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The 2-Modular Characters of Conway's Third Group

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The 2-Modular Characters of Conway's Third Group The 2-modular characters of Conway's third group Co3 Ibrahim A. I. Suleiman Department of Mathematics and Statistics, Mu'tah University, Al-Karak, P.O. Box 7 Jordan and Robert A. Wilson School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. email: [email protected] published in J. Symbolic Comput. 24 (1997), 493{506 Abstract We determine the 2-modular character table of the third sporadic simple group of Conway, up to two ambiguities. In each of these cases we give the smallest possibility for the character, which is also very likely to be the correct answer. 1 Introduction 10 7 3 The sporadic simple group Co3 of order 495; 766; 656; 000 = 2 :3 :5 :7:11:23 was discov- ered by J. H. Conway (see [2]). This group can be obtained as a subgroup of the double : cover 2 Co1 of Conway's first group Co1, which may be defined as the automorphism group of the 24-dimensional Leech lattice (see [3]). In [5] Curtis has classified certain sublattices (which he called S-lattices) of the Leech lattice and found corresponding sub- : groups of 2 Co1 preserving these sublattices. Conway's third group Co3 is the stabilizer 1 of a type 3 vector in the Leech lattice (see [5]). The fourteen conjugacy classes of maximal subgroups of Co3 were determined by Finkelstein (see [7]). The p-modular character tables for Co3 for the primes p = 7, 11, and 23 have been determined (see [10]). In this paper we find the 2-modular character table of Co3. There are two characters which we were unable to prove correct, although we have very strong evidence for them. Our methods are computational, making use mainly of the `Meat-axe' written by R. A. Parker [8]. We also used GAP [13] for a few character calculations, as well as some extensions to the `Meat-axe' written by M. Ringe of RWTH, Aachen. The calculations were performed partly on the IBM3090 in the Birmingham University Computer Centre, and partly on a SUN SPARCstation ELC provided by the School of Mathematics and Statistics in the University of Birmingham, with financial assistance from the Science and Engineering Research Council. 2 The blocks Using the ordinary character table of Co3 (see [4]), we calculate the 2-modular cen- tral characters. These central characters give the block distribution of characters. The ordinary characters are distributed in the following three blocks: 1. Block B1 = f129536a; 129536bg is of defect 1. 2. Block B2 = f896a; 896b; 20608a; 20608b; 73600; 93312; 226688; 246400g is of defect 3. The defect group is elementary abelian and of inertial index 21. 3. The principal block B0 contains the remaining thirty two ordinary characters. The theory of blocks of cyclic defect (see for example [1]) implies that the two ordinary representations 129536a, 129536b which are in block B1, have equal character values on the 2-regular classes, and that 129536 (the reduction modulo 2 of either one of them) is a 2-modular irreducible representation of Co3. 3 Elementary use of the Meat-axe To determine the decomposition matrices of the other two blocks, we use the Meat-axe, especially the method of condensation (see [12], and Section 4 below). We started with : some 24 × 24 matrices over GF (3) generating Co3, as a subgroup of 2 Co1. Then using the Meat-axe program `VP' (`vector permute'|see [14]) to find the action of Co3 on a suitable orbit of vectors, we obtained permutations on 276 points also generating Co3. Then we reduced the corresponding permutation representation modulo 2 and chopped this up with the Meat-axe to get 276 = 1 + 1 + 22 + 22 + 230: In particular, we obtained two generators for Co3 in GL22(2), as in Table 1. 2 Table 1: Generators for Co3 in GL22(2) 1101110001000001010000 0101000010111010111111 1111010111110100001011 0110010100011110110000 0000001000000100010101 0011010000111111010111 1111100110110001001110 0001101110001011010011 0101010000000010011101 1010010000100001011110 0000010000000100010101 1101000000001010100011 0010000000000100010101 1100101010001111010101 0001000011000000111111 1000110100110101010101 1110100100110100010011 0100110001010000000111 0000000000000110010101 1100000010100101010010 0000000000100100010101 0101110110011100000101 0110111111010011101111 0101111101010011111001 0000000000001100010101 1000010101010101010001 0000000000000100000101 0001010000111100100111 0000000001000100010101 0011010010111011001111 0000000000000100011101 0100110010110011111010 0001000110000010011010 1101011001111101100011 0000000000000000010101 0100101001001000100001 0000000000000101010101 1100101100001001110011 0000000000000100010100 0101110110010100000001 0000000000000100010111 0000001101111000101110 0000000000000100010001 1101101010101110000101 a b 3 Table 2: Words for classes in Co3 Class Word 9A (ab2)2 9B ab 11A bab(abab2)2 15A abab2 15B (ab)2(abab2)2(ab2)2 21A ab2ab(abab2)2 23A (ab)2(abab2)2ab2 Then these two 2-modular irreducible representations (namely 22 and 230) can be used to produce some more irreducibles in the usual way with the Meat-axe. We find that Λ2(22) = 1 + 230 Λ3(22) = 22 + 22 + 1496 22 ⊗ 230 = 22 + 22 + 1496 + 3520: On restriction to the maximal subgroup McL:2, the ordinary characters 896a, 896b remain irreducible. Using the 2-modular character table of McL:2 (see [9]) we can see that the latter remain irreducible on reduction modulo 2. It follows that 896a and 896b are two 2-modular irreducible representations of Co3. Thus we have shown that 1, 22, 230, 896a, 896b, 1496, 3520 and 129536 are eight of the sixteen 2-modular irreducible representations of Co3. Before we look for the remaining irreducibles, we calculate the character values of the irreducibles obtained so far. To do this, we find representatives for all the 2-regular classes of Co3 as words in a; b: The words for some classes are given in Table 2. The other 2-regular classes are powers of these. We then work out the character values on these classes using the program `EV' of the Meat-axe which works out the eigenvalues of a matrix. The results are given in Table 3. 4 The method of condensation The main method used in the following was condensation of permutation modules. Some- times we also constructed the invariant subspaces which contain the required represen- tations, by spinning up the corresponding subspaces in the condensed module under the group generators. This can be done using the uncondense program `UK' of the Meat-axe. These methods have been explained in detail in [14] (see also [12] and [20]). We give here a brief summary. Let G be a group, and V be a kG-module, where k is a finite field of characteristic p. 4 Table 3: The characters of some 2-modular irreducibles of Co3 @@@@@@@@@@@@@@@@ 4 349 29 4 1 95766656000 920 160 536 500 300 42 162 81 22 22 30 15 21 23 23 p power A A A A A A A A A A AA BB AC A A p' part A A A A A A A A A A AA BB AC A A 1A 3A 3B 3C 5A 5B 7A 9A 9B 11A B** 15A 15B 21A 23A B** 1111111111111111 22 -5 4 -2 -3 2 1 -2 1 0 0 0 -1 -2 -1 -1 230 14 5 2 5 0 -1 2 -1 -1 -1 -1 0 2 0 0 896 32 -4 -7 -4 1 0 2 -1 b11 ** 2 1 0 -1 -1 896 32 -4 -7 -4 1 0 2 -1 ** b11 2 1 0 -1 -1 1496 -16 2 8 -4 -4 -2 -1 -4 0 0 -1 2 1 1 1 3520 -44 10 -8 -5 0 -1 1 1 0 0 1 0 -1 1 1 129536 -64 44 8 -14 -4 1 -1 -1 0 0 1 -1 1 0 0 5 Let H be a subgroup of G such that p does not divide jHj. Define the idempotent 1 e = X h 2 kG: jHj h2H Then e:kG:e is a sub-algebra of kG known as a Hecke algebra. From any kG-module V , we obtain an e:kG:e-module V e. We say that V e is condensed from V since V e consists of the fixed points of the action of H on V . We call H the condensation subgroup. In this way we get a condensed module V e such that dim(V e) ≈ dim(V ) . Accordingly, it should be much easier to apply the `Meat-axe' to V e rather jHj than to V . Moreover, any information about V e which we obtain with the `Meat-axe' gives rise to information about V . This can be seen using the following well-known result (see [12]). Proposition 1 Let χ1; χ2; : : : ; χr be the irreducible constituents of the module V , then the irreducible constituents of V e are the non-zero members of the set fχ1e; χ2e; : : : ; χreg. The problem in practice is to prove that we have generators for the Hecke algebra. If g1; : : : ; gr generate G, it does not necessarily follow that eg1e; : : : ; egre generate e:kG:e. We choose our group generators in such a way that we believe that heg1e; : : : ; egrei (which we call the condensation algebra) is the whole of the Hecke algebra e:kG:e, but it could conceivably be smaller. Thus we could have `mirages'|subspaces of V e invariant under the condensation algebra but not under the whole of the Hecke algebra. For this reason, condensation in general only provides lower bounds for the degrees of irreducible representations. Sometimes we need more information about the constituents of V than straightfor- ward condensation can provide. One method is to use the `uncondense' program UK of the Meat-axe to calculate constituents of the permutation representation explicitly. Given an invariant subspace W e of a condensed module V e we can construct the corresponding invariant subspace W of the permutation module V as follows: 1. Use UK to express the basis vectors for W e in terms of the basis vectors for V: Each basis vector for V e is the sum of basis vectors in V over an orbit of H.
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