Characterizations of equivariant Steiner and linear isometries operads

Abstract

We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When $G$ is a finite abelian group, we prove that a $G$-indexing system is realized by a Steiner operad if and only if it is generated by cyclic $G$-orbits. When $G$ is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than $3$, we prove that a $G$-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill's horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and we develop basic tools for computing with them.