Duality index of oriented regular hypermaps

TL;DR

This paper introduces the duality index of oriented regular hypermaps, demonstrating the existence of hypermaps with arbitrary duality indices and duality coindices, expanding understanding of hypermap symmetries.

Contribution

It defines the duality index using the chirality group concept and proves the existence of hypermaps with any specified duality index and coindex.

Findings

Existence of hypermaps with any positive duality index.

Existence of hypermaps with any positive duality coindex.

Characterization of the duality group in hypermaps.

Abstract

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