Classification and Galois conjugacy of Hamming maps
Jones, Gareth A.
We show that for each ▫$d \ge 1$▫ 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 ▫$F_q$▫. We show that for each such pair ▫$(d, q)$▫ the corresponding Belyi pairs are conjugate under the action of the absolute Galois group Gal ▫$\overline{\mathbf Q}$▫, and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for ▫$q > 3$▫.
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