The symmetries of the tetrahedral Kummer surface in the Mathieu group M_24

Abstract: We provide a method, based on Nikulin's lattice gluing techniques, which identifies the symplectic automorphisms of Kummer surfaces as permutation groups on 24 elements preserving the Golay code. In other words, we explicitly realise these symplectic automorphism groups as subgroups of the Mathieu group M_24. The example of the tetrahedral Kummer surface is treated in detail, confirming the existence proofs of Mukai and Kondo, that its group of symplectic automorphisms is a subgroup of one of eleven subgroups of the sporadic group known as Mathieu group M_23. Kondo's lattice construction, which uses a different gluing technique from the one advocated here to rederive Mukai's results, is reviewed, and a slight generalisation is used to check the consistency of our results. The framework presented here provides a line of attack to unravel the role of the sporadic Mathieu group Mathieu M_24, of which M_23 is a subgroup of index 24, when searching for symmetries beyond the classical symplectic automorphisms in the context of strings compactified on a K3 surface.