Majorana algebra for the Hoffman–Singleton graph

Abstract

Majorana theory is an axiomatic tool for studying the Monster group M and its subgroups through the 196,884-dimensional Conway–Griess–Norton algebra. The theory was introduced by A. A. Ivanov in 2009 and since then it experienced a remarkable development including the classification of Majorana representations for small (and not so small) groups. The group U_3(5) is (isomorphic to) the socle of the centralizer in M of a subgroup of order 25. The involutions of this U_3(5)-subgroup are 2A-involutions in the Monster. Therefore, U_3(5) possesses a Majorana representation (based on the embedding in the Monster). We prove that this is the unique Majorana representation of U_3(5), calculate its dimension, which is 798, and obtain a description in terms of the Hoffman–Singleton graph of which the automorphism group has U_3(5) as an index 2 subgroup.