The combinatorics of $N_{\infty }$ operads for $C_{q{p}^n}$ and $D_{p^n}$

Abstract

We provide a general recursive method for constructing transfer systems on finite lattices. Using this we calculate the number of homotopically distinct $N_{\infty }$ operads for dihedral groups $D_{p^n}$, $p>2$ prime, and cyclic groups $C_{q{p}^n}$, $p\ne q$ prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful $N_{\infty }$ operads for these groups.

v2: pre-proofs version accepted in Glasgow Mathematical Journal, 21 pages, 2 figures. v1: 20 pages, 2 figures; comments welcome!