Let [Ascr ]* be the mod-2 Steenrod algebra of cohomology operations
and χ its
canonical antiautomorphism. For all positive integers f and k,
we show that the excess of the element
χ[Sq (2k−1f)·
Sq (2k−2f)…
Sq (2f)·Sq (f)] is
(2k−1)μ(f), where
μ(f) denotes the minimal number of summands in any representation
of
f as a sum of numbers of the form 2i−1.
We also interpret this result in purely combinatorial
terms. In so doing, we express the Milnor basis representation of the products
Sq (a1)…Sq (an)
and
χ[Sq (a1)…Sq (an)]
in terms of the cardinalities of certain sets of matrices.
For s[ges ]1, let
ℙs=[ ]2=
[x1, …, xs]
be the mod-2 cohomology of the s-fold product of
ℝP∞ with itself, with its usual structure
as an
[Ascr ]*-module. A polynomial P∈ℙs
is hit
if it is in the image of the action
[Ascr ]*×ℙs→ℙs,
where [Ascr ]* is the augmentation ideal of [Ascr ]*. We prove that
if the
integers e, f, and k satisfy
e<(2k−1)μ(f),
then for any polynomials E and F of degrees e
and f respectively, the product
E·F2k is hit. This
generalizes a result of Wood conjectured by Peterson, and proves a conjecture
of
Singer and Silverman.
