Classification and Galois conjugacy of Hamming maps

TL;DR

This paper characterizes when Hamming graphs have orientably regular surface embeddings, constructs these embeddings as Cayley maps, and explores their Galois conjugacy and minimal fields of definition.

Contribution

It provides a complete classification of orientably regular embeddings of Hamming graphs and their Galois conjugacy properties, including new constructions as Cayley maps.

Findings

H(d,q) has orientably regular embeddings iff q is a prime power.

Number of such embeddings is er(q-1)/e for q>2.

The Galois conjugacy of the associated Belyi pairs is established.

Abstract

We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an orientably regular surface embedding if and only if q is a prime power p^e. If q>2 there are up to isomorphism \phi(q-1)/e such maps, all constructed as Cayley maps for a d-dimensional vector space over the field of order q. We show that for each such pair d, q the corresponding Belyi pairs are conjugate under the action of the absolute Galois group, and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q>3.