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Pieces of eight: semiselfdual lattices and a new foundation for the theory of Conway and Mathieu groups. (English) Zbl 0982.20009

This remarkable paper lays down a new foundation for the theory of the Leech lattice, Golay codes, Conway and Mathieu groups from a completely new point of view. Classical treatments of the Leech lattice and Conway groups [cf. J. H. Conway, Invent. Math. 7, 137-142 (1969; Zbl 0212.07001), and Bull. Lond. Math. Soc. 1, 79-88 (1969; Zbl 0186.32304)] used orthonormal bases and built up particularly on the binary Golay code and its uniqueness. The novelty of the author’s approach is the use of semiselfdual lattices (i.e. lattices \(L\) with \(\tfrac 12L\) containing the dual lattice \(L^*\)) instead. The author starts from a construction of a Leech lattice, (i.e. any \(24\)-dimensional even unimodular Euclidean lattice without vectors of squared length \(24\)) using pieces of eight, that is, scaled copies \(\sqrt 2L_{E_8}\) of the root lattice \(L_{E_8}\). Assuming only the structure of \(L_{E_8}\) and its automorphism group and the theta series of \(L_{E_8}\) and a Leech lattice, the author subsequently deduces the uniqueness of the Leech lattice, various properties of its automorphism group \(2Co_1\), the existence and uniqueness of the binary Golay code and its automorphism group \(M_{24}\). The author’s construction also yields the existence and uniqueness of the ternary Golay code. Semiselfdual lattices, and the related notion of semiselfdual involutions introduced in the paper, which is a generalization of the classical notion of a reflection associated to a root vector, will be useful in other questions involving integral lattices and their automorphism groups as well.

MSC:

20D08 Simple groups: sporadic groups
11H06 Lattices and convex bodies (number-theoretic aspects)
11H56 Automorphism groups of lattices
20B20 Multiply transitive finite groups
94B05 Linear codes (general theory)
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References:

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