Hit Polynomials and Conjugation in the Steenrod Algebra and its Dual
Judith H. Silverman
judith@iu-math.math.indiana.edu
Let $A^*$ be the mod-2 Steenrod algebra of cohomology operations
and $\chi$ its canonical antiautomorphism. For all positive integers
$f$ and $k$, we show that the excess of the element $\chi[Sq(2^{k-1}f)
\cdot Sq(2^{k-2}f) \cdots Sq(2f) \cdot Sq(f)]$ is $(2^k-1) \mu(f)$,
where $\mu(f)$ denotes the minimal number of summands in any
representation of $f$ as a sum of numbers of the form $2^i-1$.
We also interpret this result in purely combinatorial terms. In so
doing, we express the Milnor basis representation of the products
$Sq(a_1) \ldots Sq(a_n)$ and $\chi[Sq(a_1) \ldots Sq(a_n)]$ in
terms of the cardinalities of certain sets of matrices.
For $s \geq 1$, let $P_s = F_2[x_1, \ldots, x_s]$ be the mod-2
cohomology of the $s$-fold product of $RP^{\infty}$ with
itself, with its usual structure as an $A^*$-module. A
polynomial $P \in P_s$ is {\em hit} if it is in the image of the
action $\overline{A^*} \otimes P_s \longrightarrow P_s$, where
$\overline{A^*}$ is the augmentation ideal of $A^*$. We prove
that if the integers $e$, $f$, and $k$ satisfy $e<(2^k-1)\mu(f)$, then
for any polynomials $E$ and $F$ of degrees $e$ and $f$ respectively,
the product $E \cdot F^{2^k}$ is hit. This generalizes a result of
Wood, conjectured by Peterson, and proves a conjecture of
Singer and Silverman.
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