Duality on hypermaps with symmetric or alternating monodromy group
Pinto, Daniel
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 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 the duality index of oriented regular hypermaps and we prove that for each pair ▫$n, l \in \mathbb{N}$▫, with ▫$n, l \ge 2$▫ (but not both equal to 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.
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