We write for the polynomial
ring on letters over the
field , equipped with the
standard action of , the
symmetric group on
letters. This paper deals with the problem of determining a minimal set of generators for the invariant
ring as a module over
the Steenrod algebra .
That is, we would like to determine the graded vector spaces
.
Our main result is stated in terms of a “bigraded Steenrod algebra”
. The generators
of this algebra ,
like the generators of the classical Steenrod algebra
,
satisfy the Adem relations in their usual form. However, the Adem
relations for the bigraded Steenrod algebra are interpreted so that
is not the unit of the algebra; but rather, an independent generator.
Our main work is to assemble the duals of the vector spaces
, for all
, into a
single bigraded vector space and to show that this bigraded object has the structure of an
algebra over .
Keywords
Steenrod algebra, cohomology of classifying spaces,
cohomology of the Steenrod algebra, Adams spectral
sequence, algebraic transfer, hit elements