Closed Majorana representations of {3, 4}+-transposition groups

Abstract

Abstract The paper contributes to Majorana theory. Among the eight non-trivial Norton–Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are 2A, 2B, 3C and 4B. The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article the classification is achieved for shapes restricted to 2A, 3C and 4B and for the set of generating involutions in the target group forming a single conjugacy class. Timmesfeld’s classification of {3, 4}+-transposition groups reduces to consideration of just three groups: L 3(2), G 2(2)' and 3 D 4(2). Each of these groups possesses a unique Majorana representation of the required shape. Only the representation of L 3(2), known before, is based on an embedding into the Monster.