Q-Rings and the Homology of the Symmetric Group
by Terry Bisson & André Joyal
ABSTRACT
The goal of this paper is to study the rich algebraic structure supported
by the homology mod 2 of the symmetric groups. We propose to organise the
algebra of homology operations around a single concept, that of Q-ring.
We are guided by an analogy with the representation theory of the symmetric
groups and the concept of $\lambda$-ring.
We show that $H_*\Sigma_*$ is the free Q-ring on one generator. It is a
Hopf algebra generated by its subgroup ${\cal K}$ of primitive elements.
This subgroup is an algebra (for the composition of operations) that we
call the Kudo-Araki algebra. It is closely related to the Dyer-Lashof algebra
but is better behaved: the dual coalgebra is directly representing the
substitution of Ore polynomials.
Many results on the homology of $E_\infty$-spaces can be expressed in the
language of Q-rings.
We formulate the Nishida relations by using a Q-ring structure on a semidirect
extension ${\cal A}$ of Milnor's dual of the Steenrod algebra. We show
that the Nishida relations lead to a commutation operator between ${\cal
K}$ and ${\cal A}$.