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 Λ_{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 Λ_{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 prove the following result.

Lemma 2. The vertices of a shallow hole in the Leech lattice are sets of 25 points of Λ_{24} for 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 of spherical Coxeter-Dynkin diagrams, and by dimension considerations the hole must contain at least 25 vertices. But the fundamental roots corresponding to such a diagram (whether or not it is connected) are linearly independent in the vector space they lie in, and so are affinely independent. This shows that the graph cannot contain more than 25 nodes. It is 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 corresponding result for deep holes. Thus all the shallow holes in the Leech lattice are simplices.

3. The volume formula.

Let P_{1}, P_{2}, …, P_{N} be a system of representatives for all the holes in Λ_{24} under the full automorphism group ·∞ of Λ_{24}. Let vol(P_{i}) denote the volume of P_{i} and g(P_{i}) the order of its automorphism group (i.e. the subgroup of ·∞ fixing P_{i}). Then we have the volume formula:

volume of a fundamental domain of Λ_{24} = ∑_{i=1}^{N} vol(P_{i}) × no. of images of P_{i} under · 0

= ∑_{i=1}^{N} \frac{| · 0|}{g(P_{i})}vol(P_{i}) ,

where | · 0| denotes the order of ·0.

The volume of a hole P can be expressed in terms of familiar concepts in the Lie theory (cf. [B]). For a deep hole,

vol(P) = \frac{1}{24!} h\sqrt{d} ,

and for a shallow hole

vol(P) = \frac{1}{24!} \sqrt{sd} ,

where h is the Coxeter number, d is the determinant of the Cartan matrix of the spherical Coxeter-Dynkin diagram, and s is the norm of the Weyl vector. For a connected component the values of these quantities are shown in Table 2 (note that s = h(h + 1)n/12). For a disconnected graph, h and d are the products of the values of the components, while s is the sum. One can also show that the radius of a shallow hole is \sqrt{2 − \frac{1}{s}} .

Table 2

a_{n}(or A_{n})  d_{n}(or D_{n}) e_{6}(or E_{6}) e_{7}(or E_{7}) e_{8}(or E_{8})
h :  n + 1  2n − 2  12  18  30
d :  n + 1  4  3  2  1
s :  \frac{n(n+1)(n+2)}{12} \frac{(n−1)n(2n−1)}{6}  78  \frac{399}{2}  620

4. The enumeration of the shallow holes.

We shall only give a brief description of how the shallow holes were enumerated and Theorem 1 proved. Two methods of classification were used.

Method 1. This method was used to find all the shallow holes that contain a particular spherical Coxeter-Dynkin diagram X as a component, by finding all occurrences of X as a set of points in the Leech lattice. This method was only used when X has at least seven points, for otherwise the classification becomes too complicated. We drew the graph of all points in Λ_{24} not joined to any point of X, and then deleted enough points from this graph to make the union of X with the remaining graph be a collection of spherical diagrams 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 = a_{15}. We find from Figure 23.9 of [C-S] that there are just nine types of ordered a_{15} diagrams in the Leech lattice. When we identify reversals these reduce to five distinct types, which are drawn in Figures 1(a)-(e) as heavy lines. (The background of these figures is taken from Figure 27.3 of [C-S], and shows all edges between pairs of the 35 points nearest to the center of a deep hole of type A_{24}. This is a convenient portion of the Leech lattice to work with, and in particular contains all the points of Λ_{24} not joined to the a_{15} diagrams.)

In four of figures (a)-(e) we find that in the subgraph not joined to the a_{15} diagram (indicated by shaded nodes and broken lines) it is impossible to find a union of disjoint spherical Coxeter-Dynkin diagrams with a total of 10(= 25 − 15) nodes, and so the corresponding type of a_{15} diagram cannot be part of a shallow hole. However, for the fifth type (figure (e)), the disjoined subgraph contains 11 points, and by omitting each point in turn we can break it into Coxeter-Dynkin diagrams, as shown in the figure. Three of the eleven possibilities fail (those marked with an x), since they leave a graph containing an extended diagram. The remaining eight succeed.

Thus there are eight types of shallow hole with a component of type a_{15} in their diagram: a_{15}e_{8}a_{1}^{2}, a_{15}e_{7}a_{3}, a_{15}e_{6}d_{4}, a_{15}d_{5}^{2}, a_{15}d_{6}a_{4}, a_{15}d_{7}a_{2}a_{1}, a_{15}d_{9}a_{1}, and a_{15}d_{10}.

Method 2. We take a given deep hole and find all shallow holes having a face in common with it. The environs of a deep hole are described in sufficient detail in chapter 24 of [C-S] to make this quite easy. The group orders of these holes can be found by considering the subgroup of those automorphisms of the corresponding deep hole that fix the face in question and then making due allowance for the fact that the shallow hole may touch several deep holes in the same manner.

The complete enumeration was carried out by combining the two methods and using the volume formula as a check. A second check was supplied by the fact that the quantity sd for each hole must be a perfect square (since the volume of a hole is a rational number). Most of the enumeration was performed twice: first by L.Q. and J.H.C., then independently by R.E.B who completed the list. The group orders were computed by both teams.

Remarks. (1) It turns out that (in contrast to the deep holes) the diagram of a shallow hole does not determine it uniquely. There are two holes of each of the types a_{1}7a_{8}, a_{17}d_{7}a_{1}, a_{11}d_{7}a_{3}a_{2}^{2} and a_{9}^{2}a_{4}a_{3}.

(2) There is a unique hole of type a_{1}^{24}a_{1}. This has the shape of a regular simplex with 25 vertices, although its group is not transitive on the vertices. The group is the Mathieu group M_{24} acting with orbits of size 24 and 1. This hole has the smallest radius, namely \sqrt{2 − \frac{2}{25}} , while the shallow hole of largest radius is d_{25}, with radius \sqrt{2 − \frac{1}{4900}} . The

deep holes have radius \sqrt{2}.

(3) If the closed polytopes corresponding to the deep holes of type D_{4}^{6} are deleted from R^{24}, the resulting space is disconnected.

Acknowledgment. We should like to thank R. A. Parker for several helpful discussions.

Table 1.

A list of all 307 holes in the Leech lattice. The first 23 entries are the deep holes. The entries give the name of a hole P_{i}, the order g(P_{i}) of its automorphism group, its scaled volume

svol(P_{i}) = vol(P_{i}) · \frac{24!}{| · 0|} ,

the norm s(P_{i}) of its Weyl vector, and the determinant d(P_{i}) of the Cartan matrix. The volume formula then becomes

∑_{i} \frac{svol(P_{i})}{g(P_{i})} = \frac{24!}{| · 0|} = 74613 .

The name of a hole indicates the orbits of its automorphism group on the components of the diagram. Thus the hole a_{7}^{2}a_{3}^{2}a_{3}a_{1}^{2} has two equivalent components of type a_{7}, two of type a_{1}, and three of type a_{3}, only two of which are equivalent under the automorphism group.

Table 1(a).

name  g  svol  s  d
D_{24}  2  92  4
A_{24}  10  125  25
A_{1}7E_{7}  12  1944  36
D_{16}E_{8}  2  1800  4
A_{15}D_{9}  16  2048  64
D_{12}^{2}  8  1936  16
A_{12}^{2}  52  2197  169
A_{11}D_{7}E_{6}  24  20736  144
D_{10}E_{7}^{2}  8  23328  16
A_{9}^{2}D_{6}  80  20000  400
E_{8}^{3}  6  27000  1
D_{8}^{3}  48  21952  64
A_{8}^{3}  324  19683  729
A_{7}^{2}D_{5}^{2}  256  131072  1024
E_{6}^{4}  432  186624  81
D_{6}^{4}  384  160000  256
A_{6}^{4}  1176  117649  2401
A_{5}^{4}D_{4}  3456  559872  5184
D_{4}^{6}  138240  2985984  4096
A_{4}^{6}  30000  1953125  15625
A_{3}^{8}  688128  16777216  4^{8}
A_{2}^{12}  138568320  387420489  3^{12}
A_{1}^{24}  1002795171840 68719476736  2^{24}
d_{25}  1  140 9800/2  4

a_{25}  1  195 2925/2  26^{d_{24}a_{1}  1  186 8649/2  8}^{a_{24}a_{1}  2  255 2601/2  50}^{a_{23}a_{2}  2  288 2304/2  72}^{d_{22}a_{2}a_{1}  1  282 6627/2  24}^{d_{21}a_{4}  1  240 5760/2  20}^{a_{21}a_{3}a_{1}  1  396 1782/2  176}^{d_{20}d_{5}  1  200 5000/2  16}^{a_{20}a_{5}  1  315 1575/2  126}^{d_{19}e_{6}  1  162 4374/2  12}^{a_{19}d_{6}  1  240 1440/2  80}^{a_{19}a_{4}a_{1}^{2}  2  520 1352/2  400}^{d_{18}e_{7}  1  126 3969/2  8}^{a_{18}e_{7}  1  171 1539/2  38}^{a_{18}a_{6}a_{1}  1  399 1197/2  266}^{d_{17}e_{8}  1  92 4232/2  4}^{a_{17}e_{8}  2  141 2209/2  18}^{a_{17}a_{8}  2  297 1089/2  162}^{a_{17}a_{8}  2  297 1089/2  162}^{a_{17}e_{7}a_{1}  2  222 1369/2  72}^{a_{17}d_{7}a_{1}  1  288 1152/2  144}^{a_{17}d_{7}a_{1}  2  288 1152/2  144}^{a_{17}d_{6}a_{2}  2  342 1083/2  216}^{a_{17}a_{6}a_{2}  2  441 1029/2  378}^{a_{17}a_{5}a_{3}  2  468 1014/2  432}^{a_{17}a_{4}a_{3}a_{1}  2  600 1000/2  720}^{d_{16}d_{9}  1  152 2888/2  16}^{d_{16}a_{9}  1  230 2645/2  40}^{d_{16}e_{8}a_{1}  1  122 3721/2  8}^{d_{16}a_{8}a_{1}  1  306 2601/2  72}^{d_{16}e_{7}a_{2}  1  186 2883/2  24}^{d_{16}e_{6}a_{3}  1  252 2646/2  48}^{d_{16}a_{6}a_{2}a_{1}  1  462 2541/2  168}^{d_{16}d_{5}a_{4}  1  320 2560/2  80}^{d_{16}a_{5}a_{4}  1  390 2535/2  120}^{a_{16}d_{9}  1  204 1224/2  68}^{a_{16}e_{8}a_{1}  1  187 2057/2  34}^{a_{15}d_{10}  2  200 1250/2  64}^{a_{15}d_{9}a_{1}  2  264 1089/2  128}^{a_{15}e_{8}a_{1}^{2}  2  248 1922/2  64}^{a_{15}e_{7}a_{3}  2  264 1089/2  128}^{a_{15}d_{7}a_{2}a_{1}  2  408 867/2  384}^{a_{15}e_{6}d_{4}  2  288 864/2  192}^{a_{15}d_{6}a_{4}  2  360 810/2  320}

5

a_{15}d_{5}d_{5}  2  320 800/2  256^{d_{14}d_{10}a_{1}  1  188 2209/2  32}^{d_{14}a_{10}a_{1}  1  286 1859/2  88}^{d_{14}a_{9}a_{1}a_{1}  1  380 1805/2  160}^{d_{14}e_{8}a_{2}a_{1}  1  186 2883/2  24}^{d_{14}e_{7}a_{3}a_{1}  1  256 2048/2  64}^{d_{14}a_{7}a_{2}a_{1}a_{1}  1  576 1728/2  384}^{d_{14}e_{6}a_{4}a_{1}  1  330 1815/2  120}^{d_{14}a_{6}a_{4}a_{1}  1  490 1715/2  280}^{d_{14}d_{5}a_{5}a_{1}  1  408 1734/2  192}^{a_{14}a_{9}a_{2}  2  405 729/2  450}^{a_{14}e_{8}a_{2}a_{1}  1  285 1805/2  90}^{a_{14}d_{7}a_{2}a_{2}  1  450 750/2  540}^{a_{14}a_{6}a_{3}a_{2}  1  630 630/2  1260}^{a_{14}a_{4}^{2}a_{2}a_{1}  2  825 605/2  2250}^{d_{13}d_{12}  1  136 2312/2  16}^{d_{13}a_{12}  1  208 1664/2  52}^{d_{13}a_{11}a_{1}  1  276 1587/2  96}^{d_{13}a_{9}a_{2}a_{1}  1  420 1470/2  240}^{d_{13}e_{8}a_{4}  1  160 2560/2  20}^{d_{13}a_{8}a_{4}  1  360 1440/2  180}^{d_{13}e_{7}a_{5}  1  204 1734/2  48}^{d_{13}a_{7}d_{5}  1  304 1444/2  128}^{d_{13}e_{6}a_{6}  1  252 1512/2  84}^{a_{13}a_{12}  1  273 819/2  182}^{a_{13}d_{10}a_{2}  1  294 1029/2  168}^{a_{13}a_{9}a_{3}  1  420 630/2  560}^{a_{13}e_{8}a_{4}  1  245 1715/2  70}^{a_{13}e_{8}a_{2}a_{1}a_{1}  1  378 1701/2  168}^{a_{13}e_{7}a_{4}a_{1}  1  350 875/2  280}^{a_{13}d_{7}a_{5}  1  336 672/2  336}^{a_{13}a_{7}a_{4}a_{1}  1  560 560/2  1120}^{a_{13}e_{6}d_{5}a_{1}  1  336 672/2  336}^{a_{13}a_{6}a_{6}  1  441 567/2  686}^{d_{12}^{2}a_{1}  2  180 2025/2  32}^{d_{12}d_{10}a_{2}a_{1}  1  276 1587/2  96}^{d_{12}d_{9}a_{4}  1  240 1440/2  80}^{d_{12}e_{8}d_{5}  1  136 2312/2  16}^{d_{12}d_{8}d_{5}  1  208 1352/2  64}^{d_{12}e_{7}d_{6}  1  156 1521/2  32}^{d_{12}d_{7}e_{6}  1  180 1350/2  48}^{a_{12}^{2}a_{1}  4  351 729/2  338}^{a_{12}e_{8}d_{5}  1  208 1664/2  52}^{a_{12}e_{8}a_{4}a_{1}  1  325 1625/2  130}

6

a_{12}e_{7}a_{6}  1  273 819/2  182^{a_{12}d_{7}e_{6}  1  234 702/2  156}^{d_{11}e_{8}e_{6}  1  114 2166/2  12}^{d_{11}e_{7}^{2}  2  112 1568/2  16}^{a_{11}d_{10}a_{3}a_{1}  1  408 867/2  384}^{a_{11}d_{10}a_{2}^{2}  2  432 864/2  432}^{a_{11}d_{9}a_{5}  2  324 729/2  288}^{a_{11}a_{9}d_{4}a_{1}  1  480 480/2  960}^{a_{11}e_{8}e_{6}  1  174 1682/2  36}^{a_{11}e_{8}d_{5}a_{1}  1  276 1587/2  96}^{a_{11}e_{8}a_{2}^{2}a_{1}^{2}  2  576 1536/2  432}^{a_{11}d_{8}e_{6}  2  228 722/2  144}^{a_{11}e_{7}d_{7}  1  204 867/2  96}^{a_{11}e_{7}a_{5}a_{1}^{2}  2  456 722/2  576}^{a_{11}d_{7}e_{6}a_{1}  2  300 625/2  288}^{a_{11}d_{7}a_{6}a_{1}  1  420 525/2  672}^{a_{11}d_{7}a_{5}a_{2}  2  468 507/2  864}^{a_{11}d_{7}a_{3}a_{2}^{2}  2  648 486/2  1728}^{a_{11}d_{7}a_{3}a_{2}^{2}  2  648 486/2  1728}^{a_{11}a_{7}d_{5}a_{1}a_{1}  1  576 432/2  1536}^{a_{11}e_{6}e_{6}a_{1}^{2}  2  360 600/2  432}^{a_{11}e_{6}d_{5}a_{3}  2  384 512/2  576}^{a_{11}e_{6}d_{5}a_{2}a_{1}  2  468 507/2  864}^{a_{11}e_{6}d_{4}a_{4}  2  420 490/2  720}^{a_{11}d_{6}a_{5}a_{3}  2  504 441/2  1152}^{a_{11}a_{6}d_{4}a_{2}^{2}  2  756 378/2  3024}^{a_{11}d_{5}a_{5}a_{3}a_{1}  2  672 392/2  2304}^{a_{11}d_{5}a_{4}a_{2}^{2}a_{1}  2  900 375/2  4320}^{a_{11}a_{5}a_{5}d_{4}  2  576 384/2  1728}^{d_{10}^{2}a_{3}a_{1}^{2}  2  384 1152/2  256}^{d_{10}a_{10}a_{5}  1  330 825/2  264}^{d_{10}d_{9}a_{5}a_{1}  1  312 1014/2  192}^{d_{10}a_{9}d_{6}  1  260 845/2  160}^{a_{10}a_{9}a_{3}a_{2}a_{1}  1  600 750/2  960}^{d_{10}e_{8}e_{7}  1  94 2209/2  8}^{d_{10}d_{8}d_{6}a_{1}  1  248 961/2  128}^{d_{10}a_{8}e_{7}  1  198 1089/2  72}^{d_{10}a_{8}a_{5}a_{2}  1  486 729/2  648}^{d_{10}e_{7}^{2}a_{1}  2  148 1369/2  32}^{d_{10}e_{7}d_{7}a_{1}  1  192 1152/2  64}^{d_{10}e_{7}d_{6}a_{2}  1  228 1083/2  96}^{d_{10}e_{7}a_{6}a_{2}  1  294 1029/2  168}^{d_{10}e_{7}a_{5}a_{3}  1  312 1014/2  192}^{d_{10}e_{7}a_{4}a_{3}a_{1}  1  400 1000/2  320}

7

d_{10}a_{7}d_{6}a_{2}  1  384 768/2  384^{d_{10}a_{7}a_{3}^{2}a_{1}^{2}  2  832 676/2  2048}^{d_{10}d_{6}^{2}a_{3}  2  320 800/2  256}^{d_{10}d_{6}a_{5}a_{4}  1  420 735/2  480}^{d_{10}d_{6}a_{5}a_{3}a_{1}  1  528 726/2  768}^{d_{10}a_{6}a_{5}a_{3}a_{1}  1  672 672/2  1344}^{d_{10}a_{5}^{3}  2  540 675/2  864}^{a_{10}a_{9}d_{6}  1  330 495/2  440}^{a_{10}e_{8}e_{7}  1  143 1859/2  22}^{a_{10}e_{8}e_{6}a_{1}  1  231 1617/2  66}^{a_{10}e_{8}a_{4}a_{2}a_{1}  1  495 1485/2  330}^{a_{10}e_{7}a_{7}a_{1}  1  352 704/2  352}^{a_{10}d_{7}a_{6}a_{2}  1  462 462/2  924}^{d_{9}^{2}a_{7}  2  240 900/2  128}^{d_{9}a_{9}a_{6}a_{1}  1  420 630/2  560}^{d_{9}e_{8}^{2}  2  76 2888/2  4}^{d_{9}d_{8}^{2}  2  176 968/2  64}^{d_{9}a_{8}^{2}  2  324 648/2  324}^{d_{9}a_{7}^{2}a_{1}^{2}  4  544 578/2  1024}^{a_{9}a_{9}e_{7}  2  270 729/2  200}^{a_{9}^{2}d_{7}  4  320 512/2  400}^{a_{9}^{2}d_{6}a_{1}  4  420 441/2  800}^{a_{9}^{2}d_{5}a_{1}^{2}  4  560 392/2  1600}^{a_{9}^{2}d_{4}a_{2}a_{1}  4  660 363/2  2400}^{a_{9}^{2}a_{4}a_{3}  4  600 360/2  2000}^{a_{9}^{2}a_{4}a_{3}  2  600 360/2  2000}^{a_{9}e_{8}^{2}  2  115 2645/2  10}^{a_{9}e_{8}e_{7}a_{1}  1  190 1805/2  40}^{a_{9}e_{8}d_{5}a_{2}a_{1}  1  420 1470/2  240}^{a_{9}e_{8}a_{4}^{2}  2  425 1445/2  250}^{a_{9}d_{8}e_{7}a_{1}  1  260 845/2  160}^{a_{9}a_{8}^{2}  2  405 405/2  810}^{a_{9}e_{7}d_{7}a_{2}  1  300 750/2  240}^{a_{9}e_{7}a_{6}a_{3}  1  420 630/2  560}^{a_{9}e_{7}a_{4}^{2}a_{1}  2  550 605/2  1000}^{a_{9}a_{7}d_{5}a_{3}a_{1}  1  640 320/2  2560}^{a_{9}d_{5}a_{4}^{2}a_{2}a_{1}  2  900 270/2  6000}^{a_{9}a_{4}^{2}a_{4}^{2}  4  875 245/2  6250}^{e_{8}^{3}a_{1}  6  61 3721/2  2}^{e_{8}^{2}a_{8}a_{1}  2  153 2601/2  18}^{e_{8}^{2}e_{7}a_{2}  2  93 2883/2  6}^{e_{8}^{2}e_{6}a_{3}  2  126 2646/2  12}^{e_{8}^{2}a_{6}a_{2}a_{1}  2  231 2541/2  42}^{e_{8}^{2}d_{5}a_{4}  2  160 2560/2  20}

8

e_{8}^{2}a_{5}a_{4}  2  195 2535/2  30^{e_{8}a_{8}d_{5}a_{4}  1  360 1440/2  180}^{e_{8}a_{8}e_{6}a_{2}a_{1}  1  351 1521/2  162}^{e_{8}e_{7}^{2}a_{3}  2  128 2048/2  16}^{e_{8}e_{7}a_{7}a_{2}a_{1}  1  288 1728/2  96}^{e_{8}e_{7}e_{6}a_{4}  1  165 1815/2  30}^{e_{8}e_{7}a_{6}a_{4}  1  245 1715/2  70}^{e_{8}e_{7}d_{5}a_{5}  1  204 1734/2  48}^{e_{8}a_{7}e_{6}a_{4}  1  300 1500/2  120}^{e_{8}a_{7}d_{5}^{2}  2  304 1444/2  128}^{e_{8}e_{6}^{2}a_{5}  2  207 1587/2  54}^{e_{8}e_{6}a_{6}d_{5}  1  252 1512/2  84}^{d_{8}^{3}a_{1}  6  232 841/2  128}^{d_{8}^{2}e_{7}a_{1}a_{1}  2  248 961/2  128}^{d_{8}^{2}e_{6}a_{3}  2  264 726/2  192}^{d_{8}^{2}d_{6}a_{2}a_{1}  2  360 675/2  384}^{d_{8}^{2}d_{5}d_{4}  2  288 648/2  256}^{d_{8}^{2}d_{5}a_{4}  2  320 640/2  320}^{d_{8}e_{7}^{2}a_{2}a_{1}  2  228 1083/2  96}^{d_{8}e_{7}d_{6}a_{3}a_{1}  1  320 800/2  256}^{d_{8}e_{7}a_{7}a_{2}a_{1}  1  384 768/2  384}^{d_{8}e_{7}a_{5}a_{4}a_{1}  1  420 735/2  480}^{d_{8}e_{7}a_{5}a_{3}a_{1}a_{1}  1  528 726/2  768}^{d_{8}a_{7}e_{6}a_{4}  1  360 540/2  480}^{d_{8}a_{7}d_{5}^{2}  2  352 484/2  512}^{d_{8}e_{6}a_{5}^{2}a_{1}  2  468 507/2  864}^{d_{8}d_{6}^{2}d_{4}a_{1}  2  368 529/2  512}^{d_{8}d_{6}^{2}a_{3}a_{1}^{2}  2  512 512/2  1024}^{d_{8}d_{6}d_{5}a_{5}a_{1}  1  432 486/2  768}^{a_{8}^{3}a_{1}  12  513 361/2  1458}^{a_{8}a_{7}d_{5}^{2}  2  432 324/2  1152}^{a_{8}e_{6}^{2}a_{2}^{2}a_{1}  2  567 441/2  1458}^{a_{8}e_{6}d_{5}a_{4}a_{2}  1  540 360/2  1620}^{e_{7}^{3}d_{4}  6  140 1225/2  32}^{e_{7}^{2}d_{7}a_{4}  2  200 1000/2  80}^{e_{7}^{2}e_{6}d_{5}  2  156 1014/2  48}^{e_{7}^{2}d_{6}d_{5}  2  176 968/2  64}^{e_{7}d_{7}a_{7}a_{3}a_{1}  1  416 676/2  512}^{e_{7}d_{7}d_{6}a_{5}  1  264 726/2  192}^{e_{7}d_{7}a_{6}a_{5}  1  336 672/2  336}^{e_{7}a_{7}^{2}a_{3}a_{1}  2  544 578/2  1024}^{e_{7}a_{7}a_{6}a_{4}a_{1}  1  560 560/2  1120}^{e_{7}a_{7}a_{5}a_{4}a_{1}a_{1}  1  720 540/2  1920}^{e_{7}e_{6}^{3}  6  153 867/2  54}

9

e_{7}d_{6}^{3}  3  216 729/2  128^{e_{7}a_{6}^{3}  3  441 567/2  686}^{e_{7}a_{5}^{3}a_{1}^{3}  6  936 507/2  3456}^{d_{7}a_{7}^{2}a_{3}a_{1}  4  608 361/2  2048}^{d_{7}a_{7}a_{6}a_{3}a_{2}  1  672 336/2  2688}^{d_{7}a_{7}a_{3}^{2}a_{3}a_{2}  2  960 300/2  6144}^{d_{7}d_{6}^{3}  6  256 512/2  256}^{d_{7}a_{6}^{3}  3  490 350/2  1372}^{d_{7}a_{3}^{6}  24  1408 242/2  16384}^{a_{7}^{3}a_{1}^{4}  24  1024 256/2  8192}^{a_{7}^{2}e_{6}d_{5}  2  384 384/2  768}^{a_{7}^{2}d_{6}d_{5}  4  416 338/2  1024}^{a_{7}^{2}d_{5}^{2}a_{1}  8  544 289/2  2048}^{a_{7}^{2}d_{5}a_{4}a_{1}^{2}  4  800 250/2  5120}^{a_{7}^{2}d_{5}a_{3}a_{2}a_{1}  4  864 243/2  6144}^{a_{7}^{2}a_{5}a_{3}a_{1}^{2}a_{1}  4  1152 216/2  12288}^{a_{7}^{2}a_{3}^{2}a_{3}a_{1}^{2}  8  1280 200/2  16384}^{a_{7}e_{6}^{2}a_{5}a_{1}  2  432 432/2  864}^{a_{7}d_{6}^{2}a_{3}^{2}  4  576 324/2  2048}^{a_{7}a_{6}^{3}  3  588 252/2  2744}^{a_{7}a_{5}^{3}a_{1}^{3}  6  1152 192/2  13824}^{a_{7}a_{3}^{6}  24  1536 144/2  32768}^{e_{6}^{4}a_{1}  48  225 625/2  162}^{e_{6}^{3}a_{6}a_{1}  6  315 525/2  378}^{e_{6}^{3}a_{5}a_{2}  12  351 507/2  486}^{e_{6}^{3}a_{3}a_{2}^{2}  12  486 486/2  972}^{e_{6}^{2}a_{5}^{2}a_{3}  8  504 392/2  1296}^{e_{6}^{2}a_{5}a_{4}a_{2}^{2}  4  675 375/2  2430}^{e_{6}^{2}a_{5}a_{2}^{4}  8  891 363/2  4374}^{e_{6}d_{5}a_{5}^{2}a_{2}^{2}  4  756 294/2  3888}^{e_{6}a_{5}^{3}d_{4}  12  612 289/2  2592}^{d_{6}^{4}a_{1}  24  336 441/2  512}^{d_{6}^{3}d_{5}a_{1}^{2}  6  448 392/2  1024}^{d_{6}^{3}d_{4}a_{2}a_{1}  6  528 363/2  1536}^{d_{6}^{3}a_{4}a_{3}  6  480 360/2  1280}^{d_{6}^{3}d_{4}a_{1}^{3}  6  608 361/2  2048}^{d_{6}^{2}a_{5}d_{4}a_{3}a_{1}  2  672 294/2  3072}^{d_{6}^{2}d_{4}^{2}a_{3}a_{1}^{2}  4  768 288/2  4096}^{d_{6}a_{5}^{3}d_{4}  6  648 243/2  3456}^{d_{6}d_{4}^{4}a_{1}^{3}  24  960 225/2  8192}^{a_{6}^{4}a_{1}  24  735 225/2  4802}^{a_{6}a_{5}^{3}d_{4}  6  756 189/2  6048}^{d_{5}a_{5}^{4}  16  720 200/2  5184}^{d_{5}d_{4}^{5}  120  640 200/2  4096}

10

d_{5}a_{4}^{5}  20  1000 160/2  12500
a_{5}^{4}d_{4}a_{1}  48  936 169/2  10368
a_{5}^{4}a_{2}a_{1}^{3}  48  1512 147/2  31104
a_{5}a_{4}^{5}  20  1125 135/2  18750
a_{5}a_{2}^{10}  720  3645  75/2  354294
d_{4}^{6}a_{1}  2160  832 169/2  8192
d_{4}^{5}a_{2}a_{1}^{3}  360  1344 147/2  24576
d_{4}^{4}a_{3}a_{1}^{6}  144  2048 128/2  65536
d_{4}^{4}a_{1}^{9}  432  2816 121/2  131072
d_{4}a_{3}^{7}  336  1792  98/2  65536
d_{4}a_{1}^{21}  120960  14336  49/2 8388608
a_{4}^{6}a_{1}  240  1375 121/2  31250
a_{4}a_{3}^{7}  168  1920  90/2  81920
a_{3}^{8}a_{1}  2688  2304  81/2  131072
a_{3}a_{2}^{11}  7920  4374  54/2  708588
a_{3}a_{1}^{22}  887040  16384  32/2 16777216
a_{2}^{12}a_{1}  190080  5103  49/2 1062882
a_{2}a_{1}^{23}  10200960  18432  27/2 25165824
a_{1}^{24}a_{1}  244823040  20480  25/2 33554432

Figure 1. The enumeration of shallow holes whose diagram contains a component a_{15}. For types (a)-(d) the set of points disjoined from a_{15} has no 10-point subgraph of mutually disjoined spherical diagrams, so only case (e) leads to holes.

(This figure is not yet included as I can’t be bothered to draw it in TEX.)

References.

[B] Bourbaki, Nicolas Groupes et algèbres de Lie. Chapitres 4, 5 et 6. Masson, Paris,

1981. ISBN: 2-225-76076-4 [C-N] Conway, J. H.; Norton, S. P. Monstrous moonshine. Bull. London Math. Soc. 11

(1979), no. 3, 308–339. [C-S] Conway, J. H.; Sloane, N. J. A. Sphere packings, lattices and groups Grundlehren

der Mathematischen Wissenschaften, 290. Springer-Verlag, New York, 1999. ISBN: 0-387-98585-9