Pieces of Eight: Semiselfdual Lattices and a New Foundation for the Theory of Conway and Mathieu Groups

Abstract

We give a new and easy construction of the largest groups of Mathieu and Conway, as the full automorphism groups of the Golay code and Leech lattice, respectively. In addition we get a new uniqueness proof for the Leech lattice and the Golay code. The main technique is intensive use of semiselfdual sublattices: instead of rank 1 lattices as the basis for coordinate concepts, we use scaled versions of LE8 , the E8 root lattice; the semiselfdual lattices we use most of the time are isometric to 2 LE8 . While it has been recognized for decades that one can use copies of LE8 to describe the Leech lattice (see [23, 33, 34]), our uses of it to create the theory of Conway and Mathieu groups are new. Using these ``smarter coordinates,'' properties of their automorphism groups and appropriate uniqueness theorems, we get a compact foundation of this theory (see Section 3, esp. (3.7) and (3.19)). Article ID aima.1999.1846, available online at http: www.idealibrary.com on