need only to determine the kernel of ( Sq 0 * ) (5,13.2 t −5) . To study this space, we combine our recent results in [39] with previous results by Kameko [21], Mothebe [28,29], Singer [56], Sum [60], and Tín [68].…”

On Peterson's open problem and representations of the general linear groups

“…From this event, one of the applications of the hit problem of Peterson is to study the representations of the general linear groups over Z/2. Therefrom the hit problem has attracted great interest of many algebraic topologists (see Crabb and Hubbuck [12], Crossley [13], Kameko [21], Mothebe and his collaborators [27,28,29], Nam [30], Pengelley and Williams [33,35], Priddy [54], Silverman and Singer [56], Singer [58], Peterson [38], the present author [39,40,41,42,43,49,51,52,53], Sum [61,62,63,64,65,66,67], Walker and Wood [71,72,73], Wood [76,77] and others).…”

“…We remark that for t ∈ {1, 2}, since ( Sq 0 * ) (5,13.2 t −5) : (QP 5 ) 13.2 t −5 → (QP 5 ) 13.2 t−1 −5 is an epimorphism, we need only to determine the kernel of ( Sq 0 * ) (5,13.2 t −5) . To study this space, we combine our recent results in [39] with previous results by Kameko [21], Mothebe [28,29], Singer [58], Sum [62], and Tín [70].…”