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 .
"Rings of symmetric functions as modules over the Steenrod algebra." Algebr. Geom. Topol. 8 (1) 541 - 562, 2008. https://doi.org/10.2140/agt.2008.8.541