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The Cellular Structure of the Leech Lattice



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The Cellular Structure of the Leech Lattice R E Borcherds J H Conway and L Queen This is chapter 25 of Sphere packings lattices and groups edited by Conway and Sloane We complete the classification of the holes in the Leech lattice and of the associated Delaunay cells by showing that there are precisely 284 types of shallow hole 1 Introduction In Chapter 23 of C S it was shown that there are 23 types of deep hole in the Leech lattice 24 and that these holes are in one to one correspondence with the Niemeier lattices The existence of this correspondence and the recently discovered correspondence between the conjugacy classes in the Monster group and certain modular functions C N suggested that it might be worth enumerating the shallow holes in the Leech lattice and completing the classification of its Delaunay cells in case any deep structure became apparent Although this has not yet happened the complete list of deep and shallow holes has already found several uses and it seems worth while to put it on record The main result is the following Theorem 1 There are 307 types of hole in the Leech lattice consisting of 23 types of deep hole and 284 types of shallow hole They are listed in Table 1 The neighborhood graph of the deep holes may be seen in Chapter 17 of C S 2 Names for the holes We use the notation of Chapter 23 of C S and describe sets of Leech lattice points by graphs with a node for each lattice point and where two nodes x and y are not joined if N x y 4 joined by a single edge if N x y 6 joined by two edges if N x y 8 Larger numbers of joins will not arise here It was shown in Chapter 23 of C S that the vertices of a deep hole in 24 are described by a graph that is a disjoint union of extended Coxeter Dynkin diagrams having total subscript or dimension 24 and constant Coxeter number h The Coxeter numbers are shown in Table 2 below There are just 23 possible combinations which can be seen in the first 23 lines of Table 1 Using the same graphical notation we



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