The spectra of finite 3-transposition groups

We calculate the spectrum of the diagram for each finite $3$-transposition group. Such graphs with a given minimal eigenvalue have occurred in the context of compact Griess subalgebras of vertex operator algebras.

Assuming that all three sets are nonempty, we say that G acts with rank 3 on (X) (and so also on [X]) and that (X) and [X] are a complementary pair of rank 3 graphs. (There is nothing to say if all three are empty. If two are empty, then |X| = 1. If one is empty, then (X) and [X] are a complementary pair of a complete and an empty graph, and G is 2-transitive on X; this is rank 2 action.) A strongly regular graph is a finite graph (X) with the following strong regularity property: There are constants k, λ, and µ such that, for x, y ∈ X, the number of common neighbors of x, y is k when x = y; λ when x ∼ y; and µ when x ∼ y.
Empty and complete graphs provide the degenerate cases k = 0 and k = n − 1 of this condition, where |X| = n . (3.1) Here we do not include these as strongly regular; that is, we additionally require 0 < k < n − 1 . (3.2) This graph will be connected of diameter 2 unless µ = 0. In that case, the graph is a disjoint union of complete subgraphs K k+1 . Its complementary strongly regular graph is then complete multipartite with µ = k. This pair of graphs is imprimitive. We shall only be concerned with strongly regular graphs that are not imprimitive-those that are primitive. For us, the basic observation is that a rank 3 graph is strongly regular. A strongly regular graph is, in particular, regular of degree k. One says that the strongly regular graph (X) has parameters (n, k, λ, µ). The parameters are thus nonnegative integers with n > k ≥ µ and k − 1 ≥ λ . (3.3) An elementary calculation shows that if (X) is strongly regular with parameters (n, k, λ, µ), then [X] is also strongly regular, its parameters being (n, k ′ , λ ′ , µ ′ ) for k ′ = n−k−1 , λ ′ = n−2k+µ−2 , and µ ′ = n−2k+λ . (3.4) It is usual to write the codegree as l = k ′ = n − k − 1 . (3.5) By counting paths of length two from a fixed vertex µl = k(k − 1 − λ) . (3.6) Let M be the adjacency matrix of the strongly regular graph (X) with parameters (n, k, λ, µ). Counting all directed paths of length 2 yields In particular, the restricted eigenvalues of M are the roots r and s of the monic quadratic polynomial x 2 + (µ − λ)x + (µ − k) . (3.7) As µ ≤ k the roots r and s are real. We take s ≤ r by convention. As tr M = 0 and k is a positive eigenvalue of A, s < 0. Also, −rs = k − µ ≥ 0, and so the real parameters s and r are restricted by s < 0 and 0 ≤ r ≤ k . (3.8) In particular, s < r.
These sets of parameters are highly redundant, being related by the various equations of this section. All parameters can be determined by various small subsets of the complete parameter list. In particular three parameters are enough when we have n ; one of k = l ′ or l = k ′ ; any one of λ, µ, λ ′ , µ ′ .
Of course, the more parameters that can be calculated directly, the easier the remaining calculations will be.
It is also of note that all parameters can be derived from the spectrum We have already seen that the values µ = 0 and µ = k are special-these are the imprimitive graphs. Indeed these parameters make the complementary statements that one of (X) or [X] is a nontrivial equivalence relation-a disjoint union of complete subgraphs (of fixed size m > 1)-while the other is a complete multipartite graph with all parts of size m. As an important special case, when G acts imprimitively with rank 3 on (X) and [X], these form a complementary pair of imprimitive strongly regular graphs.

3-transposition diagrams and eigenvalues
The normal set D of the group G is a set of 3-transpositions in G if it consists of elements of order 2 with the property d, e ∈ D =⇒ |de| ∈ {1, 2, 3} .
The study of such sets D and groups G was initiated by Bernd Fischer [Fi71]. Fischer's paper and the later paper [CH95] of Cuypers and Hall are our basic references on this topic.

6
If E is a subset of D in G then the diagram of E, denoted (E), is the graph with vertex set E and having an edge between the two vertices d, e precisely when |de| = 3. The commuting graph of E, or codiagram of E, is the graph complement [E] of the diagram of E.
There are two cases of primary interest. The first has E some small generating set of G; for instance, the 3-transposition group Sym(n + 1) is the Weyl group W(A n ) with diagram the n-vertex path A n . In the second case E is equal to the full class D, and we then abuse terminology by saying that the diagram of D is also the diagram of G.
The first two parts of the theorem are Fischer's basic Inheritance Properties [Fi71,(1.2)]. The second of these allows us to focus on the case G = D for the conjugacy class D of 3-transpositions. In this situation we say that (G, D) is a 3-transposition group.
The third part of the theorem is embedded in Fischer's [Fi71, Lemma (2.1.1)] and is also in [CH95,Lemma 3.16].
We say that the two 3-transposition groups (G 1 , D 1 ) and (G 2 , D 2 ) have the same central type (usually abbreviated to type) provided G 1 /Z(G 1 ) and G 2 /Z(G 2 ) are isomorphic as 3-transposition groups. Theorem 4.1(c) tells us that the 3-transposition properties of groups sharing a central type are essentially the same. In particular the two 3-transposition groups have the same type if and only if they have isomorphic diagrams (D 1 ) and (D 2 ).
A consequence of the work by Fischer [Fi71] and later Cuypers and Hall [CH95] is the classification up to isomorphism of all diagrams for all finite 3transposition groups. 1 In Section 6 we shall give the eigenvalues and spectrum of (the adjacency matrix of) each such diagram. If ((. . . , r i , . . . )) is the spectrum of AMat((D)), we also say that ((. . . , r i , . . . )) is the spectrum of (G, D) and G.
As D is a conjugacy class of the 3-transposition group (G, D), its diagram (D) is connected. Thus the degree k of the diagram is an eigenvalue of multiplicity one (associated with the eigenvector 1) and will be listed first in the spectrum. The remaining eigenvalues are restricted.
For the 3-transposition group (G, D) we write p •h , with p ∈ {2, 3}, for a normal p-subgroup N with |D ∩ dN | = p h for all d ∈ D. We call this the shape of N . For fixed H and p •h , there may be 3-transposition groups G of distinct central type with N of type p •h and G/N = H, so that they have the same diagram.
The first part of the corollary appears in Matsuo's original papers on vertex operator algebras [Ma03,Lemma 4.1.3] and [Ma05,§5].
We can continue in this fashion, so that Note that in (a) one of the r i may be zero, in which case the expected tail multiplicity (2 h −1)n H should be combined with the multiplicity m i of 2 h r i = 0; this explains the exponent ⋆, which indicates a multiplicity that is whatever is needed to exhaust all eigenvalues. Similarly, in (b) one of r i may be −1 and then the expected tail multiplicity (3 h − 1)n H is added to the multiplicity m i of 3 h (r i + 1) − 1 = −1. The notation is that of [CH95] and will be discussed in the next section. No example appears twice in the theorem. Apparent omissions within it, in the first table of the next section, and throughout the paper are explained by the following coincidences.
Fischer's theorem was extended in [CH95]. A consequence of the main theorem of that paper is: Theorem 5.3. Let (G, D) be a finite 3-transposition group. Then, for integral m and h, the group G has one of the central types below. Furthermore, for each G the generating class D is uniquely determined up to an automorphism of G.
The notation Ik and PRk of the two theorems and the tables of the next section comes from [CH95], where the first suggests that the groups act Irreducibly on their natural modules, while the second says that more general examples arise from Parabolic subgroups of the irreducible examples-specifically their subgroups generated by Reflections or transvections, as appropriate.
In the theorem (and elsewhere) A :B indicates a split group extension with normal subgroup A while A . B is a nonsplit group extension with normal subgroup A and quotient B. The related notation AB indicates that A is normal with quotient B, but the extension may or may not be split. Extensions are left-adjusted, so in A :B :C, the normal subgroup A :B is split by C while A :B has A normal and split by B.
Neither the actual structure of the normal p-subgroup nor the splitting of the extension affect the shape of the normal subgroup and so the diagram. This allows us in the theorem to bundle the exotic cases PR13-19 from [CH95] under the corresponding generic cases PR5-6 (where both split and nonsplit group extensions may occur).
In Theorem 5.3 we have rewritten shapes 2 •2h as (4) •h when all the nontrivial composition factors in the normal subgroup those factors are naturally F 4 -modules for the quotient.

Case analysis of spectra
In Theorem 5.3, each choice of parameters in each part yields a unique diagram which admits a 3-transposition group (and perhaps many). In this section we calculate the size (number of vertices) and spectrum of the diagram in each case. These are collected in two tables-one for the Irreducible examples of Theorem 5.1 and a second for the Parabolic Reflection examples of Theorem 5.3. The second of these essentially comes from combining the first with Corollary 4.4.
As Fischer noted [Fi71,Theorem 3.3.5], in each case of Theorem 5.1 (except for the triality groups PΩ + 8 (2) : Sym (3) and PΩ + 8 (3) : Sym (3)), the permutation representation of G acting on D by conjugation is primitive of rank 3. Therefore the corresponding spectrum obeys all the conditions discussed in Section 3.
The redundancy of the parameter sets is of aid here. We have n = |D|. As the codiagram is the commuting graph of D, we also have We have seen in Theorems 5.1 and 5.3 that in the pair (G, D) the group G determines the generating class D uniquely up to an automorphism. Therefore we may abuse notation by writing (G) for the diagram in place of (D).
Most of the results given here could be extracted from the literature-for instance [Hu75] and [AEB]-although the notation varies enough that translation into the form we desire can be difficult. We have recalculated everything (to our own satisfaction) but only outline the paths taken.
The first table gives the extended parameters (n, k, λ, µ ; {[r] f , [s] g }) of the rank 3 (strongly regular) codiagrams [G] and diagrams (G). Note the set notation for the eigenvalues and their multiplicities. This is because in some cases the roles of r (positive eigenvalue) and s (negative eigenvalue) may switch depending on the value of m. In these cases we use d and e for multiplicities in order to avoid misleading the reader.
The second table gives the size n and spectrum ((k; . . . , [r i ] mi , . . . )) of all diagrams (G). The eigenvalue in bold is the minimum eigenvalue. This will be of relevance in Section 7.
In Theorem 5.3 we have restricted parameters in order to minimize repetition of examples. In the second table we reverse that decision, enlarging the parameter sets to a natural level of generality. In particular, unless otherwise stated, h can be any nonnegative integer.

Moufang case
This is the situation in which the diagram (D) is a complete graph. That is, there are no D-subgroups of G isomorphic to Sym(4). The terminology comes from a connection with commutative Moufang loops of exponent 3; see [CH95].
For h ≥ 0, let N h be an elementary abelian 3-group of order 3 h . Further let d be an element of order 2 that acts on N h as inversion. Then for (2) (2)) for h ≥ 0 has size n = 3 h and spectrum The fundamental 3-transposition groups Z 2 and Sym (3) occur here as h = 0 and h = 1. Proof. This is well known, but it is also easy to calculate the basic parameters of [Sym(m)] using 3-transposition properties: ((1, 2)) has type Sym(m − 2).

Polar space cases
For us a finite polar space graph [X] has as vertex set X the isotropic 2 1-spaces for a nondegenerate reflexive sesquilinear form f i on a finite space V i = F i q with edges given by perpendicularity. In our context, the form f i is either symplectic over F 2 or hermitian over F 4 . By Witt's Theorem, the corresponding isometry group acts with rank 3 (or less). There are exactly two types of 2-spaces spanned by isotropic vectors-totally isotropic 2-spaces with q +1 pairwise perpendicular isotropic 1-subspaces and hyperbolic 2-spaces with s pairwise nonperpendicular isotropic 1-subspaces. The hyperbolic 2-spaces are precisely the nondegenerate 2-spaces containing an isotropic 1-space. A vertex is either adjacent to all those of a given totally isotropic 2-space or exactly one.
Let s i = 1 + k i + l i be the number of isotropic 1-spaces in V i . (So s = s 2 .) The decomposition V i = V 2 ⊥ V i−2 can be used to calculate the parameters. This yields recursions for the degree of [X] Here we initialize with s 1 = 0 (as nondegenerate 1-spaces contain no isotropic vectors), but s 2 will depend upon the type of form under consideration. A further consequence of the decomposition is Therefore we have the three parameters s i , k ′ i , and µ ′ i , from which it is (at least in principal) easy to calculate all parameters of [X] and (X) using the identities of Section 3. The additional identity can be seen within x, y ⊥ , where x and y are distinct perpendicular isotropic 1-spaces. This is because x, y ⊥ / x, y ∼ = V i−4 .

Symplectic over F 2
The nondegenerate form f = f 2m above is symplectic on V 2m if it is bilinear with all 1-spaces isotropic: f (x, x) = 0 for all x ∈ V 2m . Its polar graph is denoted [Sp 2m (q)].
In the special case of symplectic polar spaces over F 2 the corresponding transvection isometries D form a class of 3-transpositions in the full isometry group G = Sp 2m (2) with the codiagram [D] = [X] = [Sp 2m (2)]. In this case s i = n i , i = 2m, q = 2, and n 2 = 1 + 2 = 3.

Unitary over F 4
For finite unitary polar graphs we must have q = t 2 for some prime power t. The for all x, y ∈ V m and a, b ∈ F q . Its polar graph is denoted [SU 2m (t)].
In the special case of unitary polar spaces over  (2)) has extended parameters and spectrum Proof.
Proposition 6.13. PR12: the diagram (3 •2h :4 •1 : SU 3 (2) ′ ), for h ≥ 0, has size 36(3 2h ) and spectrum for all x, y ∈ V 2m . The vectors x with q(x) = 0 are singular and those with q(x) = 1 are nonsingular. Each of the two types of symplectic 2-spaces resolves into two types of orthogonal 2-spaces. A totally isotropic 2-space is either totally singular (q is identically 0) or is defective-it has exactly two nonsingular vectors. A symplectic hyperbolic 2-space is either orthogonal hyperbolic-a unique nonsingular vector-or is asingular-its only singular vector is 0. Thus the isometry type of a 2-space is uniquely determined by the number of nonsingular vectors it contains-respectively 0, 2, 1, 3. Up to isometry, the form q has one of two types denoted by the Witt sign ǫ, equal to + = +1 or − = −1 depending upon whether maximal totally singular spaces have dimension m or m − 1. The corresponding diagram (O ǫ 2m (2)) has as vertices the nonsingular 1-spaces x ∈ V ǫ 2m with two adjacent when not perpendicular. That is, (O ǫ 2m (2)) is the subgraph of (Sp 2m (2)) induced on the set of 1-spaces that are nonsingular for q, and correspondingly for [O ǫ 2m (2)]. The symplectic transvections centered at nonsingular vectors form a generating conjugacy class 3 D of 3-transpositions in the corresponding orthogonal group O ǫ 2m (2).
(ii) k ′ = 2 2m−2 − 1: consider [O ǫ 2m (2)] as an induced subgraph of [Sp 2m (2)]. For the nonsingular vector x, a 2-space containing x and in x ⊥ must be defective-of its two 1-spaces not containing x, one is singular and one is nonsingular. Therefore (2)] as an induced subgraph of [Sp 2m (2)]. In calculating λ ′ for the symplectic case, and more generally for the polar cases over F q as in equation 6.5, we found λ ′ [Sp 2m (2)] = (q − 1) + q 2 n 2m−4 = 1 + 4(2 2m−4 − 1) = 2 2m−2 − 3 , counting the q − 1 remaining isotropic 1-spaces of the totally isotropic 2space x, y plus the q 2 additional 1-spaces of each 3-space in x, y ⊥ on x, y , these enumerated by the 1-spaces of x, y ⊥ / x, y of dimension 4 less. Here we must restrict ourselves to nonsingular vectors, so the isotropic spaces through x and y are defective. The only singular vectors in x, y are x and y, and each 3-space in x, y ⊥ on x, y has exactly two additional nonsingular vectors. Therefore the count becomes (2)) for h ≥ 0, ǫ = ±, and m ≥ 1 has size n = 2 h (2 2m−1 − ǫ2 m−1 ) and spectrum

Nonsingular orthogonal cases over F 3
Let V = V m = F m 3 admit the nondegenerate symmetric bilinear (that is, orthogonal) form f . The diagonal of f yields the quadratic form q : V −→ F 3 given by .
In this context, the isotropic vectors (those x with q(x) = f (x, x) = 0) are called singular. As described at the beginning of Section 6.3, the singular (= isotropic) 1-spaces form the vertex set of a polar space graph which is strongly regular and indeed rank 3 (by Witt's Theorem). There are two types of nonsingular 1-spaces x ; those with q(x) = 1 are called +-spaces and x is a +-vector; those with q(x) = −1 are −-spaces and x is a −-vector.
There are two parameters of interest for the nondegenerate space V -its discriminant and its Witt index. The discriminant δ is the determinant of any Gram matrix for the space. It is either +1 = + or −1 = − and determines V up to isometry. Concretely, V has discriminant +1 if and only if it possesses an orthonormal basis.
The Witt index (introduced in the previous section for F 2 -spaces) is the maximum dimension of a totally singular subspace (q identically 0). In even dimension m = 2a the Witt index is either a or a − 1, and (again as before) we attach the Witt sign ǫ equal to + = +1 or − = −1 in these respective cases. In odd dimension m = 2a + 1, the Witt index is always a. In even dimension m always δǫ = −1 ( m+1 2 ) , and we use this identity to define the sign ǫ for odd dimension m as well. The space V may then be denoted δ V ǫ m , which is sometimes abbreviated to δ V m or V ǫ m since, once m has be fixed, the parameters δ and ǫ determine each other.
For odd m an equivalent geometric definition is that V m has sign ǫ when it is isometric to x ⊥ for a +-vector x in the even dimensional V ǫ m+1 . 4 As before, in the polar space of δ V ǫ m there are exactly two types of 2-spaces spanned by singular (= isotropic) vectors: the totally singular 2-spaces with 4 Our convention for ǫ is that of [CH92,CH95]. See [CH92] for a discussion and a comparison with other conventions from the literature. Our choice differs from that of Brouwer [AEB] where δǫ = −1 m 2 . With Brouwer's convention, for odd m and x a +-vector within Vm, the even dimensional x ⊥ is isometric to V ǫ m−1 . q + 1 = 4 pairwise perpendicular singular 1-subspaces and the hyperbolic 2spaces with s 2 = 2 nonperpendicular singular 1-subspaces plus a +-space and a perpendicular −-space. The hyperbolic 2-spaces have type − V + 2 . The 2-spaces spanned by nonsingular vectors have three types. The only nondegenerate example is the asingular space + V − 2 , which is spanned by a pair of perpendicular +-spaces and a pair of perpendicular −-spaces. The two degenerate examples are the +-tangent spaces-consisting of a singular radical of dimension 1 and three +-spaces-and the similar −-tangent spaces.
By Witt's Theorem again, the full isometry group of δ V ǫ m has rank 3 on the +-spaces. 5 The reflections with centers of +-type form a normal set of 3-transpositions, commuting pairs of reflections corresponding to asingular 2spaces + V − 2 and noncommuting pairs to +-tangent spaces and their three pairwise nonperpendicular +-spaces. We consider the 3-transposition groups (G, D): is that subgroup of the full isometry group generated by the reflection class D having centers of +-type.
and the diagram ( + Ω ǫ m (3)) has extended parameters Proof. Some of the calculations work better in terms of δ while for others ǫ may be preferred. As ǫ is the canonical parameter in even dimension, we state the final results in terms of it, remembering that always Some rules-of-thumb for a fixed δ: if m is even then dropping to m − 1 does not change ǫ, while if m is odd then dropping to m − 1 changes ǫ to −ǫ; thus any drop by 2 changes ǫ to −ǫ; is spanned by x and the unique singular 1-space of the tangent, which belongs to x ⊥ . The remaining two +-spaces of the tangent are adjacent to x in the diagram.

Initialization of the recursion is provided by
The identity δ (λ ′ ) ǫ m = δ n m−2 follows directly from δ V m = + V 2 ⊥ δ V m−2 . (iv) The parameters we have found so far are enough to calculate all remaining ones using the identities of Section 3. Some are also geometrically evident. Consider the decomposition Let x be a +-vector spanning + V 1 . If z is a nonzero singular vector in the hyperbolic − V + 2 then the 2-space x, z is a +-tangent, and within it y = x + z spans a +-space not perpendicular to x. This leads to Proposition 6.19.  Proof. See [Fi71] or [Hu75] for the basic parameters. The extended parameters can then be calculated as in Section 3 and are also given in [AEB]. While we do not repeat these calculations, the basic parameters for the codiagram (= commuting graph) appear naturally within Fischer's 3-transposition theory. Fischer [Fi71] attacked the classification by induction, noting that the 3-transposition group (G, D) is essentially determined by the codiagrams of two if its "local" 3-transposition subgroups:

Sporadic cases
for d, c ∈ D with |dc| = 3. Fischer used this local data to reconstruct the global group (G, D). This is particularly relevant for us since The additional 3-transposition subgroup L G = C D (d, e) with |de| = 2 (naturally found as the subgroup K KG of K G ) yields The local parameters k ′ , µ ′ , λ ′ then allow us, using (3.6), to calculate the global parameter  Here the global parameters (in italics) can be calculated from the local parameters. Initialization is provided by the values for [SU 6 (2)] = [PSU 6 (2)] found in Theorem 6.10 6 ; the remaining M G were identified as part of Fischer's induction.  On M we define the symmetric bilinear form · | · given by, respectively, 1A : a | a = 1 ; 2B : a | b = 0 ; We say that the algebra M is compact if the associated form is positive definite. Matsuo algebras were introduced [MM99, Ma03,Ma05] because certain compact Matsuo algebras arise as the Griess algebras of compact vertex operator algebras of OZ type, as noted by Miyamoto [Mi96]. A classification of all such Griess algebras is desirable.
The crucial observation, due to Miyamoto, is that in the Griess algebra case, for each axis a ∈ A, the permutation τ a of A given by is an automorphism and indeed {τ ai | i ∈ I} is a normal set of 3-transpositions in the automorphism group of M . It is enough to consider the case in which D = {τ ai | i ∈ I} is a class of 3-transpositions in the group G = D ≤ Aut(M ).
This property of Griess and Matsuo algebras was seen in [HRS15] to characterize 7 the axial algebras of Jordan type η. In that case the symmetric form · | · is associative in that xy | z = x | yz for all x, y, z ∈ M . An important property of every associative form is that its radical R is an ideal of the algebra M . The Gram matrix of the form with respect to A is where H is the adjacency matrix of the diagram (D), so the compact axial algebra M of Jordan type η must have where ρ is the minimum eigenvalue of (D). In the case ρ = − 2 η , the algebra M is positive semidefinite and its axial quotient M/R is again compact (which is Theorem 7.2. There are nondecreasing, nonnegative integral valued functions S(t) and I(t) defined on 2 ≤ t ∈ Z + such that the diagram (D) of a finite conjugacy class D of 3-transpositions has minimum eigenvalue ρ min ≥ −t if and only if the corresponding 3-transposition group (G, D) belongs to one of the following central type classes: (a) infinitely many groups 3 u :2 of Moufang type (the case ρ min = −1);  For a given η ≤ 2 t let S η (t) and I η (t) be the corresponding functions counting those 3-transposition groups realized by some Griess algebra for the eigenvalue η. Clearly 0 ≤ S η (t) ≤ S(t) and 0 ≤ I η (t) ≤ I(t). Matsuo's Theorem 7.1 and the results of the next subsection give 3 = S The differences are caused by Matsuo's proven restriction to symplectic type. For ρ ≥ −8 three of the four symmetric families as in Theorem 7.2(b) have symplectic type (W 3 (Ã m−1 ) does not), and Matsuo showed that all lead to Griess algebras. Similarly exactly nine of the 14 individual groups counted by I(8) are of symplectic type, and they too produce Griess algebras. Almost by definition, Sym (3) is the only 3-transposition group that is simultaneously of Moufang and symplectic type, and this is reflected in the stark difference between Theorem 7.1(a) and Theorem 7.2(a).
Very little is known about the corresponding S 1 32 (64) and I 1 32 (64). Chen and Lam [CL14] have shown that SU 3 (2) ′ can be realized for η = 1 32 . In particular Matsuo's restriction to symplectic type will not be available in this case.