The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits

Dutour Sikiric, M.

Schuermann, A.

Vallentin, F. (Frank)

English

Dutour Sikirić, M. (Mathieu)

CWI's Institutional Repository

Discrete Comput Geom (2010) 44: 904–911DOI 10.1007/s00454-010-9266-zThe Contact Polytope of the Leech LatticeMathieu Dutour Sikiric´ · Achill Schürmann ·Frank VallentinReceived: 16 June 2009 / Revised: 16 February 2010 / Accepted: 9 May 2010 /Published online: 3 June 2010© The Author(s) 2010. This article is published with open access at Springerlink.comAbstract The contact polytope of a lattice is the convex hull of its shortest vectors.In this paper we classify the facets of the contact polytope of the Leech lattice up tosymmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.Keywords Leech lattice · Contact polytope · Conway groups · Voronoi cell1 IntroductionAn n-dimensional lattice L is a discrete subgroup of the n-dimensional Euclideanspace Rn of the form L = {∑ni=1 αibi : α1, . . . , αn ∈ Z}, where b1, . . . , bn is a basisof Rn. By λ(L) we denote the Euclidean length of nonzero shortest vectors of L, andwe denote by MinL the set of shortest vectors.Every lattice comes with two important polytopes: The contact polytope of L isthe convex hull of its shortest vectorsC(L) = conv{v : v ∈ MinL},M. Dutour Sikiric´Rudjer Boskovic´ Institute, Bijenicka 54, 10000 Zagreb, Croatiae-mail: mdsikir@irb.hrA. Schürmann · F. Vallentin ()Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600 GADelft, The Netherlandse-mail: f.vallentin@tudelft.nlA. Schürmanne-mail: a.schurmann@tudelft.nlDiscrete Comput Geom (2010) 44: 904–911 905and the Voronoi cell of L is the region of points that are closer to the origin than toother lattice points,V(L) ={x ∈ Rn : x · v ≤ 12v · v for all v ∈ L}.Maybe one of the most remarkable lattices is the 24-dimensional Leech lattice24. It has 196,560 shortest vectors which is the highest possible number in di-mension 24. Its orthogonal group, i.e., the group of orthogonal transformations pre-serving the lattice is the Conway group Co0. It has 222 · 39 · 54 · 72 · 11 · 13 · 23 =8,315,553,613,086,720,000 elements and is connected to many sporadic simplegroups. We refer to the book [6] by Conway and Sloane for an extensive treatment ofthe Leech lattice.Borcherds, Conway, Parker, Queen, Sloane [6, Chaps. 23 and 25] determine thevertices of the Voronoi cell of the Leech lattice. The Voronoi cell tiles the space Rn bytranslations; this gives the Voronoi cell tiling of Rn. So, in the context of the Voronoicell it is natural to consider orbits under the isometry group (the group generated bythe orthogonal group of the Leech lattice together with lattice translations) acting onthe Voronoi cell tiling. We denote the isometry group of the Leech lattice by Co∞.There are 307 orbits of vertices in the Voronoi cell tiling under the action of Co∞.In this paper we determine the facets and their incidence relations of the contactpolytope of the Leech lattice. We get the following result.Theorem 1 There are 232 orbits of facets of C(Λ24) under Co0.The contact polytope and the Voronoi cell are related. To see this relation, weconsiderC(L)∗ ={x ∈ Rn : x · v ≤ 12v · v for all v ∈ MinL},which is the standard polar polytope scaled by a factor of 12λ(L)2. The faces ofC(L) and of C(L)∗ are in bijection. The bijection reverses the inclusion relation:k-dimensional faces of C(L) correspond to (n − k)-dimensional faces of C(L)∗. Inparticular, vertices of C(L)∗ correspond to facets of C(L). For these notions, we referto the standard literature on polytope theory, e.g., the book by Ziegler [21].We chose the scaling in the definition of C(L)∗ so that it contains V(L). In thecase of the Leech lattice some vertices of V(Λ24) and C(Λ24)∗ are shared. As a sideremark: One has the equality C(L)∗ = V(L) if and only if L is a root lattice, seeRajan and Shende [18].Theorem 2 164 orbits of vertices of C(Λ24)∗ are also orbits of vertices of V(Λ24).They are listed in Table 1 in the complete version of the paper [10]. The additional68 orbits of vertices are listed in Table 2 of [10].We classify the shared vertices in Sect. 2 and give them in Table 1 of [10]. InSect. 3 we classify the additional vertices of C(Λ24)∗ that are not vertices of V(Λ24).906 Discrete Comput Geom (2010) 44: 904–911We conclude the paper by Sect. 4, where we briefly explain our computational tech-niques.The data presented here is also electronically available from [8].2 Shared VerticesIn this section we explain the notation used in Table 1 of [10], which contains the 164orbits of shared vertices mentioned in Theorem 2.The vertices of the Voronoi cell of a lattice are centers of empty spheres, i.e.,spheres S(x,‖x‖) with center x and radius ‖x‖ which contain lattice points on theirboundary but not in their interior. The convex hull of lattice points on the boundaryof such an empty sphere is called the Delone cell of the vertex x.The Delone cells of the Leech lattice are classified by Borcherds, Conway, Parker,Queen, Sloane [6, Chaps. 23 and 25] up to the action of the isometry group Co∞. Forthis classification, they use Coxeter–Dynkin diagrams.A Coxeter–Dynkin diagram with vertex-set {1, . . . ,N} is a symmetric N ×N ma-trix (mij )1≤i,j≤N with ones on the diagonal and mij ≥ 2 if i = j and mij ∈ N ∪ {∞}.A Coxeter–Dynkin diagram is called simply laced if mij = 2, 3, or ∞. TheCartan matrix of a Coxeter–Dynkin diagram (mij )1≤i,j≤N is the matrix M =(− cos πmij)1≤i,j≤N . A Coxeter–Dynkin diagram is called spherical if its Cartanmatrix is positive definite and affine if its Cartan matrix is positive semidefinite.A Coxeter–Dynkin diagram is called decomposable if we can partition its vertex-setinto S1 ∪ S2 with mij = 2 if i ∈ S1 and j ∈ S2. It is called indecomposable otherwise.A Coxeter–Dynkin diagram D admits a unique decomposition into indecomposableCoxeter–Dynkin diagrams D1, . . . ,Dr , which we write as D = D1D2 . . .Dr . Theclassification of spherical and affine Coxeter–Dynkin diagrams is presented, for ex-ample, in Humphreys [13, Sects. 2.4 and 4.7]. Here the famous A−D −E diagramsshow up, explained, e.g., by Hazewinkel, Hesselink, Siersma, and Veldkamp [12].The spherical, simply laced, indecomposable Coxeter–Dynkin diagrams are an forn ≥ 1, dn for n ≥ 4, and en for 6 ≤ n ≤ 8. Each diagram corresponds to an indecom-posable affine diagram: An, Dn, and En. All these diagrams are pictured, e.g., in [6,Fig. 23.1].In the Leech lattice, a Coxeter–Dynkin diagram (mij )1≤i,j≤N can be associatedwith a Delone cell with vertex-set {v1, . . . , vN } bymij =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1 if ‖vi − vj‖2 = 0,2 if ‖vi − vj‖2 = 4,3 if ‖vi − vj‖2 = 6,∞ if ‖vi − vj‖2 = 8.As can been seen in Table 1 of [10], different Delone cells may have the sameCoxeter–Dynkin diagram.In Table 1 of [10] the rows are sorted first by the squared length ‖v‖2 (third col-umn) of the vertex v. Second, they are sorted by the size of the stabilizer of v withinDiscrete Comput Geom (2010) 44: 904–911 907the orthogonal group of the Leech lattice (fifth column) and then by the number ofincident facets of C(Λ24)∗ (fourth column).In the second column we give the Coxeter–Dynkin diagrams of the associatedDelone cell of v. Note that the diagrams are affine if and only if the squared length ofv equals 2, the maximum among shared vertices. In all other cases they are spherical.Furthermore, in the spherical cases the number of incident facets is always equal tothe minimum possible number of 24. These observations follow from [6, Chaps. 23and 25].In the last column we give the MOG (Miracle Octad Generator) coordinates ofrepresentatives of each orbit which one has to multiply with α (sixth column). TheMOG coordinates form a standard coordinate system for the Leech lattice. They areexplained in [6, Chap. 11].There are 307 orbits of vertices in the Voronoi cell tiling under the action of theisometry group Co∞ of the Leech lattice. Our computation shows that there are 5,297orbits of vertices of the single Voronoi cell V(Λ24) under the action of the smaller,finite orthogonal group of the Leech lattice; 164 of them are shared with C(Λ24)∗.3 Additional VerticesThere are 68 additional orbits of vertices of C(Λ24)∗ that are not vertices of theVoronoi cell of the Leech lattice. These additional vertices are characterized by thefact that the distance to a closest lattice point is strictly less than the distance ‖v‖ tothe origin.Table 2 of [10] describes these 68 orbits. Like in Table 1 of [10], the rows aresorted (in this order) by the squared length ‖v‖2 (third column), the size of the stabi-lizer of v within the orthogonal group of the Leech lattice (fifth column), and then bythe number of incident facets (fourth column).In the second column we give names for diagrams. The first row corresponds to anexceptional vertex which we explain below. The other 67 rows correspond to graphswhich we define later in Sect. 3.2.3.1 The Exceptional VertexThe first orbit of vertices is exceptional: Its squared norm 8/3 = 2.666 . . . is sub-stantially bigger than the squared norm of all other vertices which lie in the interval[1.92,2.25]. Its incidence number of 552 and the size of its stabilizer, which is theConway group Co3, are also substantially larger than the values for the other ver-tices. This orbit of vertices is a scaled copy of the vectors of Λ24, having Euclideannorm√6.In the contact polytope C(Λ24) this exceptional vertex corresponds to a facet.Since it has maximum norm among all vertices, the corresponding facet is closest tothe origin and has the largest possible circumsphere among all other facets of C(Λ24).This solves a conjecture of Ballinger, Blekherman, Cohn, Giansiracusa, Kelly, andSchürmann [2, Sect. 3.7]. We note that a similar calculation as the one presented here,solves the corresponding conjecture about the contact polytope of the 23-dimensionallattice O23, the shorter Leech lattice, which has 4,600 vertices.908 Discrete Comput Geom (2010) 44: 904–911The 23-dimensional point configuration, given by the 552 shortest vectors of theLeech lattice defining facets incident to the exceptional vertex, appears in severaldifferent contexts: It is universally optimal (Cohn and Kumar [5]), it defines 276equiangular lines (Lemmens and Seidel [15]), and it defines an extreme Delone cell(Deza and Laurent [14, Chap. 16.3]). Moreover, it contains a wealth of remarkablesubstructures (see Cohn et al. [2]), e.g., the highly-symmetric point configurationsdiscussed in the next section, and also others, e.g., the one defined by the McLaughlingraph.3.2 The Other VerticesTo the remaining 67 orbits of vertices we associate a diagram as follows. Let v be oneof these vertices, and let w1, . . . ,wN be shortest vectors of the Leech lattice definingfacets incident to v. Only the two inner products 1 and 2 occur between distinctvectors wi and wj . So we can define a graph with vertex-set {1, . . . ,N} and edge-set{{i, j} : wi · wj = 1}; the other inner product 2 defines nonedges.Here again the graphs decompose into connected components where several ofthese occurring components are highly symmetric and have been studied in othercontexts. We discuss them below; the graphs an, dn, and en are already described inthe previous section, and the remaining ones are in Fig. 1.The Higman–Sims graph HS100 is the unique strongly regular graph with parame-ters (100,22,0,6). See Brouwer, Cohen, and Neumaier [4, Chap. 13.1].The Hoffman–Singleton graph HS50 is the unique strongly regular graph with pa-rameters (50,7,0,1). See [4, Chap. 13.1].For the Johnson graph J (7,4), see [4, Chap. 9.1].A (k, g)-cage is a regular graph of valency k and girth g which attains the min-imum possible number of vertices. The (5,6)-cage (incidence graph of a projec-tive plane PG(2,4)) and the (3,8)-cage (Tutte–Coxeter graph) are unique. See [4,Chap. 6.9] and Tutte [20].The Coxeter graph Cox is the unique distance regular graph with intersection array{3,2,2,1;1,1,1,2}. See [4, Chap. 12.3].In Fig. 1 we list the remaining graphs. The vertices of these graphs only havedegree one (white circles), degree two (sitting on edges, which are not depicted; seebelow), or degree three (black circles). We have three kinds of trees: T ab c havinga + b + c + 4 vertices, T ab cde having a + b + c + d + e + 6 vertices, and T ab cdefghaving a + b + c + d + e + f + g + 8 vertices; we have 12 other graphs Gn,m withn vertices and m edges. In Fig. 1 the numbers on the edges show how many verticesof degree 2 sit on them, but in the following four cases we did not put these numbers:The graph G24,30 has one vertex of degree 2 on every edge, G25,30 is the Petersengraph which has one vertex of degree 2 on every edge, G22,22 has three vertices ofdegree 2 on every edge, and the graph G24,27 is the complete bipartite graph K3,3which has two vertices of degree 2 on every edge.4 Computational TechniquesComputing the vertices of C(Λ24)∗ from its facets is called a polyhedral represen-tation conversion problem. A direct application of standard programs like Fukuda’sDiscrete Comput Geom (2010) 44: 904–911 909Fig. 1 Diagramscdd [11] or Avis’ lrs [1] for this conversion is not feasible due to the extremelylarge number of vertices.In order to exploit the symmetries of C(Λ24)∗, we use the adjacency decomposi-tion method which is surveyed in Bremner, Dutour Sikiric´, and Schürmann [3]. Animplementation by the first author is available from [7].910 Discrete Comput Geom (2010) 44: 904–911The adjacency decomposition method computes a complete list of inequivalentvertex representatives. First, one computes an initial vertex by solving a linear pro-gram and inserts it into the list of orbit representatives. From any such representative,we compute the list of adjacent vertices, and if they give a new orbit, we insert it intothe list of representatives. After finitely many steps all orbits have been treated. Com-puting adjacent vertices is a lower-dimensional representation conversion problem.So this method can be applied recursively.For C(Λ24)∗, we had to come up with two case-specific insights:From [2] it is known that the exceptional vertex of Sect. 3.1 is indeed a vertex ofC(Λ24)∗. We used it as starting vertex of the adjacency decomposition method.For checking isomorphy and for computing stabilizers, we used the following stan-dard strategy: we characterize a vertex of C(Λ24)∗ by the set of its incident facets, andwe represent the symmetry group Co0 as a permutation group acting on the 196,560shortest vectors of the Leech lattice. Then, we use the backtracking algorithm by Leon[16, 17] implemented in [19]. This worked reasonably fast for all the cases except forthe two orbits of vertices having the same Coxeter–Dynkin diagram a251 . The stabi-lizer of the corresponding Delone cell under the isometry group Co∞ is the Mathieugroup M24. Under the action of M24, the 25 vertices of the Delone cell split into twoorbits of size 1 and 24. Hence, these two orbits correspond to two distinct orbits ofvertices of C(Λ24)∗, one having stabilizer M24 and the other having stabilizer M23.The backtracking algorithm of GAP could not decide in reasonable time whether ornot two vertices with the same Coxeter–Dynkin diagram a251 are in the same orbit. Sowe used the third method of Sect. 3.5 of [9] to resolve this problem.Acknowledgements We started this research during the Junior Trimester Program (February 2008–April 2008) on “Computational Mathematics” at the Hausdorff Institute of Mathematics (HIM) in Bonn.Then, part of this research was done at the Mathematisches Forschungsinstitut Oberwolfach during a staywithin the Research in Pairs Programme from 3 May 2009 to 16 May 2009. We thank both institutes fortheir hospitality and support. The work of the first author has been supported by the Croatian Ministry ofScience, Education and Sport under contract 098-0982705-2707. The second and the third authors weresupported by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1503/4-2.Open Access This article is distributed under the terms of the Creative Commons Attribution Noncom-mercial License which permits any noncommercial use, distribution, and reproduction in any medium,provided the original author(s) and source are credited.References1. Avis, D.: A C-implementation of the reverse search vertex enumeration algorithm. School of Com-puter Science, McGill University, Montreal, Canada 1993. http://www-cgrl.cs.mcgill.ca/~avis/C/lrs.html (Software)2. Ballinger, B., Blekherman, G., Cohn, H., Giansiracusa, N., Kelly, E., Schürmann, A.: Experimentalstudy of energy-minimizing point configurations on spheres. Exp. Math. 18, 257–283 (2009)3. Bremner, D., Dutour Sikiric´, M., Schürmann, A.: Polyhedral representation conversion up to symme-tries. In: Proceedings of the 2006 CRM Workshop on Polyhedral Computation. 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The contact polytope of the Leech lattice

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