The Graph of the Truncated Icosahedron and the Last Letter of Galois Bertram Kostant
he use of group theory by chemists to of SO(3) it is the proper symmetry group of the determine certain properties of suit- icosahedron (and the dodecahedron). It is the able molecules is a well-established unique finite subgroup of SO(3) which equals its procedure and there is a vast litera- own commutator subgroup—in fact, it is the ture on the subject. For a single mol- unique non-abelian simple finite subgroup of Tecule the group involved is the molecule’s sym- SO(3). It can be thought of as the beginning or metry group which, up to conjugacy, can be smallest member of the family of non-abelian fi- considered as a subgroup of O(3) (necessarily fi- nite simple groups. From the point of view of the nite, assuming the molecule is non-trivial). The McKay correspondence it (or more exactly its most significant part of the symmetry group is “double cover”) corresponds to the exceptional Lie its intersection, G, with SO(3) and group E8. A group isomorphic to A will be referred to G SO(3) 5 ⊂ as an icosahedral group and any structure ad- will be referred to as the molecule’s proper sym- mitting such a group as a symmetry group is said metry group. This use of the word proper in to have icosahedral symmetry. Given the unique connection with subgroups of O(3) will be main- role of the icosahedral group in group theory, tained throughout the paper. any natural structure having icosahedral symme- From the point of view of mathematics the try surely deserves special attention. In this paper groups G in question—with exactly one excep- we will be concerned with one such structure—a tion (up to conjugacy)—are not very interesting structure which seems to be appearing with in- since they are easily constructed solvable groups. creasing frequency in the scientific literature. The one exception is a group which is isomor- Prior to the discovery of the Fullerenes, around phic to the alternating group A5. As a subgroup ten years ago, the only known form of pure solid carbon was graphite and diamonds. These two Supported in part by NSF Grant No. 9307460-DMS. forms are crystalline materials where the bonds Bertram Kostant is professor emeritus of mathematics between the carbon atoms exhibit hexagonal and at the Massachusetts Institute of Technology in Cam- tetrahedral structures, respectively. In neither of bridge, MA. His e-mail address is kostant@ these two substances, however, are there isolated math.mit.edu. molecules of pure carbon. On the other hand, in
SEPTEMBER 1995 NOTICES OF THE AMS 959 Fullerene one finds for the first time a pure car- tagons and separate hexagons from pentagons. bon crystalline solid with well-defined carbon We refer to these sixty edges as pentagonal molecules. See e.g. the top of p. 58 in [6]. Math- edges. The remaining thirty edges separate hexa- ematically these molecules can be described as gons from hexagons and are referred to here as convex polyhedrons where hexagonal edges. According the faces are either hexagons to pp. 46–47 in [8] the pen- or pentagons and each ver- tagonal edges in C60 are sin- tex (carbon atom) is the end- gle carbon bonds and the point of three edges (carbon hexagonal edges are double bonds). The fact that the Euler carbon bonds. Thus each characteristic of the 2-sphere Among the hexagon, reminiscent of a is 2 easily implies that the many Fullerene benzine ring, has alternat- number of pentagonal faces ing single and double bonds. is necessarily 12. Hint: Ex- molecules the However, unlike benzine, press the number of vertices, single and double bonds in edges and faces in terms of most prominent these hexagons of C60 re- the number of pentagons and main fixed. the number of hexagons. See and the most The truncated icosahe- e.g. [8], p. 16. dron has sixty vertices. Be- Fullerenes exhibit remark- studied is C60 cause of the isolation of the able chemical and physical pentagonal faces each ver- properties (e.g. supercon- tex lies on a unique penta- ductivity, ferromagnetism, gon. In this way the pen- tremendous stability) and tagons define a natural have been the objects of a equivalence relation on the vast amount of research throughout the world. set of vertices—partitioning the set of vertices The shape of the molecules is such that they can into twelve equivalence classes where each class behave as cages, encapsulating other atoms or is the 5-element set of vertices of one of the pen- molecules. For an up-to-date account of tagons. By abuse of terminology we will gener- Fullerenes, see [7]. A catalogue of possible ally refer to these 5-element sets themselves as Fullerene applications is given in Chapter 20 of pentagons. [7] and in part III of [11]. One also notes that at each vertex there are Among the many Fullerene molecules the three edges, two pentagonal and one hexagonal. most prominent and the most studied is C60. The structure of a truncated icosahedron is com- (The term Fullerene is sometimes used to refer pletely determined by the graph of its vertices specifically to this molecule.) In the case of C60 and edges. The proper symmetry group G of the corresponding polyhedron is the truncated C60, or of the truncated icosahedron,Γ is an (60- icosahedron (see chapter 8, 4, in [5]). This poly- element) icosahedral group. The group G oper- hedron is seen on the surface§ of a soccer ball. It ates in a simple transitive way on the set V of has thirty-two faces. That is, besides the twelve vertices. Thus, given a pair of ordered vertices pentagons there are twenty hexagons. It is of sig- there is a unique proper symmetry which car- nificance that the twelve pentagons are isolated ries the first to the second. That is, the action from one another. Chemists believe that the iso- of G on V is equivalent to the action of G on lation of the pentagons in a Fullerene molecule itself by left translations. In particular the action is a requirement for stability. See e.g. bottom of of Gon V does not “see” the edge structure in p. 4 in [8]. C60 is the smallest Fullerene molecule . This, in our opinion, points to an inadequacy in which this isolation occurs. of Gin dealing with many questions about the The truncated icosahedron is also found as natureΓ of C60. In the expectation that a group- the polyhedral coat of a number of viruses. For based harmonic analysis will eventually lead to a survey of the polyhedral structure of viruses, a deep understanding of the remarkable mole- including electron microscope pictures, see [9]. cule C60, it seems to be highly desirable to be Another important biological occurrence of a able to express the full structure of group the- truncated icosahedral structure is in a sub- oretically. The edge structure in determines a stance—called a clathrin—which is concerned 60 60 adjacency matrix H and consequentlyΓ × with the release of neurotransmitters into the H would be expressed group theoretically.Γ In this synapses of neural networks. See p. 365 in [15] connection it is useful to point out that the (I thank S. Sternberg for this latter reference) and, eigenvalues of H, via what is called a Huckel ap- in more detail, in Chapter 5 of [11]. proximation, enter into the determination of There are ninety edges in the truncated icosa- the molecular energy levels of C60. See e.g. [8], hedron, sixty of which bound the twelve pen- p. 44 and 9 in [3]. §
960 NOTICES OF THE AMS VOLUME 42, NUMBER 9 If p is a prime number let Fp denote the fi- the icosahedron P as a symphony in the golden nite field of p elements. The group Sl(2,p)is the number. group of all 2 2 matrices with entries in Fp Let A SO(3) be the group of all rotations having determinant× 1 and PSl(2,p) is Sl(2,p) which stabilize⊂ the 12-element set V. Then A is modulo its (2-element if p is odd) center. The an icosahedral group. For any integer j>1, let group PSl(2,p) is simple if p 5 and PSl(2, 5) A(j) A denote the set of all elements in A of is an icosahedral group. The icosahedral≥ group order⊂ j. The A(j) = only if j =2,3,or5and PSl(2, 5) admits an embedding into PSl(2, 11) where denotes 6 set∅ cardinality and the relationship (see section headed “The || A(2) =15 Embedding of PSl(2, 5) into PSl(2, 11)) and Ga- | | lois’ Letter to Chevalier”, p. 964) between these A(3) =20 two groups is quite remarkable. This relationship | | A(5) =24 has much to do with a statement (see p. 964) | | made by Galois in his famous letter to Chevalier For any prime p (or in fact power of a prime) written on the night before his life-ending duel. the group PSl(2,p) naturally operates (transi- The set of all elements of order 11 in PSl(2, 11) tively) on p +1 points. More specifically, it op- decomposes into two conjugacy classes, each of erates on the projective line Fp over Fp which has sixty elements. The choice of the em- as the group of fractional transformations∪{∞} bedding (there are two such inequivalent em- x ax+b. The isotropy subgroup is the Borel beddings) of PSl(2, 5) in PSl(2, 11) favors one cx+d subgroup7→ of the conjugacy classes, say M. It will be proved that the conjugacy class M has a natural struc- ab ture of the graph of a truncated icosahedron. In B = aF∗,b Fp 0a1|∈p ∈ effect, the model we are proposing for C60 is (2) ( −! ) such that each carbon atom can be labeled by modulo the center an element of order 11 in PSl(2, 11) in such a fashion that the carbon bonds can be expressed where Fp∗ is the multiplicative group of invert- in terms of the group structure of PSl(2, 11). It ible elements in Fp. Thus for the case at hand will be seen that the twelve pentagons are exactly the projective line is the “flag manifold” for the intersections of M with the twelve Borel sub- PSl(2,p)—the set of all conjugates of the Borel groups of PSl(2, 11). (A Borel subgroup is any subgroup B. Since A PSl(2, 5) this means that ' subgroup which is conjugate to the group A naturally operates on six points. Geometrically PSl(2, 11) defined in (2).) In particular the pen- this is readily seen on the icosahedron P where tagons are the maximal sets of commuting ele- the six points may be taken to be six pairs of an- ments in M. The most subtle point is the nat- tipodal vertices. But, of course, A A5 so that ' ural existence of the hexagonal bonds. This will A operates on a set of five “objects”. Look- F arise from a group theoretic linkage of any ele- ing at the icosahedron P this is less obvious. Each ment of order 11 in one Borel subgroup with a edge and its antipodal edge are the edges of a uniquely defined element of order 11 in another golden rectangle defining in this way fifteen Borel subgroup. golden rectangles on P. These fifteen golden rectangles break up uniquely into five sets of The Graph of the Icosahedron and a three mutually orthogonal golden rectangles (one set of which is given by (1)). This is a geo- Conjugacy Class in A5 1+√5 metric way of seeing the five objects. Alge- Let τ = 2 be the golden number. A rectangle braically it is simpler. Define a relation on the is referred to as a golden rectangle if the ratio 15-element set A(2) where if σ,ρ A(2) then of the larger side to the smaller side is the golden σ ρ if σ commutes with ρ. Marvelously∈ in this 2 number. In R the rectangle with the four ver- case∼ the relation is an equivalence relation and tices ( 1, τ) is a golden rectangle. Let V be there are five equivalence classes each of which { ± ± } 3 the set of 12 points in R obtained from the ver- has three elements. Thus algebraically we can tices of the 3 mutually perpendicular golden take rectangles = the set of maximal commuting (3) F subsets of A(2) (1) V = ( 1, τ,0), (0, 1, τ), ( τ,0, 1) { ± ± ± ± ± ± } Of course there is a natural bijective corre- Then V is the set of vertices of an icosahedron spondence between A(2) and the set of fifteen P where c,d V define an edge of P if the golden rectangles. Each element in A(2) defines scalar product{ }⊂
SEPTEMBER 1995 NOTICES OF THE AMS 961 −1 −1 v u (4) u, v to be an edge of ,ifuv C { } ∈ Note that this is well defined since, being con- ∆ jugate elements, uv C if and only if vu C. Note also that whatever∈ graph has been defined,∈ it is necessarily invariant under the icosahedral group A. But in fact that there are exactly thirty edges and indeed one has the icosahedron. The following theorem is proved in [13]. Theorem 1. The graph is isomorphic to the graph of vertices and edges of an icosahedron. With respect to such an isomorphism∆ for any u C the vertex corresponding to u 1 is the an- u − v tipode∈ of the vertex correponding to u. If c and d are vertices of an icosahedron and c,d is an edge, we will call d a neighbor of c. Each{ }vertex c, of course, has five neighbors. The five neighbors of the antipode of c will be re- ferred to as coneighbors of c. Any vertex d not equal to c or its antipode is either a neighbor of c or a coneighbor of c, but not both. Accordingly we refer to the pair c,d as neighbors or coneighbors. The same terminology{ } of neighbor and coneighbor will be used for the conjugacy class C in A. − − With an example, we illustrate Theorem 1 for u 1v 1 the icosahedral group A5. Let C be the conju- gacy class of the permutation cycle u =(1,2,3,4,5) A . If w =(1,5,2,4,3) then Figure 1 5 one readily has ∈w C. However ∈ (1, 2, 3, 4, 5)(1, 5, 2, 4, 3) = (1, 4, 2) To what extent does A “see” the icosahedron itself? By this we mean: where in A can we find so that necessarily u, w are coneighbors. But the graph of the vertices and edges of an icosa- { } 1 then u and v must be neighbors where v = w − . hedron? It was our solution to this question, Indeed v =(1,3,4,2,5) and Theorem 1, given next, which led to the main re- sult of the paper (given in the section entitled “The Graph of the Truncated Icosahedron and uv =(1,2,3,4,5)(1, 3, 4, 2, 5) the Conjugacy Class M in PSl(2, 11)”, p. 965)— =(1,5,3,2,4) the solution of a similar question when the icosa- hedron is replaced by the truncated icosahe- and (1, 5, 3, 2, 4) C. dron. The sets A(2) and A(3) are conjugacy Remark 2. It follows∈ easily from the example classes in A. On the other hand, A(5) decom- above that if u, w C are arbitrary then u and poses into a union of two conjugacy classes, say u are coneighbors if∈ and only if uw has order 3. C and C0 each with twelve elements. Being con- Being a polyhedron, any edge of the icosahe- jugacy classes, they are, of course, A-sets with dron is the boundary of two faces. For the icosa- respect to the action of conjugation and both are hedron the faces are triangles. Hence if closed under inversion. The map C C0, u, v Care neighbors, there are exactly two σ σ 2 is an A-bijection. → other{ }⊂ elements w C which are neighbors of 7→ ∈ 1 1 1 Consider one of the two, say C A(5), con- both u and v. Since (vu)− = u− v− and ⊂ 1 1 1 jugacy classes of elements of order 5 in A. We (uv)− = v− u− satisfy these conditions they will define a graph whose set of vertices is C. must be two other elements as indicated in Fig- Normally it is not a common practice in group ure 1. theory to consider ∆whether or not the product Starting with the edge defined by u and v, and of two elements in a conjugacy class is again an choosing the orientation of the icosahedron so that 1 1 element in that conjugacy class. However such the edge u, v− u− is obtained from u, v by a consideration here turns out to be quite pro- counterclockwise{ rotation,} the five neighbors{ } of ductive. For u, v C we define u can be expressed in terms of u and v, as indi- ∈
962 NOTICES OF THE AMS VOLUME 42, NUMBER 9 − − v 1u 1 cated in Figure 2, exhibiting the five faces of the icosahedron which have u as a vertex. − Of course the five coneighbors of u are the uuv 1 inverses of the five neighbors. Remark 3. Note that a comparison of Figure 1 and Figure 2 yields the equation
2 2 1 1 (5) u vu− = u− v− u v Since u has order 5 the equation (5) is equiva- 2 lent to the condition that x = u− v has order 2. 1 By Remark 2 the element ux = u− v has order 3. These three relations involving u and x are a presentation (referred to in (12) as a standard presentation) of the icosahedral group and such a presentation will later be seen (Theorem 7) to define, as a Cayley graph, the graph of the trun- cated icosahedron. uv−1u−2 For non-solvable groups it is difficult to keep track of commutators of pairs of elements. The icosahedral group is the group of smallest order which equals its own commutator subgroup so that it is not without interest to see how com- uu2v−2 mutators in A distribute themselves. For the conjugacy class C, we will now see that this dis- Figure 2 tribution is neatly expressed in terms of the −−−− three orthogonal golden rectangles. Assume vv1u1=uu1v1 − 1 1 − v1uv u, v C are neighbors. Then u, v, u− ,v− uuv 1 are the∈ vertices of one of the fifteen{ golden rec-} −−1 −− uvu 1v tangles. On the other hand, given two elements u1v1uv in any group, there are eight ways of forming a commutator and these eight expressions de- −− −1 −1 −1−1 vvuu=1 u1 v uuv compose naturally into two sets with four ex- vvu pressions in each. For u and v these are set forth in the second and third columns of the array
uuvu1v1 uv 1u 1v −− − − v 11 1 1 u vv−u−vu vu− v− u (6) 1 1 1 1 1 u− vuv− u− v− uvu− 1 1 1 1 1 v− u− v− uv u− vuv−
Theorem 2. The elements in A defined by the twelve expressions in (6) are distinct and lie in C −1 −1 v −1−1 u —and hence in fact exhaust C. Furthermore the vuuv elements in any one of the 3-columns are the −− vertices of a golden rectangle and the three golden u1v1 rectangles are orthogonal (i.e. correspond to or- thogonal golden rectangles in the icosahedron 3 − −1 P R ). In particular the second and third golden u1vu vuv rectangles⊂ are the unique (among the fifteen − − −1−1 golden rectangles) golden rectangles which are vvu 1u1 vu vu orthogonal to the first. −− −1−1 This theorem is proved as Theorem 1.18 in vv1u1 uuv [13]. =−1−1 The following diagram (Figure 3) illustrates The- uuvv v u orem 2 and gives additional expressions (see 1 in [13]) for some of the vertices of the icosahedron§ in the terms of a pair u, v Cof neighbors. Figure 3 { }⊂
SEPTEMBER 1995 NOTICES OF THE AMS 963 The Embedding of PSl(2, 5) into PSl(2, 11) and Galois’ Letter to Chevalier A major theme in what follows is the relation- ship between the 60-element icosahedral group PSl(2, 5) and the 660-element group PSl(2, 11). Up to conjugacy the group PSl(2, 5) embeds into PSl(2, 11) as a subgroup in two different ways (which, however, are interchanged by an outer automorphism of PSl(2, 11)). That the re- lationship between the group PSl(2, 11) and its subgroup PSl(2, 5) is distinguished and extra- ordinary goes back to Galois. It will be discussed shortly. That the relationship has to do with the icosahedron is discussed in [3] and is further de- veloped in [13]. As we noted before the 6-element projective line Proj1(F5) over the field F5, as a PSl(2, 5)- set, may be identified with the six pairs of an- tipodal vertices on the icosahedron. The point is that the 12-element projective line Proj1(F11) Figure 4 over the field F11, as a PSl(2, 11)-set, may be identified with the full set of vertices V of the icosahedron in such a fashion that there exists non-trivially on a set with p elements when a partition of Proj1(F11) into six pairs of ele- p>11. This may be expressed as follows. The ments whose stabilizing subgroup of PSl(2, 11) cyclic group Zp of order p embeds uniquely in is PSl(2, 5), recovering the action of PSl(2, 5) on PSl(2,p), up to conjugacy, as the Sylow p-sub- Proj1(F5). Actually there exists two inequivalent group. It is the unipotent radical of a Borel sub- such partitions corresponding to the two em- group and pulled up to Sl(2,p) may be taken to beddings of PSl(2, 5) in PSl(2, 11). As evidence be that the bijective correspondence between Proj1(F11) and V is more than a coincidence it 1 x (7) Zp = x Fp was shown in [3] that the graph of vertices and ( 01!|∈ ) edges of the icosahedron may be neatly ex- pressed in terms of the cross-ratio in Proj (F ). 1 11 The result of Galois is that if p>11, See Theorem 1 in [3]. We are concerned here not with the graph of There exists no subgroup of PSl(2,p) the icosahedron but with the more sophisticated which is complementary to Z . graph associated with the truncated icosahe- p dron. A step in this direction will be a conse- quence of an idea inherent in Galois’ result. The By this we mean that there exists no subgroup F such that set theoretically PSl(2,p)=F Zp. three groups A4,S4, and A5 are, up to conjugacy, × the only finite subgroups of SO(3) which oper- The notation here, of course, does not mean di- ate irreducibly on R 3. They are, of course, the rect product of groups. It means that every ele- proper symmetry groups of the five Platonic ment g can be uniquely written g = fz where f F and z Zp. Implicit in Galois’ statement solids. For the tetrahedron there is A4. For the is ∈certainly the∈ knowledge that his statement is octahedron and the cube, there is S4, and for the not true for the three cases of a simple PSl(2,p) icosahedron and dodecahedron there is A5. From the point of view of the McKay corre- where p 11—namely p =5,7,11. It is surely a marvelous≤ fact, that for the three exceptional spondence the groups A4,S4, and A5 corre- spond, respectively, to the simple Lie groups cases, the groups F which run counter to Galois’ statement are precisely the symmetry groups of E6,E7, and E8. Now if p is a prime number then the group the Platonic solids. Namely one has PSl(2,p) is simple whenever p 5 and it of ≥ PSl(2, 5) = A4 Z5 course operates on the p +1-element projective × line over the field F . Galois’ result, reported in (8) PSl(2, 7) = S4 Z7 p × his letter to Chevalier (see p. 268 in [4] and PSl(2, 11) = A5 Z11 p. 214 in [10]) is that if p>11 then PSl(2,p)can- × not operate, non-trivially, on a set with fewer than These exceptional cases merit elaboration. p +1-elements. In particular it cannot operate The fact that PSl(2, 5) operates on five points
964 NOTICES OF THE AMS VOLUME 42, NUMBER 4 was discussed on page 961 (see (3)) in connec- Hadamard matrices. A Hadamard matrix is a tion with the commutativity equivalence relation square matrix all of whose elements are either on the set of elements of order 2 in PSl(2, 5). 1 or 1 and which has orthogonal columns (and − This action sets up the isomorphism of PSl(2, 5) hence has orthogonal rows). The set n of all H with A5. The group PSl(2, 7) is isomomorphic n n Hadamard matrices is clearly stable under × to PSl(3, 2). The 3 here implies that PSl(3, 2) op- the action of the group Gn of all row and col- erates on a projective plane and the 2 implies umn permutations and sign changes. If R 12 that the plane is over the field of two elements. then, first of all using sign changes, we∈H may This plane has 1+2+22 =7lines and 7 points, make the last row and the last column have only exhibiting seven objects on which PSl(2, 7) op- 1’s. If R0 is the complementary 11 11princi- erates. The plane is classically represented by the pal minor of R then, since the first 11× columns diagram in Figure 4. of R are orthogonal to the last column, each col- Of course our main interest is in the final umn of R0 must have exactly five 1’s. Taking the and most sophisticated case, PSl(2, 11). Its ac- row indices of these five 1’s defines a 5-element tion on eleven points arises from its symmetry subset of 1,... ,11 . The eleven 5-element sub- of a special and remarkable geometry. The eleven sets obtained{ in this} way must make a biplane points will be the field F11 itself—as represented geometry in order that the columns of R be or- by the integers from 1 to 11 where of course thogonal—and conversely. In particular (9) yields 11 = 0. The set of non-zero squares in F11 is the the following 12 12 Hadamard matrix. set 1, 3, 4, 5, 9 . Now using the additive × { } structure in F11 translate this five-element set by each of the elements in F11. One 1 1 1111111111 then obtains the following eleven five-ele- − − −−− − 111111111111 ment sets − −− −−− 111111111111 − −− −−− 1, 3, 4, 5, 9 111111111111 − −− −−− 111111111111 2, 4, 5, 6, 10 − −− −−− 111111111111 3, 5, 6, 7, 11 (10) R = − − −− −− 111111111111 −− − −− − 1, 4, 6, 7, 8 111111111111 −−− − −− 2, 5, 7, 8, 9 111111111111 −−− − −− (9) 3, 6, 8, 9, 10 111111111111 − −−− −− 111111111111 4, 7, 9, 10, 11 −− −−− − 111111111111 1, 5, 8, 10, 11 1, 2, 6, 9, 11 1, 2, 3, 7, 10 Remark 5. Andrew Gleason informed me that 2, 3, 4, 8, 11 it is easy to show that 12 is a single G12 orbit. This, of course, impliesH that one can obtain all If we regard these sets as lines then it may be elements in 12 by applying G12 to (10). He also H noted that any two distinct lines intersect in ex- conjectures that n =12is maximum such that actly two points and any two distinct points lie n is a single Gn-orbit, assuming of course that H in exactly two lines. That is, intersection sets up n is such that n is not empty. H a bijection between the 55-element set of all pairs of distinct points and the 55-element set The Graph of the Truncated Icosahedron of all pairs of distinct lines. This is referred to and the Conjugacy Class M in PSl(2, 11) as a biplane geometry on p. 7 in [1]. If we iden- Now returning to (8), instead of considering, as tify the symmetric group S11 with the permu- above, the 11-element PSl(2, 11)-set tation group of the set F11 then the subgroup PSl(2, 11)/H where H is the subgroup A5, con- of S11 which stabilizes the set of these eleven sider the 60-element PSl(2, 11)-set PSl(2, 11)/H lines is isomorphic to PSl(2, 11). This then yields where the subgroup H is chosen to be Z11. We an embedding of PSl(2, 11) into S11 or an action now raise the question of the possibility that of PSl(2, 11) on eleven objects. The isotropy X = PSl(2, 11)/Z11 has a natural structure of subgroup of PSl(2, 11) at a point in F11 is iso- the graph of the truncated icosahedron. Evi- morphic to PSl(2, 5). The second embedding of dence for this is that one immediately sees a PSl(2, 5) in PSl(2, 11) is obtained by taking the canonical PSl(2, 11) invariant decomposition of isotropy subgroup of a line instead of a point. X into a union of twelve “pentagons”. Namely if The biplane geometry on eleven elements is B is the normalizer of Z11 in PSl(2, 11) then directly related to the existence of 12 12 from general principles the group of B/Z11 op- ×
SEPTEMBER 1995 NOTICES OF THE AMS 965 erates (on the right) faithfully on X and the ac- Finally, A is the proper symmetry group of . tion not only commutes with the action of PSl(2, 11) but is its full centralizer in the full If we apply Theorem 7 to the coset spaceΓ group of permutations of X. But B is just that X = PSl(2, 11)/Z11 we note that the element φ Borel subgroup of PSl(2, 11) whose unipotent has been essentally given to us. It is a generator radical is Z11. Thus of the “torus” B/Z11 operating from the right on X. Explicitly what is missing is the permuta- (11) B/Z11 Z5 tion τ which with φ defines a standard pre- ' sentation of the centralizer of A5 in the per- The orbits of B/Z11 are the twelve pentagons. mutation group of X. To find τ we seek a more In addition, from (8), X is a principal homoge- natural setting for X, where there is more un- neous space for A5 which is exactly how A5 op- lying structure, and we will find it as a conju- erates on the vertices of a truncated icosahedron. gacy class M of elements of order 11 in What is missing, of course, is the most sophis- PSl(2, 11). ticated part of the graph of the truncated icosa- Let A be a fixed choice of an icosahedral sub- hedron—namely the hexagonal edges—the edges group of PSl(2, 11). (Recall there are two such which tie together the various pentagons (the subgroups, up to conjugacy.) Then if Z11 is the double carbon bonds in C60). unipotent radical of any Borel subgroup one A graph, namely a Cayley graph, is defined on has, using multiplication, the set (non-group) a group by choosing generators. In the case of direct product an icosahedral group there is a classical choice of a pair of generators. Actually, there are three (13) PSl(2, 11) = A Z11 such (which are easily related to one another) de- × pending on whether one wants the generators to If Z is the set of elements of order 11 in Z have orders 2 and 5, or 2 and 3, or 3 and 5. We 11∗ 11 then, of course, Z =10. Since there are twelve make the first choice. A presentation of the 11∗ Borel subgroups| it follows| that there are 120 el- icosahedral group—which we refer to as a stan- ements of order 11 in PSl(2, 11). Since B has two dard presentation—is given by a pair of non-triv- 5-element orbits on Z it then follows that ial group elements φ, τ satisfying the relations 11∗ there are 2 conjugacy classes M,M0 of elements of order 11 and each has sixty elements. Let M 5 2 3 (12) φ =1,τ=1, (φτ) =1 be either one of these two classes. Remark 9. Actually, as stated in Theorem 11 See e.g. Lemma 13.6, p. 120 in [14]. Non-trivial below, the choice of A will be seen to favor one elements φ, τ in an icosahedral group A0 define of these classes over the other. Any non-trivial a Cayley graph on A0. The following theorem says outer automorphism of PSl(2, 11) interchanges that this Cayley graph is the graph of a truncated the two choices of A and also interchanges the icosahedron. Actually it is more convenient for two choices of M. us to express the result in terms of a principal Since the centralizer of any element in M is homogeneous space (i.e. trivial isotropy groups) the unipotent radical of the unique Borel sub- rather than the group itself. group B which contains it, one has an isomor- Theorem 7 is just a more detailed version of phism Theorem 1 in [3]. It is proved in [3] and as The- orem 2.10 in [13]. (14) M X = PSl(2, 11)/Z11 Theorem 7. Let M be any 60-element set and ' let S60 be the full group of permutations of M. of PSl(2, 11)-sets. The twelve “pentagons” Assume that φ, τ S60 satisfy the relations (12) P M then turn out to be the subsets of the and that φ, τ and∈ φτ have no fixed points in form⊂ M. Then the subgroup A S60 generated by φ 0 ⊂ and τ is an icosahedral group and M is a prin- (15) P = B0 M ∩ cipal homogeneous space for A0. Let A be the cen- tralizer of A0 in S60 so that Ais also an icosahe- where B0 is a Borel subgroup of PSl(2, 11). dral group (A and A0 are each other’s centralizer) Remark 10. In group theoretic terms it fol- and M is also a principal homogeneous space for A. lows from (15) that the pentagons can be char- Let be the graph on M so that for any x M acterized as the maximal commutative subsets ∈ the edges (three of them) containing x are x, φx of M. Implicit in this statement is the fact that— { } , x, φΓ 1x and x, τx . Then is isomorphic to as in A(2) (see p. 961)—commutativity in M is { − } { } the graph of a truncated icosahedron where the an equivalence relation and the pentagons are two pentagonal edges containingΓ x are x, φx the equivalence classes. { } and x, φ 1x and the unique hexagonal edge If p is a prime number and M is a conjugacy { − } p containing x is x, τx . class of elements of order p in PSl(2,p) then { }
966 NOTICES OF THE AMS VOLUME 42, NUMBER 9 note that Galois’ result can be interpreted as saying that Mp cannot be a principal homoge- neous space for the conjugation action of a sub- group of PSl(2,p), in case p>11. In particular then, the Cayley graph structure on M which will be now be constructed can have no analogous generalization for Mp when p>11. The use of the term pentagon implies that there is an implicit graph structure on the orbits of the torus B/Z11 . Up to now this has not been y made clear. But in fact there are two polygonal τ Cayley graphs on the cyclic group B/Z11 = Z5— x each defining a pentagonal structure on this τy group and consequently on its orbits. These are illustrated as the single and double edge struc- ture in Figure 5. Note that n =5is the minimal value of n for which Zn has more than one polygonal Cayley graph. x The action of this Z5 can be expressed in terms of group multiplication in PSl(2, 11). For z
τz
Figure 6
ifold” of PSl(2, 11)—the twelve Borel subgroups of PSl(2, 11). Truncating a polyhedron can be viewed as a finite analogue of blowing up points in algebraic varieties. In the operation of blow- ing up points one replaces the point by objects in the tangent space at the point. Changing tan- Figure 5 gent space to cotangent space, it is a well known and standard fact that an orbit of the principal unipotent elements of a semisimple Lie group one pentagonal structure the two pentagonal embeds naturally in the cotangent bundle of its edges containing any x M are x, x3 and flag manifold. Thus regarding truncation of an x, x4 . For the other structure∈ the{ edges} are icosahedron as the operation of replacing a Borel { 9} 5 4 3 x, x and x, x —noting that x = x− and subgroup B0, regarded as an element (of the {5 } 9 { } x = x− . A choice of one these two determines “flag manifold”), by the elements of the penta- φ (see Theorem 11) up to inversion. gon B0 M is not entirely without motivation. Finding the graph of the truncated icosahe- The ∩subtle point in the construction of the dron in the conjugacy class M, group theoreti- graph of the truncated icosahedron in the con- cally, is clearly analogous, and was motivated by, jugacy class M is the determination of the per- finding, as in Theorem 1, the graph of the icosa- mutation τ of M of order 2 which with φ de- hedron in a conjugacy class C of elements of fines a standard presentation of the centralizer order 5 in the icosahedral group A. There is an- of the conjugation action of A on M. Given any other heuristic reason for taking M to be the ver- x M the pair x, τx would define the hexag- tices of the truncated icosahedron. It goes some- onal∈ edge containing{ }x. In effect, then what τ thing like this: The twelve vertices of the is to do is to assign to any x M, which, of icosahedron have been identified here, and pre- course, is a unipotent element lying∈ in a unique viously in [3], with the points of the “flag man- Borel subgroup, not only another Borel subgroup
SEPTEMBER 1995 NOTICES OF THE AMS 967 B0 but a distinguished unipotent element τx in unique choice of a (60-element) conjugacy class, that subgroup. I know of no operation for gen- M, of elements of order 11 in PSl(2, 11) such that eral groups which does that sort of thing. Geo- if x M there exists u CA which normalizes the ∈ ∈ metrically this linking together of the pentagons cyclic groups generated by x and σxxσx and is is represented locally as dotted lines in the dia- such that the pair u, ρx defines a standard pre- gram (Figure 6). sentation of A. (That is, uρx has order 3). Fur- (Of course, to represent bonds in C the sin- thermore if x M then ρxx M and a trun- 60 ∈ ∈ gle and double lines in Figure 6 should be re- cated icosahedral structure is defined on M versed). where the two pentagonal edges containing x are In the notation of the following theorem the x, x3 and x, x 3 and the Γunique hexagonal { } { − } map τ is given by putting τx = ρ x. Theorem 11 edge containing x is x, ρxx . x { } is proved as the main theorem, Theorem 3.30, Next given x M the element u CA, satis- ∈ ∈ in [13]. Although its proof makes no use of com- fying the condition above, is unique up to inver- puters, it was, however, discovered after a long sion and may be fixed by the condition that series of Maple computations involving uxu 1 = x5. As such write u = u(x). The space − P PSl(2, 11). of twelve pentagons is then parameterized by CA Theorem 11. Let A be any icosahedral sub- so that, for u CA, one has Pu where ∈ ∈P group of PSl(2, 11). Let A(2) be the set of all el- Pu = x M u = u(x) . In addition the icosahe- { ∈ | } ements of order 2 in A. Then for any dral graph structure induced on by is such P x PSl(2, 11) of order 11 there exists a unique that Pu,Pv is an edge if and only if uv CA. ∈ { } ∈ element σx A(2) such that the commutator Tables, which among∆ other things, labelΓ the 1 ∈ ρx = x− σxxσx is again in A(2). Moreover, there vertices of the truncated icosahedron and their exists a unique choice CA of a (12-element) con- neighbors by elements in M are given in [12] as jugacy class of elements of order 5 in A and a well as in [13].
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968 NOTICES OF THE AMS VOLUME 42, NUMBER 9