ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 7 (2014) 83-103
The L2 (11)-subalgebra of the Monster algebra
Sophie Decelle *
Department of Mathematics, Imperial College, 180 Queen's Gate, London, SW7 2AZ, UK Received 31 October 2011, accepted 25 August 2012, published online 15 March 2013
We study a subalgebra V of the Monster algebra, VM, generated by three Majorana axes ax, ay and az indexed by the 2A-involutions x, y and z of M, the Monster simple group. We use the notation V = ((ax,ay,az}}. We assume that xy is another 2A-involution and that each of xz, yz and xyz has order 5. Thus a subgroup G of M generated by {x, y, z} is a non-trivial quotient of the group G(5'5'5) = (x, y, z | x2, y2, (xy)2, z2, (xz)5, (yz)5, (xyz)5}. It is known that G(5,5,5) is isomorphic to the projective special linear group L2(11) which is simple, so that G is isomorphic to L2(11). It was proved by S. Norton that (up to conjugacy) G is the unique 2A-generated L2(11)-subgroup of M and that K = CM(G) is isomorphic to the Mathieu group Mi2 . For any pair {t, s} of 2A-involutions, the pair of Majorana axes {at, as} generates the dihedral subalgebra ((at, as}} of VM, whose structure has been described in [16]. In particular, the subalgebra ((at, as}} contains the Majorana axis atst by the conjugacy property of dihedral subalgebras. Hence from the structure of its dihedral subalgebras, V coincides with the subalgebra of VM generated by the set of Majorana axes {at 1t G T}, indexed by the 55 elements of the unique conjugacy class T of involutions of G = L2(11). We prove that V is 101-dimensional, linearly spanned by the set { at • as | s, t G T }, and with Cvm(K) = V © iM, where iM is the identity of VM. Lastly we present a recent result of A. Seress proving that V is equal to the algebra of the unique Majorana representation of L2 (11).
Keywords: Majorana representation, Monster group, Conway-Griess-Norton algebra. Math. Subj. Class.: 20C99, 20F05, 20C34, 20D05
1 Main result
We let (VM, • , ( , )) be the Monster algebra, a commutative non-associative algebra of dimension 196, 884 over R, as described in [2]. As an RM-module, VM = VM © 1M, where VM is the minimal faithful irreducible RM-module of dimension 196,883 and 1M
* This article was written under the supervision of Prof. Alexander A. Ivanov. E-mail address: sophie.decelle@imperial.ac.uk (Sophie Decelle)
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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is the trivial RM-module which is the R-span of the identity iM of the algebra VM. The automorphism group of (VM, • , ( , )) is M the Monster simple group ( [2], [7]). By 2A we denote the conjugacy class of involutions in M with the largest centraliser as in the Atlas [3]. For each 2A involution t of M, the centraliser CM(t) = 2.BM stabilises a 2-subspace W of VM which has two non-trivial idempotents at and iM - at. In [2], J. Conway constructed an M-invariant bijection ^ sending each 2A involution t to the nontrivial idempotent at of W with eigenvalue 1 and multiplicity 1 . We denote by at := ^(t) the image of t. In [8] A. A. Ivanov axiomatises some of the properties of the idempotents at into the definition of a Majorana axis.
A Majorana axis a of a real commutative non-associative algebra (V, • , (, )), where • associates with ( , ) in the sense that (u • v, w) = (u, v • w) for all u, v, w e V, is an idempotent of length 1, whose adjoint operator ada is semi-simple on V with spectrum {1,0, J?, 25}. The eigenspaces of ada are denoted by VM(a), with ^ an eigenvalue, and satisfy the following conditions. The 1-eigenvectors of ada are exactly the scalar multiples of a. There exists a linear transformation t(a) of V, called a Majorana involution, negating the 215 -eigenvectors, fixing the other eigenvectors and preserving both the algebra and inner products. Lastly there exists a linear transformation a (a ) of V+(a) = V(a) © v0( ) © v 1) negating the 22 -eigenvectors, fixing the 0- and 1-eigenvectors, and preserving both products on V++a). From [8], this definition is equivalent to the 'Fusion Rules'. For two eigenvectors u
VA(a) and v e VM(a) of a fixed Majorana axis a, the Fusion Rules specify in which part of the spectrum of ada the product u • v lies.
Sp	1	0	1 22	1 25
1	1	0	1 22	1 25
0	0	0	1 22	1 25
1 22	1 22	1 22	1, 0	1 25
1 25	1 25	1 25	1 25	1,0, 22
Table 1: Fusion rules
Definition 1.1. We denote by ((A)) the subalgebra of VM generated by a set A of Majorana axes.
The classification of subalgebras ((at, as)) of VM, where {as, at} is a pair of Majorana axes, was started in [2] and completed in [16]. We call them dihedral subalgebras as the corresponding pair of 2A-involutions {t, s} generates a dihedral subgroup of M. We say the dihedral subalgebra has type C if the product of involutions ts belongs to the conjugacy class C of M.
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Some subalgebras of VM generated by triples of Majorana axes are described by A. A. Ivanovetalin [11], [12], [13], [10], and [9].
In this paper, we investigate a subalgebra V = ((ax, ay, az}} of VM such that the dihedral subalgebra ((ax, ay}} has type 2 A and each of the dihedral subalgebras ((ax, az}},
((ay, az}}, and ((axy, az}} has type 5A. The vector axy is the Majorana axis ^(xy) (since a dihedral subalgebra ((as, at}} of type 2A contains the axis ast).
Keeping in mind the bijection ^ we might ask whether there exists a subgroup of M generated by a triple of 2A involutions {x, y, z} satisfying the relations:
x2 = y2 = z2 = (xy)2 = (xz)5 = (yz)5 = (xyz)5 = 1.
A group affording the presentation
(x, y, z | x2, y2, (xy)2, z2, (xz)5, (yz)5, (xyz)5}
defines the Coxeter group G(5'5,5) and from [4] it is isomorphic to the projective special linear group L2(11). From classical results on Ln(pk), [5], L2(11) is a simple group of order 660 = 22 • 3 • 5 • 11 and it has a single conjugacy class of involutions which we denote by T, and whose size is 55.
Proposition 1.2. There exists a monomorphism i: L2(11) ^ M such that i(T) C 2A and i is unique up to conjugacy in M.
Proof. In Table 5 of [17] S. Norton gives the list of simple subgroups of M having their elements of order 5 in the M-conjugacy class 5A. For i(T) it is a requirement since if a product of two 2A involutions has order 5 it belongs to the conjugacy class 5A of M [2]. By Norton's list there is only one conjugacy class of groups isomorphic to L2 (11) containing 5A elements and their involutions belong to class 2A.	□
Throughout the paper i denotes the monomorphism as in Proposition 1.2, G = L2(11) denotes the image of i, and T denotes the conjugacy class of involutions in G.
By the conjugacy property of dihedral subalgebras1, the axis atst is contained in ((at, as}}. Hence from the dihedral subalgebras of V, we can restate our aim to be the study of the subalgebra V of VM generated by the set of 55 Majorana axes {at 11 g T}. We determine the dimension of V and find a spanning set for V. In the next section we prove the following theorem.
Theorem. Let V be the subalgebra of VM generated by the set of 55 Majorana axes {at | t g T}, where T is the class of involutions of the unique 2A-generated L2(11)-subgroup G of M. Then
(1)	dim(V) = 101,
(2)	V is linearly spanned by the set {at • as 11, s g T}.
(3)	If K = Cm(G) then CyM(K) = V e im.
1Let t and s be two 2A involutions, then tst is an involution conjugate to s. Hence tst is a 2A involution with corresponding Majorana axis atst := ^(tst).
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In the last section we give some evidence towards the uniqueness of the map ^ : t ^ at, where t G T, within the class of Majorana representations of ¿2(11) satisfying conditions (2A) and (3A) (the terminology is explained in the last section)2. Lastly we state a recent result of A. Seress proving that V is equal to the algebra of the unique Majorana representation of L2 (11).
2 Some properties of L2 (11)
We present some of the standard properties of G = L2(11) used when calculating inner product values for V.
The group G is the automorphism group of the (11,5,2)-biplane, which we denote B (see [19]).
B is a 2-symmetric design with 11 points, {pi,... ,pn}, and 11 lines, {l1,..., In}, such that each line contains 5 points, each point lies on 5 lines, two lines intersect in exactly 2 points, and two points share exactly 2 lines. We call the incidence relation p^ g j a flag, which we denote ai ,j, and the relation pi G j an anti-flag, which we denote by witj.
From [14], the lines of B can be obtained by finding a difference set l1 of size 5, with elements from Z11, such that every integer modulo 11 appears exactly twice as a difference i — j mod 11 for i and j in 11. We have that l1 = {1,3,4, 5,9}, which is the set of non-zero perfect squares in Z11, and all other lines lk can be defined by lk = {1 + k, 3 + k, 4 + k, 5 + k, 9 + k}, where k G Z11 and addition is modulo 11.
The incidence matrix N of B is given below with the rows indexed by the points of B, the columns indexed by the lines, and each flag is represented by a '1' and each anti-flag by a '0'.
1	0	1	1	1	0	0	0	1	0	0
0	1	0	1	1	1	0	0	0	1	0
0	0	1	0	1	1	1	0	0	0	1
1	0	0	1	0	1	1	1	0	0	0
0	1	0	0	1	0	1	1	1	0	0
0	0	1	0	0	1	0	1	1	1	0
0	0	0	1	0	0	1	0	1	1	1
1	0	0	0	1	0	0	1	0	1	1
1	1	0	0	0	1	0	0	1	0	1
1	1	1	0	0	0	1	0	0	1	0
0	1	1	1	0	0	0	1	0	0	1
We can represent G as a permutation group on 11 letters, so that G c Sym(11), by letting G act on the indices of the points or lines such that the incidence structure of B is preserved.
The stabiliser G(aij) of a flag ai,j is isomorphic to A4, the stabiliser G(wfc i) of an anti-flag wk,l is isomorphic to D10, and the stabiliser of a line (or a point) is isomorphic to A5. We can associate to a flag ai,j a unique subgroup S(ai,j) = C2 x C2 and to an anti-flag wk,i a unique subgroup S(wk,l) = C5 such that NG(S(ai,j)) = G(ai,j) and NG(S(wk,i)) = G(wfc,;). It is easy to see that each involution t stabilises 3 flags and to deduce that CG(t) =
2When the first draft of this article was written, the author has learned that Akos Seress has written a GAP program, [6], capable of checking this uniqueness conjecture.
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D12. Similarly for (h} a subgroup of order 3 we can deduce NG((h}) = D12. There are only one class of involutions and one class of elements of order 3 in G, so we can let d be the G-invariant bijection between subgroups of order 2 and 3 sending each involution t to the unique subgroup of order 3 commuting with t. Furthermore, by [14], G contains one class of subgroups isomorphic to the Frobenius group of order 55, which we denote F55. These are the four conjugacy classes of maximal subgroups of G; two non-conjugate classes of subgroups isomorphic to A5 each of size 11 and each stabilising a point or a line, one class of subgroups isomorphic to D12, and one class of subgroups isomorphic to the Frobenius group F55.
3 The algebra V
We start this section by finding an upper bound for d«m(V) based on the work of S. Norton ([15], [16], and [17]). We then calculate the Gram matrix of a particular subset of V which provides a lower bound for d«m(V).
3.1 S. Norton's observations
The upper bound on d«m(V) stems from the following inclusion.
Lemma 3.1. V C CvM (Cm(G))
Proof. By the definition of a Majorana axis, at is fixed by CM(t) = 2.BM. Therefore
Cm(G) = CM((x, y, z}) = P| CM(t) fixes V = ((ax, ay, az}} by M-invariance of the
t=x,y,z
algebra VM.	□
We denote by K the group CM(G). The dimension of the fixed space of K in VM can be obtained by calculating the fusion of the character table of K in that of M (since the character of VM is known [3]). It is equal to the inner product of characters (xVm Ik , 1K }rK , where 1K is the trivial character of K, and xvM iK is the character of VM restricted to K. We thus need to determine the isomorphism type of K and the inclusions of the conjugacy classes of G and K into those of M.
We call an A5-subgroup H of M an A5 of type (2A, 3A, 5A) if the elements of order 2, 3 and 5 of H are in the M-conjugacy classes 2A, 3A and 5A respectively. Clearly all A5-subgroups of G are of type (2A, 3A, 5A).
Proposition 3.2. (i) For K as above, K = M12.
(ii)	All A5-subgroups H as above are conjugate in M and CM(H) = A12.
(iii)	The conjugacy classes of G fuse into those M as follows:
Class in G	1a	2a = T 3a	5a	5b	6a	11a	11b
Class in M	1A	2A 3A	5A	5A	6A	11A	11A
Proof. The result from part (i) can be read from the entry 31 of Table 3 of [15]. Part (ii) is proved in Lemma 4 of [15]. To prove (iii) we carry on from the proof of Proposition 1.1. From Table 5 of [17] we deduce the inclusion 3a c 3A. In the character table of M, given
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in [3], the information on p-powers3 of elements g G 6A gives g2 G 3A and g3 G 2A, and 6A is the unique conjugacy class of elements of order 6 with those p-powers, hence to avoid a contradiction we must have 6a c 6 A. Since M has a unique class 11A of elements of order 11 the classes 11a and 11b are subsets of 11A.	□
Proposition 3.3. For the algebra CVM (K) we have dim(CVM (K)) = 102.
Within the proof of Proposition 3.3 we determine the fusion of the conjugacy classes of K into those of M. We follow the Atlas's notation, [3], by writing the conjugacy of elements of order N in M: NA, NB,... (etc) in increasing order of the size of the class. Similarly for K we use the notation NAK, NBK, • • • (etc). The character tables used are those of the Atlas [3].
Proof. By part (i) of the previous proposition we have the inclusion of groups H c G, where H = A5 is of type (2A, 3A, 5A), which implies K c Cm(H) = A12. In A12, the elements with cycle decompositions 2218, 2414, and 26 have 8, 4, and no fixed points respectively in the natural action of A12 on 12 points, and so by Lemma 6 in [15] they are mapped to the M-conjugacy classes 2A, 2B and 2A respectively. There is a doubly transitive action of M12 on 12 points with character x1 + x11a where x1 is the trivial character of M12 and x11a is the first irreducible character of degree 11 (as in the Atlas, [3]). This character takes the value 0 for the elements in the class 2AMl2, and the value 4 for the elements in the class 2BMl2, hence 2AMl2 c 2A and 2BMl2 c 2B.
The structure class constants4 for any pair of 2A involutions in M give the number of elements in each conjugacy class of M expressible as a product of two 2A involutions. The constants can be calculated directly from the character table. For M the product of two 2A involutions lies in either of the M classes: 1A, 2A, 2B, 3A, 3C, 4A, 4B, 5A or 6A (see [2] or [15]). Similarly for K = M12 we obtain that the product of two 2AK involutions lies in either of the K classes 1Ak , 2Ak , 2Bk , 3Bk , 4Ak , 4Bk , 5Ak or 6Ak .To avoid a contradiction on the monomorphism i we have 5AK c 5A, and 6AK c 6A and 3AK is a subset of either 3A or 3C. The class 6AK has p-powers 3BK, 2AK in K, and the class 6A has p-powers 3A, 2A in M. Hence 3BK c 3A. From lemma 6 of [15] no elements of order 3 in CM(H) = A12 belongs to class 3C of M, hence 3AK belongs to either 3A or 3B. If 3Ak c 3A then 6BK c 6C and if 3BK c 3B then 6BK is in either 6B or 6E according to the relevant p-powers in K and M. We determine the fusion in M of 3AK and 6Bk at the end of the proof. The classes 4AK, 4BK contain products of 2A involutions and their squares lie in class 2BK c 2B hence 4AK, 4BK c 4A as 4A is the unique class of elements of order 4 squaring to 2B. The classes 8AK, 8BK have their squares in classes 4AK, 4BK respectively, and in M the unique conjugacy class of elements of order 8 squaring to 4A is 8B. Hence 8AK, 8BK c 8A. The class 10AK in M12 has p-powers 5Ak c 5A and 2AK c 2A and in M the class 10A is the unique class of elements of
3For a finite group L, the p-power line in the character table of L records for each conjugacy class C of L, and for each prime p dividing the order of the elements of C, to which conjugacy class the pth-power of the elements of C belongs to.
4For a finite group L, the structure class constants give the number of solutions «1,2,3 to equations in the
group of the type x1 .x2 = x3, where each xi belongs to a conjugacy class Ci of L. From the table of complex
characters of L:
= _|L|__x(xi)x(x2)x(x3)
s1,2,3 = |cL(xi)HcL(x2)|	x(l)
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order 10 with such p-powers so that 10AK c 10A. There is a unique class of elements of order 11 in M so 11AK, 11BK c 11A. If 3AK c 3A then 6BK c 6C and the completed fusion of conjugacy classes of K in those of M gives a value of (xVm Ik, 1 K}rk which is not integral, a contradiction. Hence 3AK c 3B and 6BK is in 6B or 6E. To determine which, we look at the fusion of the conjugacy classes of A := CM(H) = A12 in M. Apart from the conjugacy classes 6GA, 9Aa, 9Ba and 9CA, the fusion of the classes of A in M is straightforward using the information on p-powers and the fusion of the classes of K in M already obtained. From a calculation of S. Shpectorov in [12] we know that (xvM 4-a, 1 a}ra = 26. This can only happen if 6GA c 6B and 9Aa, 9Ba, 9Ca c 9A. In particular elements of order 6 in Ai2 cannot be subsets of 6E, hence neither can the elements of order 6 in K. Hence 6BK c 6B. We have obtained the fusion of K in M
Class in K	1Ak	2Ak	2Bk	3Ak	3Bk	4Ak	4Bk
p-powers	A	A	A	A	A	B	B
Class in M	1A	2A	2B	3B	3A	4A	4A
5Ak	6Ak	6Bk	8Ak	8Bk	10Ak	11 Ak
A	BA	AB	A	B	AA	A
5A	6B	6A	8B	8B	10A	11A
and we can now compute the inner product of real characters of K
(XVm ^K, 1K}RK = 102.
□
The following useful observation was made by S. Norton (in a more general context). Lemma 3.4. The identity iM of VM cannot be contained in V.
Proof. The groups G and K centralise each other in M and G x K is a subgroup of M. The unique conjugacy class of involutions T of G is in class 2A of M, and we have proved that the classes 2AK and 2BK are in 2A and 2B respectively.
Claim : there exist elements s G T and t G 2AK such that the element ts of G x K is in class 2B of M.
From Proposition 3.2, part (iii), the element s can be taken from a A5-subgroup X of G of type (2A, 3A, 5A). From the proof of Proposition 3.2, the elements of 2AK act fixed-point freely on 12 points, and the centraliser in M of an A5-subgroup of type (2A, 3A, 5A) is isomorphic to Ai2. By Table 4, line 7 of [17], there exists a subgroup Y C Ai2, Y = A5, that acts transitively on 12 points. Let t G Y. Then, by Table 3, line 8 of [17], the involutions in the diagonal subgroups of X x Y are in 2B. In particular ts G 2B and the claim is proved.
Since ts G 2B, the axes at and as generate a dihedral algebra of type 2B and at • as =0 (see [2] or [11]). For all z G T there is an element g G G such that z = sg, so by invariance of the algebra product (at • as )g = 0 = at • az since G normalises K. Now, V is generated by the 55 Majorana axes az for z G T and the 0-eigenspace of at is closed under the algebra product, so if the identiy iM were in V we would get the contradiction at • iM = 0. □
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The identity of a commutative algebra being unique and therefore stable under the automorphism group we have iM € CVM (K). And since iM is not in V we obtain the main result of this subsection.
Proposition 3.5. For the algebra V we have dim(V) < 101.	□
3.2 Inner product values for V
In this subsection we calculate all inner products on a well-chosen subset of V and compute the rank of the corresponding Gram matrix to bound below the dimension of V. We do so using the information on some subalgebras of V which have already been classified.
3.2.1	Dihedral subalgebras of V
The algebra V contains the dihedral subalgebras of types 2A, 3A, 5 A, and 6 A (obtained by calculating the relevant structure class constants in the character table of G). From [16], for each type of dihedral algebra, we know the dimension of the algebra, and a basis for which all algebra and inner products are known. We follow the exposition given in [8] which is now accepted as standard in the Majorana Theory and where a different scaling to [16] is used. Table 2 is taken from [8], which notation we explain below.
Each dihedral subalgebra corresponds to a dihedral subgroup D of M generated by two 2 A involutions t and s, whose product we denote by p := ts. We denote by a0, a1 and a the Majorana axes at, as and atpi in Table 2.
In the subalgebra of type 2A, we have ap = ^(p) which is also a Majorana axis, and in the types 3A and 5A the vectors up and wp are introduced to close the algebra product. They correspond to elements of order 3 or 5 in D respectively. From [2], the 1-dimensional subspace linearly spanned by the vector up or wp is invariant under the normaliser NM ((p)) which is isomorphic to 3.F24 or (D10 x F5).2 respectively. Also, in the type 3A the vector itself is stable under NM ((p)), so that up = up-1, and in the type 5 A the vector is stabilised up to negation wp = — wp2 = — wp3 = wp4.
Any element of order 3 or 5 in G can be expressed as a product of two involutions, and any two involutions correspond to a dihedral subalgebra of V. Hence to study the algebra V we can consider the span of the vectors corresponding to the cyclic subgroups of order 2, 3 and 5 of G.
We let G(i) be a set of non-trivial representatives of each cyclic subgroup of order i for i = 2, 3, 5, of size 55, 55 and 66 respectively, where for i = 5 the representatives are taken from the same conjugacy class of G. We use the notation A := {at | t € G(2)}, U := {uh | h € G(3)}, and W := {wf | f € G(5)}, and we let S := A U U U W.
3.2.2	A5 subalgebras of V
The algebra V also contains 22 A5-subalgebras of type (2A, 3A, 5A) as G contains two conjugacy classes of A5-subgroups of type (2A, 3A, 5A), of size 11 each. The structure of a subalgebra VH generated by the Majorana axes indexed by the involutions of an A5-subgroup H of type (2A, 3A, 5A) follows from [12]. We reformulate it so as to present VH as a subalgebra of VM generated by a triple of Majorana axes.
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Type
Basis
Products and angles
2A
3A
5A
6A
ao, ai, ap
a_ i, ao, ai,
Up
a—2, a_ 1, ao, ai,a,2,Wp
a—2, a—i, ao,
ai, a2, a3 ap3, up2
■»p2
ao • ai — 2i3(ao + ai — ap), ao • ap — (ao + ap — ai) (ao, ai) — (ao, ap) — (ai,ap) —
ao • ai — 25 (2ao + 2ai + a_i) — |t" ao • Up — 32(2ao — ai — a_i) + -25Up
p
(ao, ai) — -2|, (ao,Up) — 22, (up,Up) —
ao • ai — 27(3ao + 3ai — 0,2 — a_i — a_2) + Wp ao • a2 — 2r (3ao + 3a2 — ai — a_i — a_2) — Wp ao • Wp — 2^2 (ai + a_i — a2 — a_2) + ¿rWp
Wp • Wp — "2T9-(a_2 + a_i + ao + ai + a2) (ao, ai) — 27, (ao, Wp) — 0, (Wp, Wp) — |tt
• ai — 26 (ao + ai — a_2 — a_i — a2 — + ap3 ) + ^ir U ao • a2 — 25(2ao + 2a2 + a_2) — ^TrUp2 ao • Up2 — 32 (2ao — a2 — a_2) + 2"5Up2
ao • a3 — 213 (ao + a3 — ap3 ), ap3 • Up2 — 0, (ap3, Up2 ) — 0 (ao,ai) — 2"8, (ao,a2) — p, (ao,a3) — 23
p
U
p
p
Table 2: Dihedral subalgebras
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Proposition 3.6. Let VH = ((ax,ay,az}} be a subalgebra of VM where the dihedral subalgebra ((ax,ay}} has type 2A and where the dihedral subalgebras ((ax,az}}, ((ay,az}}, and ((axy, az}} have types 5A, 5A and 3A respectively. Then VH has dimension 26 and it is linearly spanned by the products of all pairs ofMajorana axes indexed by the involutions of H.	□
For explicit formulas for the algebra product in VH or a list of all inner product values, we refer the reader to [12]. In the rest of the paper we will simply refer to an A5-subgroup H to mean an A5-subgroup of type (2A, 3A, 5A).
For an A5-subgroup H, we denote by H(2), H(3) and H(5) the sets of non-trivial conjugate representatives of cyclic subgroups of order 2, 3 and 5 and in the corresponding algebra VH we denote by AH, UH and WH the sets of vectors {at 11 e H(2)}, {uh | h e H(3)}, and {wf | f e H(5)}. Let wH be the sum of all vectors in WH. By [12], the set SH := Ah U Uh U Wh is a spanning set of size 31 for VH, and VH is 26-dimensional with a basis Ah U Uh U {wH}. The five independent linear relations on SH, which can be found in [12] or [16], are called the Norton Relations.
Proposition 3.7. The Norton Relations
In the algebra VH corresponding to an A5-subgroup H of type (2A, 3A, 5A), all vectors wf e WH satisfy:
(
wf = 6WH + 27
at - Z) at I +
\i£H<2)(f )	teH(3 2)(f )
/
32.5
\
Uh - Z Uh
heH(3),	heH(3),
\o([h,f])=3	o([h,f])=5
where
h52)(/): = {t e H(2) | o(tf ) = ^, H32)(f): = {t e H(2) | o(tf) = 3}.
□
We denote by Hi = {Hi,..., Hn} and H2 = [H[,..., H[ J the two classes of A5-subgroups in G. One class corresponds to the rows of N and the other to the columns, so the intersection between A5's taken from different classes can be read directly from the entries of N.
For a given A5-subgroup Hi in G let W1 be the sum of all vectors in WHi. For a vector Wf e WHi we rewrite the Norton relation for Wf as
Wf = 6 Wi + XAi(f ) + VUi(f )	(3.1)
where the meaning of A, v, Ai(f ) and Ui( f ) is clear from Proposition 3.7.
Corollary 3.8. Let w be the sum of all vectors Wf in W Ç V. Then S' := A U U U {w} is a spanning set of size 111 for S.
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Proof. Consider an A5-subgroup H G Hi. From N, any subgroup H G H2 intersect Hi in a Di0 or an A4. If Hi n H = Di0 there exists a representative f of H(5) in Hi nH for which the Norton relations give
r 6 Wi + AAi(f) + t*Ui(f) Wf = {
{ iWi + A Aj/ (f)+ MUj/ (f)
and so Wj' is in Sp(A U U U {Wi}), the R-linear span of A U U U {Wi}. If Hi n Hi' = A4 then the situation can be visualized as the following submatrix of N, where each anti-flag has been replaced by the unique element of G(5) stabilising it, and the rows and columns are indexed with the copy of A5 stabilising the corresponding line or point of B.
Hi Hj
Hj i 1 Wg\
Hj ^Wf wk J
From the Norton relations for g G Hi n Hi/, k G Hi n Hj/ and f G Hi n Hj/, we also get Wi G Sp(A U U U{W. From N, for any subgroup Hi G Hi there are 3 elements of H2 intersecting both Hi and Hi in a Di0, with say H;/ being one of them:
Hi	Hi
H;/ (Wf	W; )
so the Norton relations for f and l give Wi G	Sp(A U U U {W. Hence there exists
v G Sp(A U U) such that
22 Wi = EheH Wi + Effi/ eH Wi/ + v,
and from N every element of G(5) is contained in exactly one element of Hi and one of H2 so that
E^em Wi + Eh4/eH2 Wi/ =2 where w is the sum of all vectors Wf in W, and hence
Wi = yjw + v' for some v' G Sp(A U U).
□
4 Inner product values
Definition 4.1. For each pair (G(i), G(j)) with i, j G {2,3, 5} we call the inner product values on G« x inner products of type (i, j).
If we let E(i) be the equivalence class of elements of order i in G belonging to the same cyclic subgroup, then the orbits of G acting by conjugation on E(i) x form a subpartition of the distinct inner products values of type (i, j) (these orbits were calculated using [1]).
We will only explain the inner products values (uk, v;) for which the subgroup (k, l) is isomorphic to F55 or to the whole of G. They arise as the solutions of equations of
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ArsMath. Contemp. 7 (2014) 94-121
intersecting subalgebras inside V, or equivalently as particular configurations of subgroups inside G, which can be read from the incidence matrix N or found using a code written in [1].
4.1 Inner products of type (2, 2)
From the dihedral subalgebras of V we know all possible inner product values of any two Majorana axes in V, see [11].
Case	o(ts) {{at, as)) {t,s) (at, as)
1 2 3 4 5	1	1A 11 2	2A D4 73 3	3A De If 5	5A D10 737 6	6A D12 78
Table 3: Inner Products of type (2, 2), with t,s e G(2)
4.2 Inner products of type (2, 3)
The value for case 5) of the inner product of type (2, 3) was computed using the following lemma.
Lemma 4.2. For t G G(2) and h G G(3) such that (t, h) = L2 (11) we have
(at, uh) = 2X5•
Proof. We fix an element h G G(3) and we let t G G(2) such that (t, h) = G. Since Ng((h)) = D12 then (h) is contained in exactly two distinct dihedral groups of order 6. Let Sh be one of the two sets of 3 involutions, Sh := {s, sh, sh2}, such that (S, h) = D6, and up to permutation of the set Sh we have
(t, s) = D6, (t, sh) = D12, (t, sh2) = D10.
In the 3A-dihedral subalgebra ((as, uh)) we have the equality
uh — 33.
ash — (2as + 2ash — ash2 )
so taking the inner product with at gives
(at , uh) — -33-15 (at , as • ash,) - 775(at , 2as + 2ash - ash2)
a
s
5
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Case	o(ht) (t,h) (at, Uh)
1 2 3 4 5	2 D6 23 A4 5 A5 2.36 C6 0 11 ¿2(11) ^
Table 4: Inner Products of type (2,3), with t e G(2) and h e G(3)
Case
o(tf ) = o(tf-1) o([t,f ]) (t,f) (at,Wf)
1	2	5	D10	0
2	3	5	A5	72 214
3	5	3	A5	72 214
4	5	5	¿2(11)	1 214
5	6	6	¿2(11)	3 212
6	11	5	¿2(11)	19 214
Table 5: Inner Products of type (2,5), with t e G(2) and f e G(5)
and by associativity of the algebra product with the inner product
(at, as ■ ash) = (as, at ■ ash).
Since (t, sh) = D12, the element p = tsh has order 6 so the algebra product at ■ ash is contained in the dihedral algebra ((at, ash)) of type 6 A, and so
(as, at ■ ash) = 26(as, at + ash - atp2 - atp3 - atp4 - atp5 + a.p3) + -^n- (a.s,Up2),
where (s, p2) = A4, so the value of (as , up3) is known to be 9 from [11]. Since all the
required inner products are now known, one can compute (at, uh) = ^^.
' ' □
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ArsMath. Contemp. 7 (2014) 83-103
4.3	Inner products of type (2, 5)
The next lemma justifies the values found in cases 4), 5) and 6). We omit its proof which is similar to the proof of Lemma 4.2.
Lemma 4.3. Let t G G(2) and f G G(5) such that (t, f} = L2(11). Then exactly one of the following holds.
(i)	There exists s G G(2) commuting with t and inverting f, and
(at, wf ) = -^rr + 2kP - 2?q, where P = 7(at, asf) + (at, asf2) and q = (ats,asf).
(ii)	There exists s G G(2) inverting f and generating with t a dihedral group of order 6, and
/	\	Q2	1	Q3 r
(at,wf) = 234 + 2rP - 3rrq, where p = (at, 5asf + asf2 + asf3 + asf4) and q = (ust, asf).
(iii)	There exists s G inverting f and generating with t a dihedral group of order 12, and
(at, wf ) = - 2T5 + 26P + 27q - |rf r, where
P = (asf, a,p3 - 2at - atp2 - atp3 - atp4 - atp5), q = (at, asf2 + asf3 + asf4)
and r = (up2, asf),
for p := ts of order 6 in (t, s} = Dr2.
□
4.4	Inner products of type (3, 3)
In the next lemma part (i) addresses cases 4) and 6) and part (ii) addresses case 5). The lemma assumes all products of type (2,3) are known.
Lemma 4.4. Let h, k G G(3) with (h,k) = L2(11). Then exactly one of the following holds.
(i)	There exists an involution t inverting both h and k, and
(uh,uk) = (5 - 23.32p + 26q), where P =(uh,atk2 ) = (uk,ath2) and
q = (ath, atk2) + (ath, atk) + (ath2 ,atk) + (ath2, atk2).
(ii)	There exists an involution t inverting h and generating with k an alternating group A4, and
(uh,uk) = 35-5(223 -P - 2q - r), where P = (ath, 3utk - 4utk - 4utk2), q = (ath2 ,uk) and r = (ath,uk).
□
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Case
{o(hk),o(hk x)} (h,k) (uh,uk)
1	{1, 3}	C3	23 5
2	{2, 3}	A4	23.17 34.5
3	{5, 5}	a5	24 34.5
4	{5, 6}	¿2(11)	23 3.52
5	{5,11}	¿2(11)	23.7 33.52
6	{6, 6}	¿2(11)	25 34.5
Table 6: Inner Products of type (3, 3), with h,k G G(3) 4.5 Inner products of type (3, 5)
Part (i) of the next lemma addresses case 3), and part (ii) addresses cases 5) and 6).
Lemma 4.5. Let h G G(3) and f G G(5) such that (h, f} = L2(11). Then exactly one of the following holds.
(i)	There exists an involution t inverting f and h, and
(uh,wf ) = -335(p - 25q), where
p = 2"2 (atf + atf4 - atf2 - atf3, ath) + 25 (wf ,ath) and q = (2at + 2ath + ath, wf).
(ii)	There exists an involution t inverting f and generating with h a subgroup isomorphic to A5, and
(uh,wf) = 27p + 27q - 24r, where p = (asf + asf4, uh), q = (asf2 + asf3, uh) and r = (asf, ush + ush2).
□
The inner product value for case 4) of the inner product of type (3, 5) can be found using the Norton relations inside some A5-subalgebras of V, see equation (1). The proof of the following lemma uses similar arguments to the proof of Corollary 3.8. We use the notation
of (1).
Lemma 4.6. Let f G and h	If there exists an element g G G(5) such that
Ai := (f, g} and A2 := (h, g} are two non-conjugate A5-subgroups of G, then
(uh, wf) = 6 (uh ,W2) + 27 (uh ,la) + |r# (uh,lu), where la = Ai(g) + Ai(f) - A2(g) and lu = Ui(g) + Ui(f) - U2(g).
□
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ArsMath. Contemp. 7 (2014) 83-103
Case
>(hf) o(hf-1) (h,f) (uh,wf)
1	2	5	A5
2	3	5	A5
3	3	6	¿2(11)
4	5	5	¿2(11)
5	6	11	¿2(11)
6	11	11	¿2(11)
-5.7 29.32
5.7 29.32
-67 29.32.5
-1 28.32.5
7
26.32.5
-7 27.32.5
Table 7: Inner Products of type (3, 5), with h € G(3) and f € G(5)
4.6 Inner products of type (5, 5)
In the next lemma, part (i) justifies the values of the inner product of type (5, 5) for the cases 2), 3) and 4), and part (ii) justifies case 6). The proof is similar to that of Corollary 3 . 8 and the notation is the same as the one used in the previous lemma.
Lemma 4.7. Let f, g € G(5) not contained in a common A5-subgroup, with f and g belonging to the pairs {Hj, H} and {Hj, Hj/ } respectively, of distinct non-conjugate A5-subgroups of G. Then exactly one of the following holds.
(i) Hj n Hj/ = D10, or Hj n Hj/ = D10, with k an element of order 5 in Hj n Hj/, say, then
(wf ,wg ) = 62 (W , Wj ) + ^ , Wj ) + 35 Wj ) + ('a ,Aj (g))
32 5	32 5	34 52
+ 21F , Uj (g^ +2^ (/u ,Aj (g^ + ^ ,Uj (g)),
where /„ = Aj(k) — Aj/ (k) + Aj/ (f) and 1„ = Uj (k) — Uj/ (k) + Uj/ (f).
(iii) Hj n Hj/ = Hj n Hj/ = A4, and there exist two elements k = l € G(5) such that k belongs to Hj and Hm/ and l belongs to Hj and Hm/, so that
(wf ,wg ) = 62 (Wj , Wj ) + ^ ('a , W ) + II ('« , W ) + ('a , Aj (g))
32 5	32 5	34 52
+ 21F ('a, Uj (g^ + ,Aj (g^ + ,Uj (g^,
where la = Aj (l) — Am/ (l) — Aj (k) + Am/ (k) + Aj (f), and 1„ = Uj (l) — Um/ (l) — Uj(k) + Um/ (k) + Uj(f).
□
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Case
{o(/g),o(/g-1)} o([/, g]) </,g> (uh)
1	{1, 5}	1	C5	53.7 219
2	{3, 5}	5	A5	7.29 219
3	{5,11}	11	F55	-11 219
4	{3, 6}	5	¿2 (11)	3.151 221
5	{2, 6}	5	¿2 (11)	157 220
6	{5,11}	2	¿2 (11)	59 220
7	{5, 5}	3	¿2 (11)	-3.41 220
Table 8: Inner Products of type (5,5), with /, g G G(5)
Corollary 4.8. The inner product values between the vector w, and the vectors at G A, uh G U, wf G W and w itself are as follows
32
(i)	(at, w) = 2TT;
32
(ii)	(uh,w) = - 27-g ;
(iii)	(wf, w) = ^^^;
(iv)	(w,w) = ^^.
□
4.7 Dependence relations in the algebra
We let VS' be the R-vector space having the subset S' = A U U U{w} of V as a basis. We turn VS into a G-module by the natural action of G on S', and we let n be the natural projection
n : VS ^ V.
Using [1] we find the rank of the Gram matrix of the set S' and give a description of the kernel of n.
We recall the bijection d introduced at the beginning of section 2 between subgroups of order 2 and 3 in G:
d :G(2) ^ G(3) <t> ^ <h>
since Vt G G(2) 3! h G G(3) where [t, h] = 1. For a fixed involution t G G(2) its normaliser NG(t) = D12 has the following orbits on G(2) (the action is conjugation):
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ArsMath. Contemp. 7 (2014) 100-121
Oi, O3, O3, O6, O6, O3, 04, O12 and O212,
where the subscript indicates the size of the orbit. If we write NG(t) = (p) x (s) then
p3 = t, so 0i = {p3}, Og = {s, sp2, sp4} and 02 = {sp, sp3, sp5} wlog. Further we can describe the orbits as follows:
og U 01 ={s e G(2)|(s, t) = 22} og U01 ={s e G(2)|(s, t) = D12} 03 U 04 ={s e G(2)|(s, t) = De}
0g2 U 0I2 ={s e G(2)|(s, t) = D10}.
For (ii, t2) in Og x Og or Og x Og the subgroup (ti, t2) in G is isomorphic to either 22 or D10. For (t1, t2) in Og x Og or Og x Og the subgroup (t1, t2) is isomorphic to either De or D12.
Proposition 4.9. (i) The rank of the Gram matrix for the set S ' is 101.
(ii) The kernel of n is 10-dimensional and consists of 10 linearly independent relations, between the vectors of A U U, taken from a set of 55 G-invariant relations R(t) indexed by the involutions of G.
For a fixed involution t in G(2), R(t) defines the following NG(t)-invariant relation:
R(t) := ^ ar - ^ as + ^^ ( ^ Uh - ^ Ufc
reTi seTg	hed(Ti) ked(T2)
where T1 and T2 can taken to be O1 U O,1 and O| U O2 respectively (or vice versa).
□
From the rank of the Gram matrix of the set S' we obtain the following proposition.
Proposition 4.10. For the algebra V we have dim(V) > 101.	□
The above, together with Proposition 3.5, proves that dim(V) = 101. Hence the set S' spans V, so that {at • as | t, s G T} also spans V. From Lemma 3.4, the identity of VM is not in V. The space CVm (K) is 102-dimensional, containing iM and having V as a subspace. Hence CVm (K) decomposes as V © iM, and we have proved our main theorem.
5 A Majorana representation of L 2 (11)
The dihedral and A5-subalgebras of V can be characterised under the axioms of Majorana theory; they are equal to the algebra of the Majorana representations of the dihedral groups D4 of type 2A, De of type 3A, D10 of type 5A, and D12 of type 6A, and of the alternating group A5 of type (2A, 3A, 5A).
Majorana theory was introduced by A. A. Ivanov in [8] to axiomatise some of the properties of VM and its Majorana axes. We refer the reader to [8] and [11] for a full description.
Definition 5.1. A Majorana representation of a finite group G is a tuple
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R = (G, T, X, ( , ), •, ^,
where T is a union of conjugacy classes of involutions generating G, and X is a commutative non-associative R-algebra endowed with an inner product ( , ) associating with its algebra product • in the sense that (u • v, w) = (u, v • w) for all u, v, w G X and satisfying the Norton Inequality
(u • u, v • v) > (u • v, u • v), for all u, v G X.
The image of the homomorphism ^ : G ^ GL(X) is an automorphism of (X, • , (, )), and the map ^ is an injection sending each involution t of T to a Majorana axis at of X, as defined in the second paragraph of section 1 (the properties of the spectrum of ada and the Fusion Rules are assumed to hold), such that ^ and ^ commute in the sense that:
ag-itg = (at)v(g) for every g G G.
We require that the algebra X be generated by the set of Majorana axes ^(T) and that it must satisfy conditions (2A) and (3A) below.
Conditions (2A) and (3A) ensure that when constructing X in the above definition we get the right number of 3A vectors uh from the Majorana axes.
(2A) Let t0,ti G T and p := toil such that
(a)	if p G T and the vectors ato, atl generate a dihedral subalgebra of type 2A then
aP = ^(p^
(b)	if pl G T for p of order 4 or 6 and the vectors at0 and atl generate a subalgebra of type 4B or 6A, then ^(p®) coincides with the axis api ;
(3A) Let to,ti,t2,t3 G T with (to,ti) = (Î2,is) = De. We let pi := toti and p2 := Î2Î3 both of order 3.
If the following two conditions are satisfied:
(i)	pi = p2 or p-i, and
(ii)	the dihedral subalgebras generated by {at0, atl} and {at2, at3} have type 3A,
then the corresponding 3A-axial vectors uPl and uP2 in the above subalgebras are equal in X.
We call d«m(X) the dimension of R, and we say that R is based on an embedding of G into M if there exists a monomorphism 1 : G ^ M with i(T) c 2A and such that R is isomorphic to the subalgebra of VM generated by the Majorana axes corresponding to i(T).
Definition 5.2. The shape of a Majorana representation R of G specifies the types of dihedral subalgebras associated with all pairs of involutions on T.
Theorem 5.3. A Majorana representation of G = L2(11) must have shape (2A, 3A, 5A, 6A).
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ArsMath. Contemp. 7 (2014) 102-121
Proof. Let R be a Majorana representation of G with associated algebra X. The group L2(11) has a single conjugacy class of involutions, 2a, and a single class 3a of elements of order 3. From the structure class constants the product of any 2a involutions is in either of the L2(11) classes 1a, 2a, 3a, 5a, 5b or 6a. Hence X contains dihedral subalgebras of type 5A and 6A since they are the only dihedral subalgebras associated with dihedral groups of order 10 and 12. By the inclusion of the dihedral subalgebras 3A ^ 6A and 2A ^ 6A the classes 3a and 2a are mapped to 3A and 2A under Hence X also contains dihedral subalgebras of type 3A and 2A and we have accounted for all possible dihedral subalgebras in X.	□
Let R be a Majorana representation of G with associated algebra X. From the above theorem, R has the same shape as the subalgebra V of VM and the same inner product values for the sets S' and S. Moreover the dihedral and A5-subalgebras of X are equal to their Majorana representations from [11] and [12].
Proposition 5.4. (i) The dihedral subalgebras of type 2A, 3A, 5A and 6A are equal to the unique Majorana representations of D4, D6, Di0, and Di2 of shape 2A, 3A, 5A and 6A respectively.
(ii) The A5-subalgebra of type (2A, 3A, 5A) is equal to the unique Majorana representation of shape (2A, 3A, 5 A) of a group A5 of type (2A, 3A, 5A), which has dimension 26. □
We would like to show that the shape of R uniquely determines the algebra product in X so that X = V .In particular it is necessary to find the closure of the algebra generated by S. This can be inspected computationally. We let S2 := {u • v | u, v e S} and S3 := {(u • v) • w, u • (v • w) | u, v, e S} and for any positive integer n the set Sn is defined in a similar way. Already for the set of vectors S U S2 the Majorana axioms yield a very large number of eigenvectors for each Majorana axis and the first computational step is to check whether or not the linear span of S U S2 over R is contained in the closure of X. In fact during the reviewing stage of this paper, the author has learned that A. Seress has proved in [18] that the system of linear equations in S U S2, obtained from the eigenvectors of the axes {at 11 e T}, has a unique solution and that dimR(X) = 101. The result was obtained computationally with an algorithm written with [6].
Theorem 5.5. The L2(11)-subalgebra of the Monster algebra VM is equal to X, the algebra corresponding to the unique Majorana representation of L2(11).	□
Acknowledgements
I would like to thank my supervisor Professor Alexander A. Ivanov for introducing me to the Monster algebra and to Majorana representations and for his guidance and advice on the case of L2 (11). My thanks also go to the EPSRC for supporting this research.
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