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 andSloane.)We complete the classification of the holes in the Leech lattice, and of the associatedDelaunay 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 theLeech lattice Λ24, and that these holes are in one-to-one correspondence with the Niemeierlattices. The existence of this correspondence, and the recently discovered correspondencebetween 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 andcompleting the classification of its Delaunay cells, in case any “deep structure” becameapparent. Although this has not yet happened, the complete list of deep and shallow holeshas already found several uses, and it seems worth while to put it on record. The mainresult is the following.Theorem 1. There are 307 types of hole in the Leech lattice, consisting of 23 types ofdeep 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 pointsby graphs, with a node for each lattice point, and where two nodes x and y arenot 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 Λ24aredescribed by a graph that is a disjoint union of extended Coxeter-Dynkin diagrams havingtotal subscript (or dimension) 24 and constant Coxeter number h. (The Coxeter numbersare shown in Table 2 below.) There are just 23 possible combinations, which can be seenin the first 23 lines of Table 1. Using the same graphical notation we prove the followingresult.Lemma 2. The vertices of a shallow hole in the Leech lattice are sets of 25 points of Λ24for which the corresponding graph is a union of spherical Coxeter-Dynkin diagrams.Proof. Theorems 5 and 6 of Chapter 23 of [C-S] show that the graph is a union ofspherical Coxeter-Dynkin diagrams, and by dimension considerations the hole must containat least 25 vertices. But the fundamental roots corresponding to such a diagram (whetheror not it is connected) are linearly independent in the vector space they lie in, and so are1affinely independent. This shows that the graph cannot contain more than 25 nodes. Itis just as easy to verify that any such set of 25 points is the vertex set of a shallow hole.The argument is similar to Theorem 7 of Chapter 23 of [C-S], which is the correspondingresult for deep holes. Thus all the shallow holes in the Leech lattice are simplices.3. The volume formula.Let P1, P2, . . . , PNbe a system of representatives for all the holes in Λ24under thefull automorphism group ·∞ of Λ24. Let vol(Pi) denote the volume of Piand g(Pi) theorder of its automorphism group (i.e. the subgroup of ·∞ fixing Pi). Then we have thevolume formula:volume of a fundamental domain of Λ24=NXi=1vol(Pi) × no. of images of Piunder · 0=NXi=1| · 0|g(Pi)vol(Pi) ,where | · 0| denotes the order of ·0.The volume of a hole P can be expressed in terms of familiar concepts in the Lietheory (cf. [B]). For a deep hole,vol(P ) =124!h√d ,and for a shallow holevol(P ) =124!√sd ,where h is the Coxeter number, d is the determinant of the Cartan matrix of the sphericalCoxeter-Dynkin diagram, and s is the norm of the Weyl vector. For a connected componentthe values of these quantities are shown in Table 2 (note that s = h(h + 1)n/12). For adisconnected graph, h and d are the products of the values of the components, while s isthe sum. One can also show that the radius of a shallow hole isq2 −1s.Table 2an(or An) dn(or Dn) e6(or E6) e7(or E7) e8(or E8)h : n + 1 2n − 2 12 18 30d : n + 1 4 3 2 1s :n(n+1)(n+2)12(n−1)n(2n−1)67839926204. The enumeration of the shallow holes.We shall only give a brief description of how the shallow holes were enumerated andTheorem 1 proved. Two methods of classification were used.Method 1. This method was used to find all the shallow holes that contain a par-ticular spherical Coxeter-Dynkin diagram X as a component, by finding all occurrences ofX as a set of points in the Leech lattice. This method was only used when X has at least2seven points, for otherwise the classification becomes too complicated. We drew the graphof all points in Λ24not joined to any point of X, and then deleted enough points fromthis graph to make the union of X with the remaining graph be a collection of sphericaldiagrams with a total of 25 points. The group orders of such holes, found by inspection,are usually 1 or 2.As an example we consider the diagram X = a15. We find from Figure 23.9 of[C-S] that there are just nine types of ordered a15diagrams in the Leech lattice. When weidentify reversals these reduce to five distinct types, which are drawn in Figures 1(a)-(e) asheavy lines. (The background of these figures is taken from Figure 27.3 of [C-S], and showsall edges between pairs of the 35 points nearest to the center of a deep hole of type A24.This is a convenient portion of the Leech lattice to work with, and in particular containsall the points of Λ24not joined to the a15diagrams.)In four of figures (a)-(e) we find that in the subgraph not joined to the a15diagram(indicated by shaded nodes and broken lines) it is impossible to find a union of disjointspherical Coxeter-Dynkin diagrams with a total of 10(= 25 − 15) nodes, and so thecorresponding type of a15diagram cannot be part of a shallow hole. However, for the fifthtype (figure (e)), the disjoined subgraph contains 11 points, and by omitting each point inturn we can break it into Coxeter-Dynkin diagrams, as shown in the figure. Three of theeleven possibilities fail (those marked with an x), since they leave a graph containing anextended diagram. The remaining eight succeed.Thus there are eight types of shallow hole with a component of type a15in theirdiagram: a15e8a21, a15e7a3, a15e6d4, a15d25, a15d6a4, a15d7a2a1, a15d9a1, and a15d10.Method 2. We take a given deep hole and find all shallow holes having a face incommon with it. The environs of a deep hole are described