Duality index of oriented regular hypermaps

Abstract

By adapting the notion of chirality group, the duality group of $\mathcal{H}$ can be defined as the the minimal subgroup $D(\mathcal{H})⊴Mon(\mathcal{H})$ such that $\mathcal{H}/D(\mathcal{H})$ is a self-dual hypermap (a hypermap isomorphic to its dual). Here, we prove that for any positive integer $d$, we can find a hypermap of that duality index (the order of $D(\mathcal{H})$), even when some restrictions apply, and also that, for any positive integer $k$, we can find a non self-dual hypermap such that $|Mon(\mathcal{H})|/d=k$. This $k$ will be called the \emph{duality coindex} of the hypermap.

13 pages, 1 figure