/^creative ^commor ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 11 (2016) 175-187 The 2 A-Majorana representations of the Harada-Norton group Clara Franchi Dipartimento di Matematica e Fisica, Universita Cattolica del Sacro Cuore, Via Musei 41, I-25121 Brescia, Italy Alexander A. Ivanov Department of Mathematics, Imperial College, 180 Queen's Gt., London, SW7 2AZ, UK Mario Mainardis Dipartimento di Matematica e Informatica, Universita di Udine, Via delle Scienze 206, I-33100 Udine, Italy Received 28 May 2015, accepted 18 September 2015, published online 11 October 2015 Abstract We show that all 2A-Majorana representations of the Harada-Norton group F5 have the same shape. If R is such a representation, we determine, using the theory of association schemes, the dimension and the irreducible constituents of the linear span U of the Majorana axes. Finally, we prove that, if R is based on the (unique) embedding of F5 in the Monster, U is closed under the algebra product. Keywords: Majorana representations, association schemes, Monster algebra, Harada-Norton group. Math. Subj. Class.: 20D08, 20C34, 05E30, 17B69. 1 Introduction Let (W, ■) be a real commutative algebra endowed with a scalar product (, )W and denote with Aut(W) the group of algebra automorphisms of W that preserve the scalar product. We shall assume that, for every u,v,w e W, (M1) (, )W is associative , that is (u ■ v, w) = (u, v ■ w), (M2) the Norton Inequality, (u ■ u,v ■ v) > (u ■ v, u ■ v), holds. E-mail addresses: clara.franchi@unicatt.it (Clara Franchi), a.ivanov@imperial.ac.uk (Alexander A. Ivanov), mario.mainardis@uniud.it (Mario Mainardis) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 175 Ars Math. Contemp. 11 (2016) 101-106 Recall that a Majorana axis of W (see [10, Definition 8.6.1] or, equivalently, [9, p. 2423]) is a vector a e W such that (M3) a has length 1, (M4) the adjoint endomorphism ad(a), induced by multiplication by a on the R-vector space W, is semisimple with spectrum contained in {1,0,2-2, 2-5}, (M5) a spans linearly the eigenspace relative to the eigenvalue 1 of ad(a), (M6) the linear transformation aT: W ^ W, that inverts the eigenvectors of ad(a) relative to 2-5 and centralises the other eigenvectors, preserves the algebra product, (M7) the linear transformation a 26 Vp := ao + ai + 33(a_i + a_i) —3ao • ai, Wp := — 27(3ao + 3ai — a_i — a_i — a_2) + ao • ai, Up2 := P5 (2ao + 2a_i + a_2) — I35ao • a_i. The indexing with powers of p is justified by the fact that, in the action of M on WM, for 3 < N < 5, the NA-axes are essentially determined (up to the sign in the 5A-case) by the cyclic groups (p) in M of order N (see [9, p. 2450]). It is not clear if that property follows from Axioms (M1)-(M7), therefore axiom (M8)(b) was added in [3] in the definition of Majorana representations. The vectors ap, ap2, resp. ap3 appearing in the algebras of type 2A-, 4B-, resp. 6A are further Majorana axes. As above, the indexing is suggested by the action of M on WM since, in that case, whenever ao and ai generate a subalgebra of type 2A, the product p = ag ai is the Majorana involution corresponding to ap. As in the previous paragraph, that property will be axiomatised in (M8)(a). Finally, by the Norton-Sakuma Theorem (see [9, Lemma 2.20 (iv) and (v)]), ao and a2 (resp. ao and a3) generate a subalgebra of type 2 A in the algebra of type 4B (resp. 6 A) and, for i e {2,3}, by the definition of a®, the product a5aT is equal to p®. The Norton-Sakuma Theorem inspired the definition of Majorana representations, introduced by A. A. Ivanov in [10] in order to provide an axiomtic framework for studying 178 Ars Math. Contemp. 11 (2016) 101-106 the actions of 2AM-generated subgroups of M on WM. Let G be a finite group, T a G-invariant set of involutions generating G, ^: G ^ Aut(W) a faithful representation of G on W, and ^: T^ A be an injective map such that for every g G G and t G T, (tf)T := t* (1.1) and (tf / = (g-1tg)f. (1.2) The quintet R :=(G, T,W,^) is called a Majorana representation (or, to put evidence on the set T, a T-Majorana representation) of G, if R satisfies the following condition (see [3, Axiom M8]): (M8) (a) For t1 and t2 in T, the Norton-Sakuma algebra generated by tf and tf has type 2A if and only if t1t2 G T. (b) Suppose that t1, t2, t3, and t4 are elements of T such that t1t2 = t3t4 and the subalgebras generated by tf, tf and tf, tf have both type 3A, 4A, or 5 A. Then U(tl = U(i3i4)^ , V(ili2}^ = V(i3i4)^ , or W(ili2}^ = W(i3i4)^ , respectively. Axiom (M8)(a) and Norton-Sakuma Theorem (see [9, Lemma 2.20]) imply that, if tf and tf generate a Norton-Sakuma subalgebra of W of type 2A, 4B, or 6A, then t1t2, (t1t2)2 , or (t1t3)3 belongs to T, and (t1t2)f, ((t1t2)2)f, or ((t1t3)3)f coincides with a^^, a((tlt2)2)^, or a((t1t2)3)^, respectively. An immediate consequence of that definition is that, given a Majorana representation R :=(G, T,W,^) of a group G and a nonempty subset 7o of T, such that 7o is (7o) -invariant, the quintet R(T„> :=((To),To,W^|(To)^|To) (1.3) is a To-Majorana representation of (To). Further, if we replace W with the subalgebra Wt„ generated by the set of Majorana axes T^f in the quintet (1.3), we still have a Majorana representation of (To) provided (To) acts nontrivially on Wt„ (which is the case, e.g., when (To) has trivial centre). In particular, if e is an embedding of a group H in M and He is generated by a subset T of 2AM, then H inherits a (T n He)e -Majorana representation Re obtained by composing e with the restriction of RM to He. In that case, the Majorana representation Re of H is said to be based on the embedding e. In this paper, whenever a Majorana representation of a group G is based on an embedding e in the Monster, we shall always identify G with Ge. For a pair (a, b) of elements in W, denote the subalgebra they generate with ((a, b)). Let R be as above, the shape of R is a function shR from the set of the nondiagonal orbitals of G on T to the set of types of the Norton-Sakuma algebras so that C. Franchi, et al.: The 2A-Majorana representations of the Harada-Norton group 179 1. shR((t, s)G) = NX if and only if ts has order N and the algebra ((t^1, s^}} is a Norton-Sakuma algebra of type NX. 2. shR must respect the embeddings of the algebras: 2A 4B, 2A 6A, 2B 4A, 3A 6A in the sense that, for t,ri,r2 € T, if ((t^}} < ((t^,rf}} < ((t^}}, then (sM(t, ri)G), shR((t, r2)G)) € {(2A, 4B), (2A, 6A), (2B, 4A), (3A, 6A)}. Remark: Clearly, if T0 is a (70} -invariant nonempty subset of T, the shape of R(t0) is the restriction of shR to 70 x 70. Majorana representations of several groups have already been investigated (see [9, 11, 12, 13, 14, 5, 3, 6]). In this paper we study the 2A-Majorana representations of the Harada-Norton group F5, where 2A is the set of the involutions of F5 whose centraliser is (2HS) • 2, the double cover of the Higman-Sims group extended by its outer automorphism group of order 2. We shall show that every 2A-Majorana representation of F5 has the same shape as the Majorana representations of F5 based on its embedding into M as the subgroup generated by the set of involutions in 2AM that centralise an element of type 5 A (here 2A = 2AM nF5, see [4]). By [18, Theorem 21], that one is the unique embedding of F5 into M (up to conjugation in M), hence, since F5 is transitive on 2A, there is (up to conjugation in M) only one Majorana representation of F5 based on an embedding in M. We prove the following result. Theorem 1.1. Let W be as above and R := (F5,2A, W, —) be a 2A-Majorana representation of F5 on W. Then (i) R has the shape given in Table 3; (ii) The R-linear span (2A^} of 2A^ has dimension 18 316; (iii) (2A^} is the direct sum of three irreducible R[F5]-submodules of dimensions 1, 8910 and 9405, respectively; (iv) if R is based on the embedding of F5 in M, then W2A = (2A^}. Unless explicitly stated, for the remainder of this paper we shall stick to the notations introduced in this section. We shall also set T := 2A. 2 The First Eigenmatrix By [4, p. 166], we have |T| = 1539000, and it seems hard, at present, to perform a direct computation of the dimension of the linear span of T^. We therefore apply the theory of association schemes as in [14] and [6] to reduce ourselves to a more manageable situation. The first step is to compute the first eigenmatrix of the association scheme relative to the permutation action of F5 on T (see [1, pp. 59-60]). For that purpose, we need to recover some information about the action F5 induces by conjugation on T. 180 Ars Math. Contemp. 11 (2016) 101-106 Let n := |T| and let t1,..., tn be the distinct elements of T, so that B := (ti,...,tn) is an ordered basis for the complex permutation module V of F5 on T. With respect to B, we identify EndC(V) with the set of n x n matrices with complex entries. Let T0,... ,T8 be the orbitals of F5 on T and, for every k e {0,..., 8}, let Ak be the adjacency matrix associated to the orbital Tk, that is (Ak)ij = 1 if the pair (ti, tj) is in Tk 0 otherwise. By [1, Theorem 1.3], the 9-tuple (A0,..., A8) is a basis for the centralizer algebra C := Endc[F5](V). For i, j, k e {0,..., 8}, let pj be the number of elements z in T such that for a fixed pair (x, y) in Tk we have (x, z) e Ti and (z, y) e Tj. By definition, the pj's are all non negative integers and, by [1, §2.2], they are the structure constants of C relative to the basis ( A0 , . . . , A8 ) , that is pikj k=0 AiAj = > vjAk. (2.1) The matrix Bi of size 9 whose j, k entry is pj is called ith intersection matrix. Clearly, Bt is the matrix associated to the endomorphism induced by Ai on C via left multiplication with respect to the basis (A0,..., A8), in particular Bi has the same eigenvalues as Ai. By [8, Lemma 2.18.1(ii)] we may choose the indexes of the orbitals T0,..., T8 in such a way that T0 is the diagonal orbital (hence B0 is the identity matrix), Ti is the non-diagonal orbital of smallest size, and the first intersection matrix Bi is as follows: ! Bi V By [1, Theorem 3.1], we have that V decomposes into the direct sum V = Vo e ... e V8 (2.2) of nine irreducible C[F5]-submodules. Since F5 is transitive on T, the subspace linearly spanned by the sum of all elements of T is the unique trivial submodule of V. As usual, we shall denote it by V0. Since the action of F5 on T is multiplicity free (see [8, Lemma 2.18.1.(ii)]), the Vj's are minimal common eigenspaces for the adjacency matrices Ai. It follows that there is a complex invertible matrix D that simultaneously diagonalises the matrices Ai's. By the definition of the adjacency matrices, we have that, for each i, the 0 1 0 0 0 0 0 0 0 1408 53 32 18 4 2 0 0 0 0 50 0 2 12 0 2 0 0 0 450 32 100 32 50 32 0 0 0 350 672 112 160 100 92 160 0 0 504 0 504 288 356 312 320 0 0 0 672 672 552 650 720 640 1280 0 0 0 0 360 250 240 288 0 0 0 0 0 0 0 10 0 128 C. Franchi, et al.: The 2A-Majorana representations of the Harada-Norton group 181 sums (say kj) of the entries in each row of the matrices Ai are constant, whence V0 is a k^-eigenspace for Ai, for each i. For i,j e {0,..., 8}, let pij be the eigenvalue of Aj on Vj. The 9 x 9 matrix P := (pij) is called the first eigenmatrix of the association scheme (T, {T0,..., T8}). Lemma 2.1. With the above notations, 11 1408 2200 35200 123200 354816 739200 277200 5775 1 128 200 0 1600 -2304 0 0 375 1 28 -50 -50 -100 396 -750 450 75 1 16 4 -56 -136 -288 504 0 -45 1 -32 40 -80 80 576 -240 -360 15 1 -47 -50 250 350 -504 0 0 0 1 -112 300 1000 -2200 -864 -1800 3600 75 1 208 -50 2200 -2800 2016 4200 -6300 525 1 208 100 1000 1400 2016 -4200 0 -525 Proof. Note that, since A0 is the identity matrix, pi0 = 1 for all i's. Straightforward computation shows that the eigenvalues of Bi are 1408, 128, 28, 16, -32, -47, -112, 208, and 208, giving the first two columns of P. Set (A0,..., A8) = (1408,128, 28,16, -32, -47, -112, 208, 208). For each h e {0,..., 8}, let Sh be the linear system (Bi - Ah/d) 4(1, Ah, xa,..., x8) = 0 (2.3) in the indeterminates x2,..., x8. Taking i =1 in Equation (2.1) and multipling each term by D on the right and by D-i on the left, we get 8 (D-lAiD)(D-lAj D) = E Phj (D-iAhD). (2.4) h=0 Since the matrices D-iAhD are diagonal with eigenvalues pkh on the common eigenspaces Vk, for each h e {0,..., 8}, from Equation (2.4) we obtain that the relation 8 Ak Pkj = E Pij Pkh (2.5) h=0 holds for every k e {0,..., 8}. Note that the second member is the jth entry of the vector Bit(1, Ak ,pk2,... ,pk8), therefore Equation (2.5) implies that the 9-tuple (1, Ak,pk2, . . . ,pk8) is an eigenvector for Bi relative to the eigenvalue Ak, for every h e {0,..., 8}. Since, for k = 7,8, the eigenvalue Ak has multiplicity 1, it follows that the first seven rows of the matrix P can be obtained computing the unique solution (pk2,... ,pk8) of the system Sk for each k e {1,..., 6}. We are now left with the last two rows of the matrix P, corresponding to the eigenvalue 208 of Bi. The set of solutions of the system S7, (Bi - 208/d) 4(1, 208, x2,..., x8) = 0, 182 Ars Math. Contemp. 11 (2016) 101-106 is {(25 - X, 1600 + y, -700 - 4x, 2016, 8x, -3150 - 6x, x) | where x G R}. Therefore, for suitable x, y G R, we can write the last two rows of the matrix P as follows x 8x 1, 208, 25 --, 1600 +--, -700 - 4x, 2016, 8x, -3150 - 6x, x 77 1, 208, 25 - y, 1600 + 8y, -700 - 4y, 2016, 8y, -3150 - 6y, y. Set mj = dimR(Vj). Then m0 = 1 and, for 1 < i < 6, can be computed from the rows of P using the following formula (see [1, Theorem 4.1]): V8 V 2^j=o kj Pij from which we get mi = 16929, m2 = 267520, m3 = 653125, m4 = 365750, m5 = 214016, m6 = 8910, whence 6 m7 + m8 = n - ^^ mi = 12749. i=0 Comparing that value with the decomposition of the permutation module of F5 on T into irreducible submodules given in [8, Lemma 2.18.1.0'/)], we obtain that, modulo interchanging the indices 7 and 8, m7 = 3344 and m8 = 9405. By the Column Orthogonality Relation of the first eigenmatrix, 8 "^muPkiPhj = nkjSjj fc=0 (see [1, Theorem 3.5]), applied with (i, j) = (0, 8) and (i, j) = (8, 8), we get the quadratic system 3344x + 9405y = -3182025 3344x2 + 9405y2 = 3513943125 whose solutions are (x, y) = (525, -525) or (x, y) = (1575/61, 62475/61). By [2, Theorem 3.5(b)], the matrices Ai's are symmetric, since, by [4], the Frobenius-Schur indices of the irreducible constituents of the permutation character of F5 on T is +1 (and the action is multiplicity free). Thus, recalling that the phj's are all non negative integers, in order to determine which of the two solutions is the right one, we may use the formula 1 nkh Phj = -rir(AiAj Ah) (2.6) (see [1, Theorem 3.6(ii)]). Since the trace is invariant by matrix conjugation, tr(AiAjAh) can be obtained by multiplying, entry-wise, the ith, jth, and hth columns of the matrix P and adding the entries of the resulting column. In that way, we get that the entries phj are integers only in the case when (x, y) = (525, -525). □ n C. Franchi, et al.: The 2A-Majorana representations of the Harada-Norton group 183 3 The shape We continue with the notations of the last section. The next lemma recalls some known facts about conjugacy classes in M and F5 (see [16, 15]). For the remainder of this paper let H be the centraliser in M of an A5-subgroup of type (2A, 3A, 5A). By [15], we have that H = A12 and we may w.l.o.g. assume that F5 centralizes a 5A-element in that A5-subgroup, in particular H < F5. Lemma 3.1. Denoting the conjugacy classes of M and F5 as in [4], the correspondences between the conjugacy classes of the elements of order less or equal to 6 in M, F5 and H are as in Table 2. Table 2: Correspondences between the conjugacy classes of the elements of order at most 6 in M, F5, and H. Conj. class in M 2A 2B 3A 4A 4B 5A 6A Conj. class in F5 2A 2B 3A 4A 4B 5A 6A Cycle type in H 22, 26 24 3, 32,34 42, 42 • 22 4 • 2, 4 • 22 5, 52 3 • 22, 6 • 23, 62, 32 • 22 Let (t1,..., tn) be as in the previous section. For i, j e {1,... n}, set Yij := (tf ,t|)w. Lemma 3.2. If (ti, tj) and (th, tk) belong to the same orbital of F5 on T, then Yj = Yhk. Proof. That follows immediately from Equation (1.2) and the definition of y j. □ Thus, we can set, for k e {0,..., 8} and (t, s) e Tk, Yk :=(t^)w. (3.1) Lemma 3.3. For every x e {22,3,4 • 2, 24, 5} there are pairs of involutions of type 22 in A12 such that their product has cycle type x. Every element of cycle type 42 • 22 in A12 is the product of two elements of cycle type 26. Proof. That is an elementary computation (note that two elements of cycle type 26 whose product has cycle type 42 • 22 are explicitely given in the proof of Lemma 3.4). □ Lemma 3.4. With the above notations, for every k e {0,..., 8} and (t, s) e Tk, the scalar products Yk's are given in Table 3. Proof. The first two columns of Table 3 follow from Lemma 2.1. The correspondence that associates to each orbital Tk of F5 on T the F5-conjugacy class xk of the products ts, where (t, s) e Tk, has been determined by Segev in [20], giving the third column. 184 Ars Math. Contemp. 11 (2016) 101-106 Table 3: Valencies, shapes, and scalar products related to the orbitals of F5 on the set of its 2A-involutions. k |tCF5 (s)| (st)F5 shK(Tfc) Yk 0 1 1 - 1 1 1408 5A 5A 3/27 2 2200 2A 2A 1/23 3 35200 3A 3A 13/28 4 123200 4B 4B 1/26 5 354816 5E 5A 3/27 6 739200 6A 6A 5/28 7 277200 4A 4A 1/25 8 5775 2B 2B 0 Assume sMTfc) = NX, where N € {1,..., 6} and X G {A, B, C}. By the definition of shape, for (t, s) G Tk, we have that |st| = N .In particular, for k equal to 1, 5 and 6, we have that shR(Tk) is equal to 5A, 5A, and 6A, respectively. Let k G {2, 3,4,8}. By the second and third rows of Table 2 and Lemma 3.3 there are involutions s and t of cycle type 22 in Tn H such that st G xk, whence, by the first and third columns of Table 3, (s,t) G Tk n (H x H). By the remark in the introduction, we have that shR(Tfc) = sh^H ((s,t)H), whence Lemma 8 and Table 10 in [6] give the entry in the fourth column corresponding to k. Assume now k = 7. Choose the elements s = (1, 2)(3, 4)(5, 6)(7, 8)(9,10)(11,12) and t = (1, 3)(2,4)(5, 7)(6, 9)(8,11)(10,12) in H. Then st has cycle type 4222. ByTable2, s and t are contained in T and (st)F5 = 4A, hence, by the third column of Table 3, (s, t) G T7 and, by the Norton-Sakuma Theorem, shR(Tr) G {4A, 4B}. By Equation (1.2), (t^ )(ts)* = (tts)^ = (t3)^, so we have that t^ and (ts)^ are contained in the subalgebra generated by t^ and s^, which is (s, t)-invariant. Since tts has cycle type 24, by Table 2 it belongs to the class 2B of F5, whence, by the third column of Table 3, (t, ts) g T8 and the subalgebra generated by t^, (ts)^ is of type 2B, by the previous paragraph. By the second condition of the definition of the shape, shR(T7) = 4A. Finally, the last column follows from Table 1. □ C. Franchi, et al.: The 2A-Majorana representations of the Harada-Norton group 185 4 Closure Lemma 4.1. Suppose that R is based on the embedding of F5 in M. Then (T*) = Wr. Proof. Let H be the subgroup of F5 isomorphic to A12 defined as in the previous section. Let t, s be distinct elements of T, set p = (ts)^ and let N be the order of p. Let U be the Norton-Sakuma algebra generated by t* and s*, and let NX be its type. By Table 1, if NX is contained in {2A, 2B, 4B}, then U is linearly spanned by elements in T*, otherwise, by Lemma 3.4, NX g {3A, 4A, 5A, 6A} and U has a basis all of whose elements but the NX-axis are Majorana axes. Therefore, with the notations of Table 1, we may assume that NX g {3A, 4A, 5A, 6A} and show that, in all those cases, the NX-axes up, vp, wp, up2 are contained in (T* ) . If ts has order 3,4, or 5, then, by Lemma 3.1, there is g g F5, depending on ts, such that ts is an element of cycle type respectively 3, 42 • 22, and 5 in Hg. By Lemma 3.3, there are elements t' and s' of cycle type 22 or 26 in Hg such that ts = t's'. By Lemma 3.1, (t')* and (s')* generate a Norton-Sakuma algebra of the same type as U, thus, by Axiom (M8)(b), we have that up = u(t,s,)^, vp = v(t,s,)^, and wp = w(t,s,)^, respectively. Assume NX = 3A. By [3, Corollary 3.2], u(t,s,)^ is a linear combination of elements of (T n Hg )* and we are done. Similarly, assume NX = 4A (resp. NX = 5A). By [3], second formula in the abstract, or Section 6 (resp. Lemma 5.1), we have that v(t,(resp. w(t,s,)^) is a linear combination of elements in (Tn Hg )* and 3A-axes, and we are done by the previous case. Finally assume NX = 6A. Then, by the remarks after Table 1, up2 is a 3A-axis and again we are done by the 3A case. □ Note that in the previous proof we require that R is based on the embedding of F5 in M only to deal with the case 4A, all the other cases following from results of [3] that depend only on the shape of that representation of Ai2. 5 Proof of Theorem 1.1 The first claim of Theorem 1.1 follows from Lemma 3.4 and the last is the content of Lemma 4.1. To prove the second and the third claims, let r = (Yj) be the Gram matrix of (, )W associated to the n-tuple (t*,..., t*). By an elementary result on Euclidean spaces, we have that rank(r) =dimR((t* 11 gT)). (5.1) Since T0,..., T8 is a partition of T x T and, by Equation (3.1), Yk = Yj, for (tj, tj) g Tk, we have that 8 r = ^ Yk Ak. (5.2) k=0 186 Ars Math. Contemp. 11 (2016) 215-229 Let D be as in Section 2. From Equation (5.2) we get: 8 r := D-1rD = ^ YkD-1AkD, (5.3) k=0 where all the matrices r, and Ak := D-1AkD for k G {0,..., 8}, are diagonal. Now, clearly, the rank of r is equal to the rank of r, hence (being r diagonal) to the number of nonzero entries of r. By Lemma 3.4 (Table 3), Equation (5.3) becomes — —r 1 _ 1-, -, r = Ao + 27 Ai + 8 A2 + 28 A3 + 26 A4 + 27 A5 + 28 Ae + 25 A7 + 0A8, which, by Lemma 2.1, gives the eigenvalues 70875/2,0,0, 0, 0,0, 875/8,0, 225/4 of r on the subspaces V0,..., V8, respectively. Hence dimR((T^)) = m0 + me + m8 = 1 + 9405 + 8910 = 18 316. References [1] E. Bannai and T. Ito, Algebraic Combinatorics I. Association Schemes, Benjamin-Cummings Lect. Notes, Menlo Park, 1984. [2] P. J. Cameron, Permutation Groups, London Mathematical Society Student Texts 45, Cambridge Univ. Press, Cambridge, 1999. [3] A. Castillo-Ramirez and A. A. Ivanov, The Axes of a Majorana Representation of A12, in: Groups of Exceptional Type, Coxeter Groups and Related Geometries, Springer Proceedings in Mathematics & Statistics 82 (2014), 159-188. [4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Simple Groups, Clarendon Press, Oxford, 1985. [5] S. Decelle, The l2(11)-subalgebra of the Monster algebra. J. Ars Math. Contemp. 7.1 (2014), 83-103. [6] C. Franchi, A. A. Ivanov, M. Mainardis, Standard Majorana representations of the symmetric groups, preprint (2014). [7] The GAP Group: Gap - Groups Algorithms and Programming, Version 4.4.12. http://www.gap-system.org (2008). [8] A. A. Ivanov, S. A. Linton, K. Lux, J. Saxl, and L. H. Soicher, Distance transitive graphs of the sporadic groups, Comm. Alg. 23 (1995), 3379-3427. [9] A. A. Ivanov, D. V. Pasechnik, A. Seress and S. Shpectorov, Majorana representations of the symmetric group of degree 4, J. Algebra 324 (2010), 2432-2463. [10] A. A. Ivanov, The Monster Group and Majorana Involutions, Cambridge Tracts in Mathematics 176, Cambridge Univ. Press, Cambridge, 2009. [11] A. A. Ivanov and A. Seress, Majorana representations of a5, Math. Z. 272 (2012), 269-295. [12] A. A. Ivanov, On Majorana representations of A6 and A7, Comm. Math. Phys. 306 (2011), 1-16. [13] A. A. Ivanov, Majorana representations of A6 involving 3c-algebras, Bull. Math. Sci. 1 (2011), 356-378. C. Franchi, et al.: The 2A-Majorana representations of the Harada-Norton group 187 [14] A. A. Ivanov and S. Shpectorov, Majorana representations of L3(2), Adv. Geom. 14 (2012), 717-738. [15] S. P. Norton, F and other simple groups, PhD Thesis, Cambridge, 1975. [16] S. P. Norton, The uniqueness of the monster, in: J. McKay (ed.), Finite Simple Groups, Coming of Age, Contemp. Math. 45, AMS, Providence, RI 1982, 271-285. [17] S. P. Norton, The Monster algebra: some new formulae, in Moonshine, the Monster and Related Topics, Contemp. Math. 193, AMS, Providence, RI 1996, pp. 297-306. [18] S. P. Norton and R. A. Wilson, Anatomy of the Monster II, Proc. London Math. Soc. 84 (3) (2002), 581-598. [19] S. Sakuma, 6-transposition property of t-involutions of vertex operator algebras, Int. Math. Res. Not. (2007), doi:10.1093/imrn/rmn030. [20] Y. Segev, The uniqueness of the Harada-Norton group, J. Algebra 151 (1992), 261-303.