Duality on hypermaps with symmetric or alternating monodromy group

TL;DR

This paper investigates the duality index in oriented regular hypermaps, demonstrating the existence of hypermaps with extreme duality index for any given duality-type {l,n}, within symmetric or alternating monodromy groups.

Contribution

It establishes the possibility of constructing hypermaps with extreme duality index for any duality-type {l,n}, restricted to symmetric or alternating monodromy groups.

Findings

Existence of hypermaps with extreme duality index for all {l,n} with {l,n} geq 2.

Construction methods within symmetric or alternating monodromy groups.

Analysis of properties of the duality index in regular hypermaps.

Abstract

Duality is the operation that interchanges hypervertices and hyperfaces on oriented hypermaps. The duality index measures how far a hypermap is from being self-dual. We say that an oriented regular hypermap has \emph{duality-type} $\{l,n\}$ if $l$ is the valency of its vertices and $n$ is the valency of its faces. Here, we study some properties of this duality index in oriented regular hypermaps and we prove that for each pair $n$, $l \in \mathbb{N}$, with $n,l \geq 2$, it is possible to find an oriented regular hypermap with extreme duality index and of duality-type $\{l,n \}$, even if we are restricted to hypermaps with alternating or symmetric monodromy group.