For example, the mod 2 cohomology of the classifying space of the exceptional Lie group [E.sub.6] is generated by two generators of degree 4 and of degree 32 as an algebra over the mod 2
Steenrod algebra, and Toda pointed out that the generator of degree 32 could be given as the Chern class of an irreducible representation [[rho].sub.6] : [E.sub.6] [right arrow] SU(27) in [12].
P(n) is a module over the mod-2 Steenrod algebra A according to well-known rules.
Silverman, "Hit polynomials and conjugation in the dual Steenrod algebra," Mathematical Proceedings of the Cambridge Philosophical Society, vol.
This conjecture is an extension of the main conjecture of [HT04] (case W = [G.sub.n], l = 1), which itself is a q-analogue of a conjecture of Wood [Woo98, Woo01] on the "hit polynomials" for the rational Steenrod algebra S := K[[P.sub.1,d] | d [greater than or equal to] 1].
Indeed, and as far as we know (see the discussion in [HT04, Section 7.1]), there is no natural analogue of the elementary symmetric polynomial inside the rational Steenrod algebra [S.sub.q].
The result is effective in computing the cohomology rings of groups and Hopf algebras, and the
Steenrod algebra in particular.
This will be carried out by employing techniques in the category of unstable modules over the mod two
Steenrod algebra A [15].
In the case of G = [E.sub.6] or [E.sub.7], as an algebra over the
Steenrod algebra, the [E.sub.2]-term of the spectral sequence is generated by only two elements, one is of degree 4 and the other is of degree 32 or 64, respectively.