Power operations in the homology of infinite loop spaces, and H∞ or E∞ ring
spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for
a prime p, Dyer-Lashof operations interact with Steenrod operations via the Nishida relations,
but for many purposes this leads to complicated calculations once iterated applications of these
functions are required. On the other hand, the homology coaction turns out to provide tractable
formulae better suited to exploiting multiplicative structure.
We show how to derive suitable formulae for the interaction between power operations and
homology coactions in a wide class of examples; our approach makes crucial use of modern
frameworks for spectra with well behaved smash products. In the case of mod <i>p</i> homology,
our formulae extend those of Bisson & Joyal to odd primes. We also show how to exploit our
results in sample calculations, and produce some apparently new formulae for the Dyer-Lashof
action on the dual Steenrod algebra
