THE BRAUER CHARACTERS OF THE SPORADIC SIMPLE

HARADA-NORTON GROUP AND ITS AUTOMORPHISM GROUP IN CHARACTERISTICS 2 AND 3

GERHARD HISS, JÜRGEN MÜLLER, FELIX NOESKE, AND JON THACKRAY

Dedicated to the memory of Herbert Pahlings

Abstract. We determine the 2-modular and 3-modular character tables of the sporadic simple Harada-Norton group and its automorphism group.

1. Introduction

The present paper is a contribution to the modular ATLAS project [31]. The aim of the project is to classify the modular representations of the simple groups and their bicyclic extensions given in the ATLAS [2], by computing their Brauer characters, or equivalently, their decomposition numbers.

In particular for the sporadic simple groups this is a challenging task. In lieu of a unified theory for these groups, the solution of this problem has always relied on computational methods, and especially on the MeatAxe (see [25]) developed by Parker and Thackray in the late 1970’s. However, even though the MeatAxe has evolved into a host of programmes to analyse modules of matrix algebras, the open problems in the modular ATLAS project defy a direct application: owing to the magnitude of the computational problems, memory and time constraints render a naive approach utterly infeasible.

To regain computational tractability, Parker and Thackray established the condensation method (see [28]). The idea of condensation is that given a module V for an algebra A, one considers the condensed module V e of the algebra eAe for some suitably chosen idempotent e ∈ A. The condensation functor from mod-A to mod-eAe has a number of interesting properties, among the most important of which is that it maps simple modules to simple modules or zero, and composition series to series whose layers are simple or zero. For a detailed introduction to condensation see [27], for example. Unfortunately, the use of condensation introduces the generation problem (see [20], for example): for computational purposes we need to work with a feasibly small generating set for the condensed algebra eAe, yet such a set is not known in general, and we therefore devote time and energy to verify our condensation results.

Results. In this paper we determine the decomposition numbers of the sporadic simple Harada-Norton group HN and its automorphism group HN.2 in characteristics 2 and 3. More precisely, the decomposition matrices of the 2-blocks of positive defect of HN and HN.2 are determined in Section 2, where the results for HN are given in Tables 3 and 6, and those for HN.2 are given in Tables 7 and 8. For the reader’s convenience we give the degrees of Brauer characters lying in the principal

1  132  132  760  2 650
2 650  3 344 15 904 31 086 31 086
34 352  34 352 43 416 43 416 177 286
217 130 1 556 136

Table 1. Brauer character degrees in the principal 2-block of HN

1  133  133  760  3 344
8 778  8 778  9 139 12 264 12 264
31 768 31 768 137 236 147 061 255 037
339 702 339 702 496 924 496 924 783 696

Table 2. Brauer character degrees in the principal 3-block of HN

2-block of HN in Table 1. The characteristic 3 case is dealt with in Section 3, where the results for HN are given in Tables 9, 16 and 17, and those for the principal block of HN.2, being the only block not Morita equivalent to a block of HN, are given in Table 19. The degrees of the irreducible Brauer characters in the principal 3-block of HN are restated in Table 2.

The results obtained here are also accessible in [31] and will be through the CTblLib database [1] in GAP [4]. Since the Brauer trees of HN and HN.2 have already been determined in [7] and the characteristic 5 case has been tackled in [14], this completes the modular ATLAS project for HN. Moreover, since the decomposition matrix of the non-principal 3-block of HN of defect 2 was known prior to the completion of the present paper, it was the starting point of further investigations into this block carried out in [9].

The proofs for the two characteristics are quite different:

In characteristic 2, the non-principal block may be treated entirely theoretically. It possesses a semi-dihedral defect group of order 16, and we may therefore use Erdmann’s classification [3] of possible decomposition matrices for such blocks to determine its decomposition matrix. The true challenge lies in the principal block. Here we use condensation, but since there are no condensation subgroups readily available which allow the application of [20, Theorem 2.7], we are forced to verify the condensation results. As the dimensions of the simple modules with the exception of the last are small enough, we construct bases for these simple modules in order to show that the degrees of the Brauer atoms produced (see [6, Definition 3.2.1]), which are lower bounds for the degrees of the Brauer characters, are maximal, i.e. the atoms are in fact characters. We can then show that the last atom is a character, too.

For the automorphism group the result is immediate from the ordinary character table.

In characteristic 3, we start by investigating the block of defect 2 by choosing a basic set of projective characters and improving it pursuing a strategy first employed in [18]: We consider the endomorphism ring E of a suitably chosen projective lattice, and by Fitting correspondence compare the decomposition numbers of E with those of the block in question (see [10, Ch.II.12]). ¿From the practical side, we consider a direct summand of a projective permutation module, so that the regular representation of E is just the associated condensed module. Actually, here we are better off with the generation problem: since the regular action of E is faithful we are able to determine the dimension of the condensed algebra in question. Alone, this strategy turns out to be computationally tractable only for projective modules leading to non-faithful trace idempotents, so that after this analysis we are still left with several possible cases for the decomposition matrix.

These cases are then greatly reduced in parallel to our treatment of the principal block, which relies entirely on condensation. There are several condensation subgroups available which are normal in maximal subgroups, making them ideal candidates for [20, Theorem 2.7]. However, we still do not find any single suitable one whose trace idempotent is faithful. Therefore, we choose two condensation subgroups such that no simple module is annihilated by both condensation idempotents. This introduces a new problem: given the simple modules of one condensed algebra, we have to match them to the simple modules of the other condensed algebra, in order to gain the complete picture of the composition factors of the original module. We use [22] to overcome this problem (see also [21]).

For the automorphism group HN.2, a simple application of Clifford theory already gives the bulk of its Brauer characters. The remaining ones are once more determined through condensation, reusing the condensation infrastructure we have established for the simple group.

Background. In the sequel, we assume the reader to be familiar with the basics of ordinary and modular group representation theory, and in particular its computational aspects, as are for example exposed in [6, 15]. We employ the particular choice of Brauer lifts using Conway polynomials as in [8]. Character theoretic computations are carried out in GAP, whose facilities are also used to explicitly compute with permutation and matrix groups; in particular we use the orb package [19]. Module theoretic computations are done using the MeatAxe and its extensions [26], and as a computational workhorse we apply condensation of tensor products, respectively homomorphism spaces, as described in [12] and [16]. For a general introduction to condensation we refer the reader to [27], and to [13] for a treatise on peakword condensation. As a source of explicit data we use the database CTblLib in GAP, as well as the web-based database [29], also accessible via the GAP package AtlasRep [30]. Next to matrix and permutation representations for many of the groups in [2], given in standard generators in the sense of [32] where available, the latter in particular also contains straight line programmes to find maximal subgroups and conjugacy class representatives.

Let us fix some notation which will hold throughout the paper. Let G denote the Harada-Norton group HN. Characters, representations, and modules are given by their degrees, respectively dimensions. By a slight abuse of notation, we therefore do not distinguish between a module, the representation it affords, and the associated Brauer character. If there is more than one character of the same degree we affix subscripts to distinguish them. The ordinary characters are numbered as they are numbered in the character tables of CTblLib, which generally corresponds to the numbering given in [2]. In the decomposition matrices we also indicate this numbering next to the character degrees if necessary. If χ is an ordinary character we let χ′ denote its restriction to the p-regular classes, where p is the characteristic of the field currently considered. Moreover, we stick to group theoretic notation as is used in [2].

¿From the ordinary character table of G and G.2, see [2], we see that all character fields are 2-elementary abelian, thus by [8, Section I.5] the quadratic extension GF(p^{2}) of any finite prime field GF(p) is a splitting field for G and G.2. We let F denote GF(p^{2}) in the sequel, and let (Q, R, F) denote a p-modular splitting system for G and G.2.

2. Characteristic 2

The 2-modular characters of G are distributed over three blocks, two of which have positive defect: the principal block B_{0} of defect d = 14, having k = 45 ordinary and l = 17 modular characters, and the block B_{1} of defect d = 4, having k = 8 ordinary and l = 3 modular characters. The block of defect 0 consists of the ordinary character χ_{46}.

2.1. Proof for the block B_{1}. By [11, Section 9] the defect group of Block B_{1} is a semidihedral group of order 16. In [3, Section 11] Erdmann lists all possible decomposition matrices for such blocks, such that we are left with to decide which of the four cases in Lemmas 11.4, 11.6, 11.9, and 11.11 holds:

In Erdmann’s notation [3, Theorem 11.1], work of Olsson [24] implies that B_{1} has four characters χ_{1}, …, χ_{4} of height zero, three characters χ_{1}^{∗}, …, χ_{3}^{∗} of height one for which χ^{∗′} := χ_{1}^{∗′} = χ_{2}^{∗′} = χ_{3}^{∗′}, and a single character χ̂ of height two. Moreover, χ_{1}, …, χ_{4} can be chosen such that

\begin{aligned}δ_{1}χ′_{1} + δ_{2}χ′_{2} & = & −δ_{3}χ′_{3} − δ_{4}χ′_{4} & = & χ^{∗′}, \\ δ_{2}χ′_{2} + δ_{4}χ′_{4} & = & −δ_{1}χ′_{1} − δ_{3}χ′_{3} & = & χ̂′, \end{aligned}

where δ_{1}, …, δ_{4} ∈ {±1}. In particular, choosing any χ_{i} ∈ {χ_{1}, …, χ_{4}} and χ_{r}^{∗} ∈ {χ_{1}^{∗}, …, χ_{3}^{∗}}, we conclude that {χ′_{i}, χ_{r}^{∗′}, χ̂′} is a basic set of Brauer characters. Actually, as is easily seen, the above conditions determine χ_{1}, …, χ_{4} and δ_{1}, …, δ_{4} uniquely up to the permutation interchanging δ_{1}χ_{1} ↔ −δ_{4}χ_{4} and δ_{2}χ_{2} ↔ −δ_{3}χ_{3}, and by the positivity of character degrees we are left with the four cases

[δ_{1}, …, δ_{4}] ∈ {[1, 1, −1, −1], [1, −1, −1, 1], [−1, 1, −1, 1], [1, 1, −1, 1]},

corresponding to the Lemmas cited above.

By computing the actual heights of the characters in B_{1}, we conclude that {χ_{1}, …, χ_{4}} = {χ_{17}, χ_{37}, χ_{45}, χ_{49}}, and {χ_{1}^{∗}, …, χ_{3}^{∗}} = {χ_{34}, χ_{35}, χ_{36}}, and χ̂ = χ_{44}. Decomposing the restrictions of the ordinary characters in B_{1} into the basic set {χ′_{17}, χ′_{34}, χ′_{44}} we obtain the matrix of Table 3, where the characters belonging to the basic set are indicated by bold face. This shows that we have

\begin{aligned}χ_{1} & = & χ_{37}, & χ_{2} & = & χ_{17}, & χ_{3} & = & χ_{49}, & χ_{4} & = & χ_{45}, \\ δ_{1} & = & 1, & δ_{2} & = & −1, & δ_{3} & = & −1, & δ_{4} & = & 1, \end{aligned}

that is, by [3, Lemma 11.6] the decomposition matrix of B_{1} is as given in Table 3.

2.2. Proof for the principal block B_{0}. Our proof for the principal block relies on the tried and tested formula of applying the MeatAxe in conjunction with condensation. Owing to the modest dimensions of the first few irreducible representations of G, the first five nontrivial ones are available through [29]. This way we obtain 132_{1}, 132_{2}, 760, 2 650_{1}, and 2 650_{2}; where 2 650_{1} yields 2 650_{2} by taking

\begin{matrix}χ & χ(1) & Φ_{1} & Φ_{2} & Φ_{3} \\ 17 & 214 016 & 1 & . & . \\ 34 & 1 361 920 & . & 1 & . \\ 35 & 1 361 920 & . & 1 & . \\ 36 & 1 361 920 & . & 1 & . \\ 37 & 1 575 936 & 1 & 1 & . \\ 44 & 2 985 984 & . & . & 1 \\ 45 & 3 200 000 & 1 & . & 1 \\ 49 & 4 561 920 & 1 & 1 & 1 \end{matrix}

Table 3. The decomposition matrix of the 2-block B_{1} of HN

the contragredient. Also available from the same source is the smallest non-trivial irreducible representation of the sporadic Baby Monster group B in characteristic 2. The restriction of this 4370-dimensional module yields the composition factors

4 370↓_{G} = 3 344 + 760 + 132_{1} + 132_{2},

thus adding 3 344 to our collection of simple modules we have available for further computations. Also owing to the small size of these modules, we may compute their Brauer characters explicitly by lifting eigenvalues, where the necessary straight line programmes to restrict from B to HN and to find conjugacy class representatives are also available at [29].

The eighth Brauer character in the principal block may also be computed with a plain application of the MeatAxe: Chopping the tensor product 132_{1} ⊗ 132_{2} gives the composition factors

132_{1} ⊗ 132_{2} = 15 904 + 2 × 760

and lets us derive the Brauer character of 15 904 from the tensor product of the Brauer characters. As the direct analysis of even larger tensor products with the MeatAxe is considerably slowed and ultimately rendered infeasible due to the increased module dimensions, we determine the remaining nine simple modules and their Brauer characters with the help of condensation.

As our condensation subgroup K we choose a Sylow 3-subgroup of order 243 of the alternating group on 12 letters which in turn is the largest maximal subgroup of G, and condense with the corresponding trace idempotent e; here, a straight line programme to find the largest maximal subgroup of G is available in [29], and Sylow subgroups of permutation groups can be computed in GAP. Unfortunately, for our choice of condensation subgroup, there are at present no computationally efficient methods available to deal with the generation problem (see [20]). Here we arbitrarily choose a few elements g_{i} ∈ G, and have to allow for the possibility that the corresponding condensed elements eg_{i}e generate a proper subalgebra C of the condensed group algebra eF Ge. We apply the condensation of tensor products technique described in [12, 16].

The results are presented in Table 4, in which we display the multiplicities of the composition factors of the condensed tensor products given, where we denote the simple C-modules by their dimension, preceded by the letter ‘k’. We also condense

1 132_{1}132_{2}760 2 650_{1} 2 650_{2} 3 344 132_{1} ⊗ 132_{2} 132_{1} ⊗ 760 132_{2} ⊗ 760 760 ⊗ 760 132_{1} ⊗ 2 650_{1} 132_{1} ⊗ 3 344 132_{2} ⊗ 3 344 760 ⊗ 2 650_{1} 760 ⊗ 3 344 2 650_{1} ⊗ 2 650_{2}
k1 1 . . . . . . . 2 2 20 6 6 6 40 12 46
k2_{1} . 1 . . . . . . 2 2 6 3 4 4 22 6 23
k2_{2} . . 1 . . . . . 2 2 6 4 4 4 22 6 23
k6_{1} …. 1 . . . . . 8 … 5 2 8
k6_{2} . . . . . 1 …. 8 1 . . 5 2 8
k12 … 1 … 1 . . 3 1 2 2 10 5 11
k28 . . . . . . 1 … 10 … 2 4 8
k62 . . . . . . . 1 . . 4 1 2 2 12 4 10
k118_{1} . . . . . . . . . 1 . 1 …. 2
k118_{2} . . . . . . . . 1 . . . . . . . 2
k160_{1} . . . . . . . . . . . 1 1 1 7 1 5
k160_{2} . . . . . . . . . . . 1 1 1 6 1 5
k164_{1} . . . . . . . . 2 … 1 2 1 2 4
k164_{2} . . . . . . . . . 2 . 1 2 1 1 2 4
k706 . . . . . . . . . . . 1 . . 4 . 3
k922 . . . . . . . . . . 2 . 1 1 2 2 2
k6 344 . . . . . . . . . . . . . . . . 1
k908 . . . . . . . . . . . . . . . 2 .
k5 652 . . . . . . . . . . . . . . . 1 .
k14 072 . . . . . . . . . . . . . . . . 1

Table 4. Condensation results for p = 2 for HN

the known simple F G-modules of the principal block individually in order to match them to simple C-modules. Note that we condense the tensor product 132_{1} ⊗ 132_{2}, even though we were able to compute its decomposition directly: it is more efficient to condense the tensor product than to deal with the module 15 904 alone.

Our next task is to determine which simple C-modules are composition factors of simple eF Ge-modules coming from simple F G-modules of its principal block. To this end, it turns out that the restrictions of the ordinary characters indicated by bold face in Table 6 are a basic set of Brauer characters for the principal block. A basic set of Brauer characters for block B_{1} was already given in Table 3, and adding χ′_{46} we obtain a complete basic set of Brauer characters.

Then, by decomposing the Brauer characters of the tensor products considered into this basic set, we see that only the last two contain composition factors which lie outside of the principal block. As a manner of speaking, we say that a simple C-module belongs to a block of F G if it is a composition factor of the restriction to C of a condensed simple F G-module lying in that block.

By computing the multiplicity of the trivial character of K in the restrictions to K of the basic set characters and the known Brauer characters, we determine the dimensions of the corresponding condensed modules. Of course, these are upper bounds for the dimensions of their simple C-constituents.

Also, decomposing the characters of the last two tensor products in the basic set of B_{0} and the irreducible Brauer characters of the other blocks, gives that 214 016 occurs twice, and 1 361 920 occurs once as a constituent of 760 ⊗ 3 344. The single defect zero module 3 424 256 appears once in 2 650_{1} ⊗ 2 650_{2}.

We compute the dimensions of the condensed modules whose Brauer characters constitute the basic set. From this it is immediate that k14 072 is a composition factor of the condensed defect zero module 3 424 256 restricted to C: it is simply to large to fit into any of the other modules. As the dimension of the C-module is equal to the dimension computed of 3 424 256 condensed, we conclude that k14 072 is the irreducible restriction of the latter module, and therefore belongs to the defect zero block.

The simple 1 361 920-dimensional modules of B_{1} condense to modules of dimension 5 652. By summing up the dimensions of the composition factors by their multiplicities as given in the second to last column of Table 4, it is plain to see that k5 652 appears in one of the condensed simple modules of the same dimension, as the sum obtained is less. Thus k5 652 too is an irreducible restriction of a condensed simple module lying in B_{1} and therefore belongs to B_{1}.

For k908 we verify computationally that it also originates from a module outside of the principal block: employing the MeatAxe and peakword techniques, we find a uniserial submodule of 760 ⊗ 3 344 condensed with head and socle isomorphic to k908 and containing k5 652 as the only additional composition factor. Therefore there is a non-trivial extension of k5 652 by k908, and as k5 652 belongs to B_{1} so must k908.

Thus all appearing C-modules, except k908, k5 652, and k14 072, belong to the principal block. Hence the matrix A comprising the first 17 rows of Table 4 describes the condensed modules of the B_{0}-components of the modules given at its top. It is immediate that A has full rank. Therefore, as B_{0} has l = 17 modular characters, by [5, Proposition 1] condensation with respect to our chosen subgroup K defines a faithful functor from mod-B_{0} to mod-eB_{0}e. Furthermore, by [14, Theorem 4.1] taking as b the B_{0}-components of the Brauer characters afforded by the modules indexing the columns of Table 4, the product bA^{−1} gives a basic set of Brauer atoms for B_{0}.

As the possibility that C is a proper subalgebra of eF Ge remains, the degrees of the Brauer atoms are lower bounds for the degrees of the irreducible Brauer characters. Therefore in order to verify that the Brauer atoms are in fact the Brauer characters, we need to show that the simple F G-modules possess these dimensions. This is achieved by constructing bases for almost all simple modules through uncondensation: With the help of the MeatAxe we can determine the socle series of the condensed tensor products considered, and may therefore seek a suitably small submodule U of the condensed module V e which contains a condensed version of the simple module S we would like to construct. The subspace embedding of V e into V therefore embeds a basis of U into V, and with standard MeatAxe functionality we may construct a basis of the smallest G-invariant subspace of V containing the embedded basis of U. Ideally, we have U = Se, and we are done after having

\begin{matrix}k118_{2} \\ k2_{2} \\ k1 \\ k2_{1} ⊕ k164_{1} \end{matrix}  \begin{matrix}k922 \\ k1 ⊕ k62 \\ k2_{1} ⊕ k2_{2} ⊕ k28 \\ k1 ⊕ k1 \\ k6_{1} ⊕ k6_{2} \\ k6_{1} ⊕ k6_{2} \\ k28 \end{matrix}  \begin{matrix}k706 \\ k1 \\ k2_{2} \end{matrix}  \begin{matrix}k2_{1} ⊕ k12 ⊕ k160_{2} \\ k164_{2} \end{matrix}

Table 5. Socle series of select submodules of some tensor products

constructed a basis of S. However, if Se does not lie in the socle of V, we need to iterate this procedure by working along an ascending composition series of U.

Recall that we have already chopped up the tensor product 132_{1} ⊗ 132_{2} completely, hence we are done with that. Within the condensed tensor products 132_{1} ⊗ 760, 760 ⊗ 760, 132_{1} ⊗ 2 650_{1}, and 132_{1} ⊗ 3 344, respectively, we find suitable submodules whose socle series we give in Table 5. Note that for every pair of modules conjugate under the outer automorphism, which of course is seen on the level of characters, it is sufficient to construct a basis of only one member. Therefore the four modules of Table 5 allow us to verify the dimensions of almost all Brauer characters except the one corresponding to k6 344. For the practical realisation of these computations, it is important to note that we do not need to construct a representation of the action of G on any of the huge tensor product spaces: using the natural isomorphism M ⊗ N ≅ Hom_{F} (M^{∗}, N) (see [17, Lemma 2.6]) we only need to work with the available representations of the tensor factors.

With only the last Brauer atom of degree 1 556 136 left to prove to be a Brauer character, we can avoid the unwieldy construction of a basis for this module: We additionally condense the tensor product V := 2 650_{1} ⊗ 3 344 instead. Let S denote the simple F G-module such that its Brauer character φ contains the Brauer atom α of degree 1 556 136 as a summand. Since φ is a constituent of the basic set character χ′_{39}, and the latter has degree 2 031 480, we conclude that φ contains α as a summand with multiplicity precisely one.

A calculation with the MeatAxe gives that head and socle of V e↓_{C} are isomorphic, and have structure

k160_{1} ⊕ k6 344 ⊕ k12 288.

By the above we have 43 416_{1}e = k160_{1} and 2 985 984e = k12 288, and Se↓_{C} has k6 344 as a composition factor. Recall that e is a faithful condensation idempotent, and that the Brauer atom α alone already accounts for the condensed dimension 6 344, that is φ = α is equivalent to Se = k6 344, but due to the generation problem this does not immediately follow from the above statement, and neither it is clear how socle and head of V e and thus of V look like. But, with the possible exception of φ, we have proved that all simple eF Ge-modules restrict irreducibly to C, and that two such eF Ge-modules are isomorphic if their restrictions to C are. Hence we conclude that both the head and the socle of V have at most the constituents 43 416_{1}, 2 985 984 and S; in particular, the constituents 132_{1/2} and 760 do not occur.

We now additionally choose another condensation subgroup, namely the normal subgroup 5_{+}^{1+4} of the sixth maximal subgroup M_{6} := 5_{+}^{1+4} : 2_{−}^{1+4}.5.4. To distinguish this condensation from the previous one, we denote the trace idempotent of 5_{+}^{1+4} by e′ and prefix the simple e′F Ge′-modules by the letter ’c’. By [20, Theorem 2.7] we obtain a generating system of e′F Ge′ of 128 elements, if we condense the 126 non-identity double coset representatives of M_{6}\G/M_{6} and the two generators for M_{6}, obtained through a straight line programme available at [29].

As we know all irreducible Brauer characters except φ, we can determine the dimensions of their condensed modules with respect to e′. This shows that e′ is not faithful; indeed, precisely the simple modules 132_{1/2} and 760 condense to the zero module. We identify the possible composition factors of the 2 772-dimensional condensed module V e′ by applying the MeatAxe. We again find that head and socle of V e′ are isomorphic, and have structure

c8 ⊕ c1 152 ⊕ c960,

where 43 416_{1}e′ = c8, 2 985 984e′ = c960 and Se′ = c1 152. Again, a consideration of condensed dimensions shows that α alone accounts for the dimension 1 152, which implies that φ − α is a sum of multiples of 132_{1/2} and 760.

Moreover, as we look at V e′ as an e′F Ge′-module, uncondensing (in theory, not in practice!) the submodule of V e′ isomorphic to c1 152 yields a submodule U of V having a simple head isomorphic to S, whose radical has only constituents condensing to the zero module, that is amongst 132_{1/2} and 760. But since none of the latter occur in the socle of V we conclude that U is isomorphic to S, in other words S occurs in the socle of V, and similarly in the head of V as well.

We now return to our established faithful condensation setup and reconsider V e: Assume that φ ≠ α. Then, as α is a summand of φ with multiplicity one, hence k6 344 occurs only once as a constituent of Se↓_{C}, we conclude that the head or the socle of V e↓_{C} (or both) contain a condensed module of at least one of 132_{1/2} and 760, which is not the case, and thus is a contradiction. Hence we conclude that the last Brauer atom α coincides with φ, thus is an irreducible Brauer character.

The decomposition matrix of the principal block is given in Table 6.

2.3. The Automorphism Group in Characteristic 2. The 2-modular characters of G.2 are distributed over three blocks, each covering precisely one of the 2-blocks of G: the principal block B_{0} of defect d = 15, having k = 63 ordinary and l = 12 modular characters, the block B_{1} of defect d = 5, having k = 13 ordinary and l = 3 modular characters, and the block B_{2} of defect d = 1, having k = 2 ordinary and l = 1 modular characters.

The task of determining the decomposition numbers of G.2 in characteristic 2 is easy: every Brauer character of G invariant under the outer automorphism gives just one Brauer character of G.2, while the pairs of Brauer characters of G whose members are interchanged by the outer automorphism induce to one Brauer character of G.2. Therefore, using the results of the previous section we can immediately give the three decomposition matrices in Tables 7 and 8.

3. Characteristic 3

In characteristic 3, the simple group G possesses three blocks of positive defect: the principal block B_{0} of defect d = 6, having k = 33 ordinary and l = 20 modular characters, the block B_{1} of defect d = 2, having k = 9 ordinary and l = 7 modular

10 MODULAR CHARACTERS OF THE HARADA-NORTON GROUP

\begin{matrix}χ & χ(1) & φ_{1} & φ_{2} & φ_{3} & φ_{4} & φ_{5} & φ_{6} & φ_{7} & φ_{8} & φ_{9} & φ_{10} & φ_{11} & φ_{12} & φ_{13} & φ_{14} & φ_{15} & φ_{16} & φ_{17} \\ 1 & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 2 & 133 & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 3 & 133 & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 4 & 760 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 5 & 3 344 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 6 & 8 778 & 2 & . & 1 & 1 & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 7 & 8 778 & 2 & 1 & . & 1 & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 8 & 8 910 & 2 & 1 & 1 & 1 & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 9 & 9 405 & 1 & . & . & 1 & 1 & 1 & 1 & . & . & . & . & . & . & . & . & . & . \\ 10 & 16 929 & 1 & 1 & 1 & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . \\ 11 & 35 112 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 & . & . & . \\ 12 & 35 112 & . & . & . & . & . & 1 & . & . & . & . & . & . & 1 & . & . & . & . \\ 13 & 65 835 & 1 & 2 & 1 & . & . & . & . & . & 1 & . & . & . & . & 1 & . & . & . \\ 14 & 65 835 & 1 & 1 & 2 & . & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . \\ 15 & 69 255 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & . & . & 1 & . & . & . & . & . & . \\ 16 & 69 255 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & . & . & . & 1 & . & . & . & . & . \\ 18 & 267 520 & 6 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & . & . & . & . & . & . & . & 1 & . \\ 19 & 270 864 & 6 & 2 & 2 & 2 & 2 & 1 & 3 & 2 & . & . & . & . & . & . & . & 1 & . \\ 20 & 365 750 & 8 & 3 & 3 & 2 & 2 & 3 & 1 & 1 & . & . & 1 & 1 & 1 & 1 & 1 & . & . \\ 21 & 374 528 & 6 & 2 & 2 & 1 & 1 & 2 & . & . & . & . & 1 & 1 & 2 & 1 & 1 & . & . \\ 22 & 374 528 & 6 & 2 & 2 & 1 & 1 & 2 & . & . & . & . & 1 & 1 & 1 & 2 & 1 & . & . \\ 23 & 406 296 & 6 & 4 & 4 & . & . & 1 & . & 2 & . & . & 1 & 1 & 1 & 1 & . & 1 & . \\ 24 & 653 125 & 13 & 7 & 7 & 2 & 2 & 3 & 2 & 4 & . & . & 2 & 2 & . & . & 1 & 1 & . \\ 25 & 656 250 & 12 & 6 & 6 & 1 & 2 & 3 & 1 & 3 & . & . & 1 & 2 & 1 & 1 & 1 & 1 & . \\ 26 & 656 250 & 12 & 6 & 6 & 2 & 1 & 3 & 1 & 3 & . & . & 2 & 1 & 1 & 1 & 1 & 1 & . \\ 27 & 718 200 & 14 & 8 & 9 & 2 & 2 & 2 & 2 & 4 & . & 1 & 2 & 2 & 1 & . & 1 & 1 & . \\ 28 & 718 200 & 14 & 9 & 8 & 2 & 2 & 2 & 2 & 4 & 1 & . & 2 & 2 & . & 1 & 1 & 1 & . \\ 29 & 1 053 360 & 26 & 12 & 12 & 6 & 6 & 7 & 5 & 6 & . & . & 3 & 3 & 1 & 1 & 2 & 1 & . \\ 30 & 1 066 527 & 19 & 11 & 11 & 2 & 2 & 5 & 1 & 5 & 1 & . & 3 & 3 & 1 & 2 & 2 & 1 & . \\ 31 & 1 066 527 & 19 & 11 & 11 & 2 & 2 & 5 & 1 & 5 & . & 1 & 3 & 3 & 2 & 1 & 2 & 1 & . \\ 32 & 1 185 030 & 20 & 12 & 12 & 2 & 2 & 5 & 1 & 4 & 1 & 1 & 3 & 3 & 3 & 3 & 2 & 1 & . \\ 33 & 1 354 320 & 24 & 14 & 14 & 2 & 2 & 6 & 1 & 8 & . & . & 4 & 4 & 1 & 1 & 2 & 2 & . \\ 38 & 1 625 184 & 30 & 16 & 16 & 4 & 4 & 7 & 4 & 10 & . & . & 4 & 4 & 1 & 1 & 2 & 3 & . \\ 39 & 2 031 480 & 14 & 8 & 8 & 2 & 2 & 2 & 2 & 4 & . & . & 2 & 2 & . & . & . & 1 & 1 \\ 40 & 2 375 000 & 20 & 8 & 8 & 4 & 4 & 6 & 2 & 2 & . & . & 3 & 3 & 2 & 2 & 2 & . & 1 \\ 41 & 2 407 680 & 50 & 26 & 26 & 8 & 8 & 13 & 6 & 14 & . & . & 7 & 7 & 2 & 2 & 4 & 3 & . \\ 42 & 2 661 120 & 48 & 28 & 28 & 5 & 5 & 8 & 4 & 14 & 1 & 1 & 7 & 7 & 2 & 2 & 4 & 4 & . \\ 43 & 2 784 375 & 25 & 14 & 14 & 3 & 3 & 5 & 2 & 5 & 1 & 1 & 4 & 4 & 2 & 2 & 2 & 1 & 1 \\ 47 & 3 878 280 & 48 & 27 & 27 & 7 & 7 & 10 & 6 & 12 & 1 & 1 & 7 & 7 & 3 & 3 & 3 & 3 & 1 \\ 48 & 4 156 250 & 52 & 29 & 29 & 7 & 7 & 11 & 5 & 13 & 1 & 1 & 8 & 8 & 3 & 3 & 4 & 3 & 1 \\ 50 & 4 809 375 & 65 & 36 & 36 & 9 & 9 & 14 & 7 & 17 & 1 & 1 & 10 & 10 & 3 & 3 & 5 & 4 & 1 \\ 51 & 5 103 000 & 68 & 38 & 38 & 8 & 8 & 14 & 6 & 18 & 1 & 1 & 10 & 10 & 4 & 4 & 5 & 5 & 1 \\ 52 & 5 103 000 & 68 & 38 & 38 & 8 & 8 & 14 & 6 & 18 & 1 & 1 & 10 & 10 & 4 & 4 & 5 & 5 & 1 \\ 53 & 5 332 635 & 71 & 41 & 41 & 8 & 8 & 16 & 5 & 17 & 2 & 2 & 11 & 11 & 6 & 6 & 6 & 4 & 1 \\ 54 & 5 878 125 & 81 & 47 & 47 & 9 & 9 & 16 & 7 & 22 & 2 & 2 & 12 & 12 & 5 & 5 & 6 & 6 & 1 \end{matrix}

Table 6. The decomposition matrix of the principal 2-block B_{0} of HN

characters, and the block B_{2} of defect d = 1, having k = 3 ordinary and l = 2 modular characters. Moreover, there are nine blocks of defect 0, consisting of the ordinary characters {χ_{15}, χ_{16}, χ_{30}, χ_{31}, χ_{44}, χ_{47}, χ_{51}, χ_{52}, χ_{53}}.

The decomposition matrix of the 3-block B_{2}, reproduced in Table 9, follows immediately from the Brauer-Dade theory of blocks with cyclic defect groups, and has previously been published in [7].

3.1. Proof for the block B_{1}, part 1. We begin with a basic set of projective characters displayed in Table 10, where we also give a basic set of Brauer characters, comprising of the restrictions of the ordinary characters indicated by bold face. The origin of the projective characters is given there as well, where 1_{B_{1}} denotes the associated block idempotent, M_{6} = 5_{+}^{1+4} : 2_{−}^{1+4}.5.4 is the sixth maximal subgroup of G (see [2]), which is a 3′-subgroup, the character φ_{2} is the linear character of M_{6}

GERHARD HISS, JÜRGEN MÜLLER, FELIX NOESKE, AND JON THACKRAY 11

χ  χ(1) Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5} Φ_{6} Φ_{7} Φ_{8} Φ_{9} Φ_{10} Φ_{11} Φ_{12}
1  1  1  .  .  .  .  .  .  .  .  .  .  .
2  1  1  .  .  .  .  .  .  .  .  .  .  .
3  266  2  1  .  .  .  .  .  .  .  .  .  .
4  760  .  .  .  1  .  .  .  .  .  .  .  .
5  760  .  .  .  1  .  .  .  .  .  .  .  .
6  3 344  .  .  .  .  1  .  .  .  .  .  .  .
7  3 344  .  .  .  .  1  .  .  .  .  .  .  .
8  17 556  4  1  2  .  2  .  .  .  .  .  .  .
9  8 910  2  1  1  .  1  .  .  .  .  .  .  .
10  8 910  2  1  1  .  1  .  .  .  .  .  .  .
11  9 405  1  .  1  1  1  .  .  .  .  .  .  .
12  9 405  1  .  1  1  1  .  .  .  .  .  .  .
13  16 929  1  1  .  1  .  1  .  .  .  .  .  .
14  16 929  1  1  .  1  .  1  .  .  .  .  .  .
15  70 224  .  .  .  2  .  .  .  .  1  .  .  .
16  131 670  2  3  .  .  .  .  1  .  1  .  .  .
17  138 510  6  4  2  2  2  2  .  1  .  .  .  .
20  267 520  6  2  2  1  2  2  .  .  .  .  1  .
21  267 520  6  2  2  1  2  2  .  .  .  .  1  .
22  270 864  6  2  2  1  3  2  .  .  .  .  1  .
23  270 864  6  2  2  1  3  2  .  .  .  .  1  .
24  365 750  8  3  2  3  1  1  .  1  1  1  .  .
25  365 750  8  3  2  3  1  1  .  1  1  1  .  .
26  749 056 12  4  2  4  .  .  .  2  3  2  .  .
27  406 296  6  4  .  1  .  2  .  1  1  .  1  .
28  406 296  6  4  .  1  .  2  .  1  1  .  1  .
29  653 125 13  7  2  3  2  4  .  2  .  1  1  .
30  653 125 13  7  2  3  2  4  .  2  .  1  1  .
31 1 312 500 24 12  3  6  2  6  .  3  2  2  2  .
32 1 436 400 28 17  4  4  4  8  1  4  1  2  2  .
33 1 053 360 26 12  6  7  5  6  .  3  1  2  1  .
34 1 053 360 26 12  6  7  5  6  .  3  1  2  1  .
35 2 133 054 38 22  4 10  2 10  1  6  3  4  2  .
36 1 185 030 20 12  2  5  1  4  1  3  3  2  1  .
37 1 185 030 20 12  2  5  1  4  1  3  3  2  1  .
38 1 354 320 24 14  2  6  1  8  .  4  1  2  2  .
39 1 354 320 24 14  2  6  1  8  .  4  1  2  2  .
45 1 625 184 30 16  4  7  4 10  .  4  1  2  3  .
46 1 625 184 30 16  4  7  4 10  .  4  1  2  3  .
47 2 031 480 14  8  2  2  2  4  .  2  .  .  1  1
48 2 031 480 14  8  2  2  2  4  .  2  .  .  1  1
49 2 375 000 20  8  4  6  2  2  .  3  2  2  .  1
50 2 375 000 20  8  4  6  2  2  .  3  2  2  .  1
51 2 407 680 50 26  8 13  6 14  .  7  2  4  3  .
52 2 407 680 50 26  8 13  6 14  .  7  2  4  3  .
53 2 661 120 48 28  5  8  4 14  1  7  2  4  4  .
54 2 661 120 48 28  5  8  4 14  1  7  2  4  4  .
55 2 784 375 25 14  3  5  2  5  1  4  2  2  1  1
56 2 784 375 25 14  3  5  2  5  1  4  2  2  1  1
63 3 878 280 48 27  7 10  6 12  1  7  3  3  3  1
64 3 878 280 48 27  7 10  6 12  1  7  3  3  3  1
65 4 156 250 52 29  7 11  5 13  1  8  3  4  3  1
66 4 156 250 52 29  7 11  5 13  1  8  3  4  3  1
69 4 809 375 65 36  9 14  7 17  1 10  3  5  4  1
70 4 809 375 65 36  9 14  7 17  1 10  3  5  4  1
71 5 103 000 68 38  8 14  6 18  1 10  4  5  5  1
72 5 103 000 68 38  8 14  6 18  1 10  4  5  5  1
73 5 103 000 68 38  8 14  6 18  1 10  4  5  5  1
74 5 103 000 68 38  8 14  6 18  1 10  4  5  5  1
75 5 332 635 71 41  8 16  5 17  2 11  6  6  4  1
76 5 332 635 71 41  8 16  5 17  2 11  6  6  4  1
77 5 878 125 81 47  9 16  7 22  2 12  5  6  6  1
78 5 878 125 81 47  9 16  7 22  2 12  5  6  6  1

\begin{matrix}χ & χ(1) & Φ_{1} & Φ_{2} & Φ_{3} & Φ_{1} \\ 18 & 214 016 & 1 & . & . & . \\ 19 & 214 016 & 1 & . & . & . \\ 40 & 1 361 920 & . & 1 & . & . \\ 41 & 1 361 920 & . & 1 & . & . \\ 42 & 2 723 840 & . & 2 & . & . \\ 43 & 1 575 936 & 1 & 1 & . & . \\ 44 & 1 575 936 & 1 & 1 & . & . \\ 57 & 2 985 984 & . & . & 1 & . \\ 58 & 2 985 984 & . & . & 1 & . \\ 59 & 3 200 000 & 1 & . & 1 & . \\ 60 & 3 200 000 & 1 & . & 1 & . \\ 67 & 4 561 920 & 1 & 1 & 1 & . \\ 68 & 4 561 920 & 1 & 1 & 1 & . \\ 61 & 3 424 256 & . & . & . & 1 \\ 62 & 3 424 256 & . & . & . & 1 \end{matrix}

Table 8. The decomposition matrices of the 2-blocks B_{1} and B_{2} of HN.2

\begin{matrix}χ & χ(1) & Φ_{1} & Φ_{2} \\ 23 & 406 296 & 1 & . \\ 38 & 1 625 184 & . & 1 \\ 39 & 2 031 480 & 1 & 1 \end{matrix}

Table 9. The decomposition matrix of the 3-block B_{2} of HN

of order 2, the character φ_{3} is one of the two linear characters of order 4, and φ_{7} is one of two the rational valued characters of degree 5. Note that the character of the B_{1}-component of Ind_{M_{6}}^{G}(φ_{7}) decomposes into the ordinary characters of B_{1} with even coefficients, hence dividing by 2 still yields a projective character. The remaining projective characters are products of irreducible characters of G, one factor of which is a character of defect zero.

We now pursue a strategy first employed in [18]. Let M′_{6} denote the unique subgroup of M_{6} of index 2. Hence the RG-permutation module on the cosets of M′_{6} is a projective RG-lattice. Put

E := End_{RG}(Ind_{M′_{6}}^{G}(R) · 1_{B_{1}});

note that (Q, R, F) is a splitting system for E. The B_{1}-component of the associated permutation character Ψ := Ind_{M_{6}}^{G}(φ_{1} + φ_{2}), where φ_{1} is the trivial character, decomposes into the ordinary characters of B_{1} as given in Table 10. In particular, Q ⊗_{R} E has eight irreducible characters.

χ  χ(1) Ψ_{1} Ψ_{2} Ψ_{3} Ψ_{4} Ψ_{5} Ψ_{6} Ψ_{7} Ψ
8  8 910 1 . . . . . . 3
10  16 929 . 1 . . . . . 1
19 270 864 . . 1 . . . . .
32 1 185 030 1 1 . 3 1 1 . 5
33 1 354 320 2 2 . 2 2 1 . 4
37 1 575 936 1 2 1 2 1 1 . 1
43 2 784 375 2 2 1 5 3 2 . 5
49 4 561 920 1 2 2 9 6 4 1 3
50 4 809 375 1 3 3 9 5 4 1 4

Ψ_{1} = Ind_{M_{6}}^{G}(φ_{2}) · 1_{B_{1}}, Ψ_{3} = Ind_{M_{6}}^{G}(φ_{3}) · 1_{B_{1}}, Ψ_{5} = (Ind_{M_{6}}^{G}(φ_{7}) · 1_{B_{1}})/2

Ψ_{2} = (χ_{2}·χ_{31})·1_{B_{1}}, Ψ_{4} = (χ_{2}·χ_{47})·1_{B_{1}}, Ψ_{6} = (χ_{5}·χ_{15})·1_{B_{1}}, Ψ_{7} = (χ_{4}·χ_{15})·1_{B_{1}}

Table 10. A first basic set of projective characters for B_{1}

We consider Ē := F ⊗_{R} E, and proceed to determine its regular representation. In practice, a straight line programme to find M_{6} is available in [29]. Using the facilities provided by GAP it is then easy to find M′_{6}, and the permutation representation of G on the 273 030 912 cosets in M′_{6}\G, realised as a G-orbit of vectors in a suitable matrix representation of G. Then the regular representation of Ē is obtained by a direct condense technique, using the orb package [19]; for details see [18].

It turns out that Ē has five simple modules 3a, 1a, 1b, 2c, 2a of the respective dimensions, and the Cartan matrix C of Ē is found to be as given in Table 11. Let D ∈ ℕ_{0}^{8×5} be the decomposition matrix of E. It is easy to check that the matrix equation D^{t}D = C has, up to a permutation of rows, the matrix given in Table 11 as its only solution. (We have already chosen the ordering with some foresight such that a later reordering of rows or columns is unnecessary.)

Hence it remains to show how the rows of the matrix in Table 11 correspond to rows of Table 10. The regular character of E shows that

\begin{matrix}\{1_{1}, 1_{2}\} ↔ \{χ_{10}, χ_{37}\}, & \{3_{1}, 3_{2}\} ↔ \{χ_{8}, χ_{49}\}, \\ \{4_{1}, 4_{2}\} ↔ \{χ_{33}, χ_{50}\}, & \{5_{1}, 5_{2}\} ↔ \{χ_{32}, χ_{43}\}, \end{matrix}

while χ_{19} is not met. The basic set of Table 10 shows in particular that χ′_{8} is an irreducible Brauer character, and the associated projective indecomposable character Φ_{1} is contained three times as a summand in the projective character Ψ. Thus Φ_{3a} is the Fitting correspondent of Φ_{1}, and part of the matching is

3_{1} ↔ χ_{8} and 3_{2} ↔ χ_{49}.

This yields 8 remaining cases. To be a valid matching, the projective indecomposable characters of E have to decompose into the basic set of projective characters in Table 10 with integral coefficients. It is easily checked using GAP that this necessary

\begin{aligned}& Φ_{3a} & Φ_{1a} & Φ_{1b} & Φ_{1c} & Φ_{2a} \\ & 3a & 1a & 1b & 1c & 2a \\ 3_{1} & 1 & . & . & . & . \\ 1_{1} & . & 1 & . & . & . \\ 5_{1} & 1 & 1 & 1 & . & . \\ 4_{1} & 1 & . & . & 1 & . \\ 1_{2} & . & . & . & 1 & . \\ 5_{2} & 1 & . & 1 & 1 & . \\ 3_{2} & . & . & 1 & . & 1 \\ 4_{2} & . & 1 & 1 & . & 1 \end{aligned} C := ( \begin{matrix}4 & 1 & 2 & 2 & . \\ 1 & 3 & 2 & . & 1 \\ 2 & 2 & 4 & 1 & 2 \\ 2 & . & 1 & 3 & . \\ . & 1 & 2 & . & 2 \end{matrix} )

Table 11. Decomposition matrix and Cartan matrix of E

χ  χ(1) Φ_{1} Φ_{2} Ψ_{3} Φ_{4} Φ_{5} Ψ_{6} Φ_{7} Ψ′
8  8 910 1 . . . . . . .
10  16 929 . 1 . . . . . .
19 270 864 . . 1 …. 1
32 1 185 030 1 1 . 1 . 1 . 1
33 1 354 320 1 … 1 2 . .
37 1 575 936 . . 1 . 1 1 . 1
43 2 784 375 1 . 1 1 1 3 . 2
49 4 561 920 . . 2 1 . 6 1 2
50 4 809 375 . 1 3 1 . 5 1 3

Table 12. A second basic set of projective characters for B_{1}

condition is only fulfilled for

1_{1} ↔ χ_{10}, 1_{2} ↔ χ_{37}, 4_{1} ↔ χ_{33}, 4_{2} ↔ χ_{50}, 5_{1} ↔ χ_{32}, 5_{2} ↔ χ_{43}.

This gives five of the seven projective indecomposable characters for block B_{1}, and together with Ψ_{3} and Ψ_{6} we obtain the basic set of projective characters displayed in Table 12.

The projective character Ψ′ given in the last column of Table 12 is obtained as (χ_{2} · χ_{30}) · 1_{B_{1}}. We have

Ψ′ = Ψ_{3} + Φ_{4} − Φ_{7}.

Since Φ_{7} and Φ_{4} are projective indecomposable, this relation implies that

Ψ′_{3} := Ψ_{3} − Φ_{7}

is a projective character. We obtain the new basic set of projective characters displayed in Table 13. ¿From this table it is evident that in order to obtain the projective indecomposable character Φ_{3} contained in Ψ′_{3}, we may subtract Φ_{7} at

\begin{matrix}χ & χ(1) & Φ_{1} & Φ_{2} & Ψ′_{3} & Φ_{4} & Φ_{5} & Ψ_{6} & Φ_{7} \\ 8 & 8 910 & 1 & . & . & . & . & . & . \\ 10 & 16 929 & . & 1 & . & . & . & . & . \\ 19 & 270 864 & . & . & 1 & . & . & . & . \\ 32 & 1 185 030 & 1 & 1 & . & 1 & . & 1 & . \\ 33 & 1 354 320 & 1 & . & . & . & 1 & 2 & . \\ 37 & 1 575 936 & . & . & 1 & . & 1 & 1 & . \\ 43 & 2 784 375 & 1 & . & 1 & 1 & 1 & 3 & . \\ 49 & 4 561 920 & . & . & 1 & 1 & . & 6 & 1 \\ 50 & 4 809 375 & . & 1 & 2 & 1 & . & 5 & 1 \end{matrix}

Table 13. A third basic set of projective characters for B_{1}

most once from Ψ′_{3}. Likewise, we consider the decomposition

Φ_{6} = Ψ_{6} − a · Φ_{4} − b · Φ_{5} − c · Φ_{7}, where 0 ≤ a, b ≤ 1 and 0 ≤ c ≤ 5 − a.

This leaves 44 possibilities for the decomposition matrix of B_{1}. We postpone the proof which of these possibilities holds until after our treatment of the principal block, as with little effort we can eliminate three quarters of these possibilities in parallel to determining the Brauer characters of B_{0}.

3.2. Proof for the principal block B_{0}. As in our treatment of G in characteristic 2, we begin by constructing several small dimensional representations. In [29] we readily find 133_{1} and 133_{2}, which are conjugate under the outer automorphism of G, and 760. Furthermore, restricting the smallest non-trivial irreducible representation of the sporadic Baby Monster group B also available from [29] gives the composition factors

4 371↓_{G} = 1 + 133_{1} + 133_{2} + 760 + 3 344,

all of which are liftable. As the tensor products 133_{1} ⊗ 133_{1} and 133_{1} ⊗ 133_{2} allow a direct treatment with the MeatAxe, we also chop these modules obtaining

133_{1} ⊗ 133_{1} = 1 + 8 778_{1} + 8 910
133_{1} ⊗ 133_{2} = 760 + 16 929

in the process. Choosing basic sets of Brauer characters for B_{0} and B_{1} as indicated in bold face in Tables 13 and 16, respectively, we see that 8 910 and 16 929 afford two of the irreducible Brauer characters of B_{1} already known. Hence, we readily obtain the Brauer character of 8 778_{1}, and applying the outer automorphism to 8 778_{1} gives 8 778_{2}.

To determine the remaining Brauer characters, we employ condensation again. As condensation subgroups we choose K_{1} ≅ 2^{3}.2^{2}.2^{6} of order 2 048, which is the largest normal 2-subgroup of the ninth maximal subgroup M_{9} ≅ 2^{3}.2^{2}.2^{6}.(3×L_{3}(2)) of G, and K_{2} ≅ 5^{2}.5.5^{2} of order 3 125, which is the largest normal 5-subgroup of the tenth maximal subgroup M_{10} ≅ 5^{2}.5.5^{2}.4A_{5} of G (see [2]). A straight line programme to find M_{10} is available in [29]. Following the construction given in [23, Section 3, p. 365] we can similarly procure a straight line programme for the ninth maximal subgroup by computing its intersection with the maximal subgroup A_{12}. The normal subgroups in question are then easily found using the facilities to deal with permutation groups provided by GAP.

The reason for choosing two condensation subgroups becomes evident, if we calculate the dimensions of the condensed simple modules known via their Brauer characters: we see that the corresponding trace idempotent e_{1} annihilates the pairs (133_{1}, 133_{2}) and (8 778_{1}, 8 778_{2}), whereas the trace idempotent e_{2} of the second condensation subgroup K_{2} annihilates 760 and 3 344. Condensing every module with both idempotents allows us to counteract the blind spots produced by condensing with just either of the two.

By [20, Theorem 2.7] condensing the non-identity representatives of the double cosets in M_{9}\G/M_{9} and M_{10}\G/M_{10} together with generators for the maximal subgroups gives generating sets for e_{1}F Ge_{1} and e_{2}F Ge_{2}, consisting of 387 and 643 elements, respectively. Words for the double coset representatives can be computed using orb, realising the permutation representations of G on the 264 515? 625 cosets in M_{9}\G, and the 364 041 216 cosets in M_{10}\G, as G-orbits of vectors in suitable matrix representations of G; again we spare the details of the actual computations.

As working with 1030 generators simultaneously is cumbersome to impossible, we adopt the strategy to consider the algebras C_{1} ≤ e_{1}F Ge_{1} and C_{2} ≤ e_{2}F Ge_{2}, each generated by the condensed elements corresponding to the following six elements

(1)  a, b, ab, ba, a^{2}, aba,

where a and b are standard generators of G. We apply the condensation of tensor products technique described in [12, 16] to the modules listed in Table 14, where the simple module 12 264_{1} which we also use will be constructed in the process, and chop the condensed modules using the six group elements of (1).

For any F G-module M the idea now is to consider a composition series of Me_{i}↓_{C_{i}} for i = 1, 2, which can be obtained with the MeatAxe. Let B_{i} be a basis of Me_{i} adapted to the composition series computed, i.e. with respect to B_{i} the action of elements of C_{i} on Me_{i}↓_{C_{i}} is given by lower block-triangular matrices, for which the block-diagonal gives the representations on the composition factors. We can prove that every composition factor of Me_{i} restricts irreducibly to C_{i} by checking if the lower block-diagonal structure is preserved by all generators of e_{i}F Ge_{i} with respect to the basis B_{i}. Moreover, we check that two composition factors are isomorphic as e_{i}F Ge_{i}-modules if they are as C_{i}-modules. This is applied to the tensor products 760⊗8 778_{1} and 3 344⊗3 344. Thus all C_{1}- and C_{2}-modules appearing as composition factors in the modules can be uniquely extended to e_{1}F Ge_{1}- and e_{2}F Ge_{2}-modules, respectively.

For the identification which simple e_{1}F Ge_{1}-module corresponds to which simple e_{2}F Ge_{2}-module, we now adopt the improved matching method detailed in [22].

Using our basic sets of B_{0} and B_{1}, together with the Brauer characters of B_{2} and the characters of defect zero, it is plain to see which composition factors outside of B_{0} and B_{1} appear in the tensor products 760 ⊗ 8 778_{1} and 3 344 ⊗ 3 344. The dimensions of the their condensed modules are also readily determined. These dimensions are almost sufficient to identify these modules, and thus to omit them immediately from the condensation results, except that there is a constituent of dimension 96 for the condensation subgroup K_{2} being the condensed module of the irreducible Brauer character 406 296 in the block B_{2} of defect 1. Running the matching algorithm shows that this constituent matches with a constituent of dimension 243 for the condensation subgroup K_{1}, hence can be uniquely identified. Moreover, the decompositions of both tensor products into our basic sets of B_{0} and B_{1} (see Table 15) show that the character χ′_{49} of the basic set of Table 13 does not occur in either decomposition. Hence by the results obtained in 3.1, the irreducible Brauer character corresponding to the projective indecomposable Φ_{7} is not a constituent of either of these tensor products. Thus their restrictions to B_{0} ⊕ B_{1} contain at most 26 distinct constituents.

The result of the matching algorithm run on the components in the condensed blocks of B_{0} and B_{1} is given in the second and third columns of Table 14. More precisely, we indeed find the matching between the constituents as shown, where only k1_{1} and k_{9} for the condensation subgroup K_{1}, and c3_{1/2}, c8_{1/2} and c16_{1/2} for the condensation subgroup K_{2} remain ‘unmatched’. This means that either we have missed existing matchings, or there is none because of non-faithful condensation. Since we have already constructed the simple modules 133_{1/2}, 760, 3 344, 8 778_{1/2}, and will do so for the simple modules 12 264_{1/2} later on, we can identify all of the ‘unmatched’ constituents as condensed simple modules, and thus show by using their Brauer characters that there indeed is no matching for them.

Thus we count 26 non-isomorphic simple modules occurring as ‘matched composition factors’ in the condensation results. Hence, all simple modules of the principal block B_{0}, and all but one simple module of B_{1} arise as composition factors in the two tensor products. Furthermore, we have a bijection between these simple F G-modules and the above ‘matched composition factors’.

The strategy now is as follows: We proceed through the columns of Tables 14 and 15, and compare the multiplicities of the ‘matched composition factors’ with the decomposition of the Brauer characters of the tensor products in question into the basic sets of B_{0} and B_{1}. This way we collect successively the information how the basic set characters decompose into irreducible Brauer characters. Using this information, it is straightforward to determine the Brauer characters which correspond to the ‘matched composition factors’. We therefore omit the elementary calculations and only state the individual results:

▶ Condensing the simple modules, which are explicitly known as matrix representations, first reveals

1e_{1} ↦ k1_{2}, 760e_{1} ↦ k1_{1}, 3 344e_{1} ↦ k9, 133_{1}e_{2} ↦ c3_{2}, 133_{2}e_{2} ↦ c3_{1}.

▶ 133_{1} ⊗ 133_{1} and 133_{1} ⊗ 133_{2}: These are condensed to match the composition factors, which we have already determined but not computed as explicit matrix representations:

8 778_{1}e_{2} ↦ c16_{1}, 8 778_{2}e_{2} ↦ c16_{2}, 8 910e_{1} ↦ k27_{1}, 16 929e_{1} ↦ k27_{2}.

▶ 133_{1} ⊗ 3 344: This yields the Brauer characters 31 768_{1} and its conjugate 31 768_{2}:

31 768_{1}e_{1} ↦ k26_{1}, 31 768_{2}e_{1} ↦ k26_{2}.

▶ 133_{1}⊗760: The only basic set character needed to express the Brauer character of 133_{1}⊗760 whose modular constituents are not yet known is χ_{13}, of degree 65 835.

K_{1}  K_{2} 133_{1} ⊗ 133_{1} 133_{1} ⊗ 133_{2} 133_{1} ⊗ 3 344 133_{1} ⊗ 760 760 ⊗ 760 1_{A_{12}}^{G} 760 ⊗ 3 344 1_{2.HS.2}^{G} 133_{1} ⊗ 8 778_{1} 3 344 ⊗ 3 344 133_{1} ⊗ 12 264_{1} 760 ⊗ 8 778_{1}
760  k1_{1}  − . 1 . 1 . 2 2 3 5 16 1 3
1  k1_{2}  c1 1 . . 1 1 2 1 4 1 6 . 3
133_{2}  −  c3_{1} … 1 1 2 . 2 . 4 . 1
133_{1}  −  c3_{2} …. 1 2 . 2 2 4 . 1
9 139  k8  c7 …. 2 3 1 7 2 13 1 3
3 344  k9  − . . 2 1 2 5 4 6 2 15 . 3
12 264_{1}  −  c8_{1} … 1 . 1 . 2 3 10 1 2
12 264_{2}  −  c8_{2} … 1 . 1 . 2 3 10 1 2
31 768_{1} k26_{1}  c8_{3} . . 1 . . 2 2 3 2 9 1 3
31 768_{2} k26_{2}  c8_{4} … 2 . 2 2 3 1 9 . 4
8 778_{1}  − c16_{1} 1 … 1 … 4 4 . .
8 778_{2}  − c16_{2} … 1 1 …. 4 . .
137 236 k50 c40 . . . . . . 1 2 . 4 1 2
147 061 k89 c33 . . . . . . . 1 . 2 1 2
339 702_{2} k138_{1} c96_{2} . . . . . . . . 2 2 . .
339 702_{1} k138_{2} c96_{1} . . . . . . . . . 2 . .
255 037 k173 c89 …. 1 2 . 3 1 6 . .
496 924_{1} k260_{1} c168_{1} . . . . . . . . . . 1 1
496 924_{2} k260_{2} c168_{2} . . . . . . . . . . . 1
783 696 k322 c272 . . . . . . . . . 1 1 1
40 338 k18  c6 . . . . . . . . . . . 1
8 910 k27_{1} c18 1 … 1 1 . 1 . 2 . 1
16 929 k27_{2}  c9 . 1 … 1 . 1 . 2 . 1
270 864 k105 c96 …. 1 . 2 . . 2 . 1
1 159 191 k618 c387 . . . . . . . . . 1 . .
1 305 072 k702 c384 . . . . . . 1 . . 1 . 2

Table 14. Condensation results for p = 3 for HN

Hence we obtain

(2)  χ′_{13} = 1 + 760 + 8 778_{1} + c8_{1} + c8_{2} + 31 768_{1},

where we denote the unknown Brauer characters in this decomposition by their corresponding condensed modules. To determine these Brauer characters, we analyse

\begin{matrix}χ & χ(1) & 133_{1} ⊗ 133_{1} & 133_{1} ⊗ 133_{2} & 133_{1} ⊗ 3 344 & 133_{1} ⊗ 760 & 760 ⊗ 760 & 1_{A_{12}}^{G} & 760 ⊗ 3 344 & 1_{2.HS.2}^{G} & 133_{1} ⊗ 8 778_{1} & 3 344 ⊗ 3 344 & 133_{1} ⊗ 12 264_{1} & 760 ⊗ 8 778_{1} & 3 344 ⊗ 9 139_{1} \\ 1 & 1 & 1 & . & . & . & 1 & 1 & . & 1 & . & 1 & 1 & 1 & 2 \\ 2 & 133 & . & . & . & . & . & 1 & . & . & 1 & . & −2 & −1 & −1 \\ 3 & 133 & . & . & . & 1 & . & 1 & . & . & −1 & . & −2 & −1 & −1 \\ 4 & 760 & . & 1 & . & . & . & 1 & 1 & . & . & . & −1 & . & −2 \\ 5 & 3344 & . & . & 1 & . & 1 & . & 1 & . & −1 & −2 & −1 & −2 & −2 \\ 6 & 8778 & 1 & . & . & −1 & 1 & −1 & . & −1 & 1 & −1 & 1 & . & 1 \\ 7 & 8778 & . & . & . & 1 & 1 & . & . & . & . & 1 & −2 & −1 & −3 \\ 9 & 9405 & . & . & . & . & 1 & 1 & . & 2 & 1 & 4 & 2 & 2 & 3 \\ 11 & 35112 & . & . & 1 & −1 & . & 1 & 1 & . & 1 & 3 & 3 & 2 & 3 \\ 12 & 35112 & . & . & . & 2 & . & 2 & 1 & 1 & 1 & 5 & −1 & 2 & −1 \\ 13 & 65835 & . & . & . & 1 & . & 1 & . & 1 & 1 & 2 & −3 & −1 & −4 \\ 17 & 214016 & . & . & . & . & . & . & 1 & 1 & . & 1 & . & 1 & . \\ 18 & 267520 & . & . & . & . & 1 & 2 & . & 2 & 1 & 4 & . & . & . \\ 21 & 374528 & . & . & . & . & . & . & . & . & . & 2 & 1 & . & 2 \\ 22 & 374528 & . & . & . & . & . & . & . & . & 2 & 2 & 1 & . & 2 \\ 24 & 653125 & . & . & . & . & . & . & . & 1 & . & 2 & . & . & 3 \\ 25 & 656250 & . & . & . & . & . & . & . & . & . & . & 1 & 1 & 1 \\ 26 & 656250 & . & . & . & . & . & . & . & . & . & . & 1 & 1 & 1 \\ 29 & 1053360 & . & . & . & . & . & . & . & . & . & 1 & 1 & 1 & 1 \\ 35 & 1361920 & . & . & . & . & . & . & . & . & . & . & −1 & . & . \\ 8 & 8910 & 1 & . & . & . & 1 & 1 & . & 1 & . & 1 & . & . & . \\ 10 & 16929 & . & 1 & . & . & . & 1 & . & 1 & . & 1 & . & 1 & . \\ 19 & 270864 & . & . & . & . & 1 & . & 1 & . & . & 1 & . & . & 1 \\ 32 & 1185030 & . & . & . & . & . & . & . & . & . & 1 & . & . & 1 \\ 33 & 1354320 & . & . & . & . & . & . & . & . & . & . & . & 1 & . \\ 37 & 1575936 & . & . & . & . & . & . & 1 & . & . & 1 & . & 1 & 1 \\ 49 & 4561920 & . & . & . & . & . & . & . & . & . & . & . & . & 1 \end{matrix}

Table 15. Decompositions into basic sets for B_{0} and B_{1}

the condensed tensor product with the MeatAxe. Its socle series is revealed to be

(3)  \begin{matrix}c8_{4} \\ c8_{2} \\ c16_{2} \\ c8_{1} ⊕ c1 \\ c8_{4} ⊕ c3_{1} \end{matrix} .

Analogously to our approach in Section 2.2, we construct a basis for the uncondensed submodule of Loewy length two, with head c8_{1} and socle c8_{4} ⊕ c3_{1}. With it we construct a representation of the module condensing to c8_{1}, giving 12 264_{1}. As 133_{1} ⊗ 760 is self-dual, but the contragredient of 12 264_{1} gives a new simple module 12 264_{2}, which can be checked with the representation, we can derive from the socle series in (3) that

12 264_{1}e_{2} ↦ c8_{1}, 12 264_{2}e_{2} ↦ c8_{2}.

¿From (2) we explicitly obtain the Brauer character of the sum 12 264_{1} +12 264_{2}. Let σ denote the trivial extension of the nontrivial Galois automorphism of ℚ(\sqrt{5}) to the algebraic number field containing all values of the ordinary characters. The character table of G (see [2]) reveals that the outer automorphism acts on the ordinary characters as the product of σ and complex conjugation. Now, since 12 264_{2} is the contragredient of 12 264_{1}, and both are interchanged by the action of the outer automorphism (which can again be checked with the representations obtained and a straightline programme for the action of the outer automorphism available in [29]), their characters are invariant under the action of the Galois group of ℚ(\sqrt{5}). It therefore follows that 12 264_{1} and 12 264_{2} are rational on all classes except the pairs 19A/B and 40A/B (note that irrationalities on 35A/B only appear as values of defect zero characters). Since we have constructed representations for 12 264_{1}, and straight line programs for conjugacy class representatives of G are available in [29], we can compute representatives for 19A and 40A, and the Brauer character value of the respective representative. We find that 12 264_{1} assumes the value b19 on 19A and i10 on 40A, using the notation in [2]. The values on all classes with rational values are simply obtained by dividing all the values of the character sum 12 264_{1} + 12 264_{2} by 2. Thus, this construction yields the Brauer characters 12 264_{1} and 12 264_{2}.

▶ 760 ⊗ 760 and 1_{A_{12}}^{G}: Considering both columns of Tables 14 and 15 gives the equations

χ′_{9} + χ′_{18} = 133_{1} + 133_{2} + 3 344 + 2 × k8 + k173,
χ′_{9} + 2χ′_{18} = 133_{1} + 133_{2} + 2 × 3 344 + 3 × k8 + 2 × k173.

This allows us to derive the constituents of χ′_{9} and χ′_{18} as

χ′_{9} = 9 139 + 133_{1} + 133_{2},
χ′_{18} = 255 037 + 9 139 + 3 344,

and considering the component of 760 ⊗ 760 in B_{1} we recover the Brauer character 270 864, hence we see that

9 139 e_{1} ↦ k8, 255 037 e_{1} ↦ k173, 270 864 e_{1} ↦ k105.

▶ 760 ⊗ 3 344 and 1_{2.HS.2}^{G} : From 1_{2.HS.2}^{G} we obtain the relation

\begin{aligned}k50 + k89 = χ′_{24} & − 1 − 2 × 9 139 − 2 × 3 344 − 12 264_{1} − 12 264_{2} \\ & − 31 768_{1} − 31 768_{2} − 255 037. \end{aligned}

In particular we derive from this decomposition that both k89 and k50 belong to the principal block.

Now considering 760 ⊗ 3 344, from the decomposition into the basic sets in Table 15 we infer that, since k50 lies in B_{0}, the other occurring unknown simple module k702 lies in B_{1}. With the help of the basic sets we may separate the character of the tensor product restricted to B_{0} and B_{1} into its block components, and obtain via

k50 = χ′_{17} − 1 − 760 − 3 344 − 31 768_{1} − 31 768_{2} − 9 139,
k702 = χ′_{37} − 270 864

the Brauer characters 137 236 of B_{0} and 1 305 072 of B_{1}.

Reconsidering the above relation derived from 1_{2.HS.2}^{G} yields the Brauer character 147 061 of B_{0}, thus

137 236 e_{1} ↦ k50, 147 061 ↦ k89, 1 305 072 e_{1} ↦ k702.

▶ 133_{1} ⊗ 8 778_{1}: Here we obtain the decomposition

χ′_{22} = 2 × 760 + 12 264_{1} + 12 264_{2} + 8 778_{1} + k138_{1}.

Hence we get an irreducible Brauer character of degree 339 702, which we call 339 702_{2} as it appears in the second basic set character of degree 374 528. Applying the outer automorphisms gives 339 702_{1}, hence

339 702_{2} e_{1} ↦ k138_{1}, 339 702_{1} e_{1} ↦ k138_{2}.

▶ 3 344 ⊗ 3 344: Again separating the character into its block components with our basic sets, we see that this tensor product yields one yet unknown simple module of B_{0} and likewise one yet unknown simple module of B_{1}. Thus we obtain a Brauer character 783 696 of B_{0}, and another Brauer character 1 159 191 of B_{1}. Computing the dimensions of the corresponding simple condensed modules with these characters, we see that

783 696 e_{1} ↦ k322, 1 159 191 e_{1} ↦ k618.

▶ 133_{1} ⊗ 12 264_{1}: This tensor product gives the relation

χ′_{25} + χ′_{26} − χ′_{35} = k260_{1} + 1 + 12 264_{1} + 147 061.

Therefore we have a new Brauer character 496 924_{1}. ¿From the latter we obtain its contragredient 496 924_{2}, hence

496 924_{1} e_{1} ↦ k260_{1}, 496 924_{2} e_{1} ↦ k260_{2}.

Thus we have determined all irreducible Brauer characters of B_{0} at this stage, and we give the 3-modular decomposition matrix of this Block in Table 16.

▶ 760⊗8 778_{1}: Finally, considering the last column of Table 14 we obtain another Brauer character of B_{1}: we derive the decomposition

χ′_{33} = k18 + 1 305 072 + 8 910,

which immediately yields the Brauer character 40 338, hence

40 338 e_{1} ↦ k18.

22 MODULAR CHARACTERS OF THE HARADA-NORTON GROUP

χ  χ(1) Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5} Φ_{6} Φ_{7} Φ_{8} Φ_{9} Φ_{10} Φ_{11} Φ_{12} Φ_{13} Φ_{14} Φ_{15} Φ_{16} Φ_{17} Φ_{18} Φ_{19} Φ_{20}
1  1  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
2  133  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
3  133  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
4  760  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
5  3 344  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
6  8 778  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .
7  8 778  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .
9  9 405  .  1  1  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .
11  35 112  .  .  .  .  1  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .
12  35 112  .  .  .  .  1  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .
13  65 835  1  .  .  1  .  1  .  .  1  1  1  .  .  .  .  .  .  .  .  .
14  65 835  1  .  .  1  .  .  1  .  1  1  .  1  .  .  .  .  .  .  .  .
17  214 016  1  .  .  1  1  .  .  1  .  .  1  1  1  .  .  .  .  .  .  .
18  267 520  .  .  .  .  1  .  .  1  .  .  .  .  .  .  1  .  .  .  .  .
20  365 750  1  1  1  1  1  .  .  2  1  1  1  1  .  .  1  .  .  .  .  .
21  374 528  .  .  .  2  .  .  1  .  1  1  .  .  .  .  .  1  .  .  .  .
22  374 528  .  .  .  2  .  1  .  .  1  1  .  .  .  .  .  .  1  .  .  .
24  653 125  1  .  .  1  2  .  .  2  1  1  1  1  1  1  1  .  .  .  .  .
25  656 250  1  .  .  .  .  .  .  .  1  .  .  .  .  1  .  .  .  1  .  .
26  656 250  1  .  .  .  .  .  .  .  .  1  .  .  .  1  .  .  .  .  1  .
27  718 200  .  .  1  2  2  .  1  2  1  1  1  1  .  .  1  1  .  .  .  .
28  718 200  .  1  .  2  2  1  .  2  1  1  1  1  .  .  1  .  1  .  .  .
29 1 053 360  .  .  .  3  .  1  1  .  2  2  1  1  1  .  .  .  .  .  .  1
34 1 361 920  .  .  .  3  3  .  .  2  2  2  1  1  1  1  1  1  1  .  .  .
35 1 361 920  .  .  .  2  1  .  .  1  1  1  .  .  .  1  .  1  1  .  1  .
36 1 361 920  .  .  .  2  1  .  .  1  1  1  .  .  .  1  .  1  1  1  .  .
40 2 375 000  2  1  1  3  3  .  .  3  2  2  1  1  .  2  1  1  1  1  1  .
41 2 407 680  2  1  1  1  1  .  .  2  2  2  2  2  1  2  .  .  .  1  1  1
42 2 661 120  2  1  1  4  2  .  .  4  3  3  1  1  .  2  2  1  1  1  1  .
45 3 200 000  1  1  1  3  2  .  .  2  3  3  1  1  1  3  .  1  1  1  1  1
46 3 424 256  2  1  1  5  2  1  1  3  4  4  2  2  1  2  1  1  1  1  1  1
48 4 156 250  1  .  .  7  4  1  1  2  5  5  2  2  2  3  .  2  2  1  1  1
54 5 878 125  2  1  1  9  5  1  1  3  6  6  2  2  1  4  .  3  3  2  2  1

\begin{matrix}χ & χ(1) & Φ_{1} & Φ_{2} & Φ_{3} & Φ_{4} & Φ_{5} & Φ_{6} & Φ_{7} \\ 8 & 8 910 & 1 & . & . & . & . & . & . \\ 10 & 16 929 & . & 1 & . & . & . & . & . \\ 19 & 270 864 & . & . & 1 & . & . & . & . \\ 32 & 1 185 030 & 1 & 1 & . & 1 & . & . & . \\ 33 & 1 354 320 & 1 & . & . & . & 1 & 1 & . \\ 37 & 1 575 936 & . & . & 1 & . & 1 & . & . \\ 43 & 2 784 375 & 1 & . & 1 & 1 & 1 & 1 & . \\ 49 & 4 561 920 & . & . & . & 1 & . & 1 & 1 \\ 50 & 4 809 375 & . & 1 & 1 & 1 & . & . & 1 \end{matrix}

Table 17. The decomposition matrix of the 3-block B_{1} of HN

3.3. The proof for the block B_{1}, part 2. At the end of Section 3.1 we were left with 44 possible decomposition matrices. Having determined all but one irreducible Brauer character of B_{1}, it is now elementary that a = 1 = b, and thus only ten cases remain.

To obtain more information on the last unknown Brauer character of B_{1}, we condense the tensor product 3 344 ⊗ 9 139, which may be done over the field with three elements, making the computation more efficient. As we have complete knowledge of all other blocks, it is trivial to only consider the part of the tensor product lying in B_{1}. It is furthermore sufficient to only condense with respect to the condensation subgroup K_{2}. We obtain the composition factors

8 910, 16 929, 40 338, 2 × 270 864, 2 × 1 159 191, 1 305 072, c1 047,

where c1 047 may possibly be just a composition factor of the condensed last simple module restricted to C_{2}. A comparison with the basic set of B_{1} (see the last column of Table 15) yields the relation

χ′_{49} = 40 338 + 1 159 191 + c1 047,

from which we obtain a Brauer atom of degree 3 362 391. The latter is a lower bound for the degree of the last irreducible Brauer character. Inspecting its possible degrees given by the ten possible decomposition matrices, we see that the obtained lower bound is in fact maximal, that is we have c = 4 and Φ_{3} = Ψ′_{3} − Φ_{7}. Hence the atom is the character sought, and we give the decomposition matrix of B_{1} in Table 17.

3.4. The Automorphism Group in Characteristic 3. The characters of the automorphism group G.2 of G fall into seventeen 3-blocks, five of which have positive defect: the principal block B_{0} of defect d = 6, having k = 42 ordinary and l = 22 modular characters, two blocks of defect d = 2, each having k = 9 ordinary and l = 7 modular characters, covering the block B_{1} of defect 2 of G, and two blocks of defect d = 1, each having k = 3 ordinary and l = 2 modular characters, covering the block B_{2} of defect 1 of G.

As G is a normal subgroup of G.2, by Clifford’s theorem the irreducible Brauer characters of G.2 come in two flavours: if an irreducible Brauer character of G is not invariant under the action of the outer automorphism, then induction to G.2 yields an irreducible Brauer character; if the character is invariant, then it possesses two extensions to G.2, where one extension may be derived from the other by changing the signs of its values on the outer classes. As it turns out, we may distinguish both extensions by the sign of their value on the outer class 2C. Hence, denoting a Brauer character by its degree, we additionally affix a superscript + or − to identify the extension.

As each of the blocks B_{1} and B_{2} of G is covered by two blocks of the automorphism group G.2 of the same defect, restriction defines a Morita equivalence between such a block of G.2 and B_{1} respectively B_{2} of G. Therefore the decomposition matrix for either block of defect 2 is given in Table 17, and the decomposition matrix of either block of defect 1 is given in Table 9. We may therefore focus on the principal block B_{0} of G.2.

Owing to the abundance of pairs of characters conjugate under the outer automorphism, most Brauer characters of B_{0} are immediately obtained through induction. We must therefore only determine the extensions of the invariant Brauer characters.

¿From Section 3 it is immediate that the invariant Brauer characters 1, 760, and 3 344 of G are liftable. Furthermore, we may deduce that the restriction of the ordinary character 9 405^{±} has the constituents 266 and 9 139^{±}. Therefore we only have to establish the extensions of the four Brauer characters 137 236, 147 061, 255 037, and 783 696 of G. Of course, one extension determines the other, and therefore it suffices to just determine one member of every pair.

This is achieved by using condensation again. For the automorphism group, we choose the same setup as for the simple group, which allows us to use the computational infrastructure already in place. In other words, owing to our previous work in Section 3.2, we can again use the six elements of (1) in G plus one element α of the class 2C of G.2, which is a member of a set of standard generators of G.2 (see [29]). Let D_{i} ≤ e_{i}F Ge_{i} be the algebra generated by these elements, hence we have C_{i} ≤ D_{i}. As our condensation subgroups are contained in G we have V e_{i}↓_{e_{i}F Ge_{i}} = (V ↓_{G})e_{i}↓_{e_{i}F Ge_{i}} for any F [G.2]-module V. Hence, the dimensions of condensed modules are clear from the onset.

Moreover, if a simple F G-module S is not invariant, that is there is a simple F Gmodule T ≅ S̸ being the image of S under the outer automorphism, and S↑^{G.2} is simple, then we have ((S↑^{G.2})e_{i}↓_{D_{i}})↓_{C_{i}} = Se_{i}↓_{C_{i}} ⊕ T e_{i}↓_{C_{i}}, where Se_{i}↓_{C_{i}} ≅ T̸ e_{i}↓_{C_{i}}. Thus (S↑^{G.2})e_{i}↓_{D_{i}} either is simple, of dimension twice the dimension of Se_{i}↓_{C_{i}}, or has two non-isomorphic constituents extending Se_{i}↓_{C_{i}} and T e_{i}↓_{C_{i}}, respectively. If S is invariant and it extends to G.2 as S^{±}, then (S^{±}e_{i}↓_{D_{i}})↓_{C_{i}} = Se_{i}↓_{C_{i}} implies that both S^{±}e_{i}↓_{D_{i}} are simple. Moreover, since the actions of α on S^{±} differ just by sign, we conclude that the actions of e_{i}αe_{i} on S^{±}e_{i} also differ by sign only. Hence e_{i}αe_{i} acts by the zero map on S^{+}e_{i} if and only if it does so on S^{−}e_{i}, and in this case we have S^{+}e_{i}↓_{D_{i}} ≅ S^{−}e_{i}↓_{D_{i}}. Otherwise we have S^{+}e_{i}↓_{D_{i}} ≅ S̸^{−}e_{i}↓_{D_{i}}, as by Schur’s Lemma any isomorphism is scalar and hence commutes with the action of e_{i}αe_{i}. Thus to decide which case occurs we only have to consider the action of e_{i}αe_{i} on either of S^{±}e_{i}.

The modules considered are given in Table 18, all of whose restrictions to G have already been considered in Table 14, where in Table 18 we only give the composition factors lying in the principal block, since restriction to B_{0} of the condensation results

K_{1} K_{2} 760^{+} ⊗ 760^{+} 760^{+} ⊗ 3 344^{+} 1_{4.HS.2}^{G.2} 3 344^{+} ⊗ 3 344^{+}
1^{+}  k1′_{1} c1′_{1} 1 . 4 5
1^{−}  k1′_{2} c1′_{2} . 1 . 1
266  −  c6′ 1 . 2 4
760^{+}  k1′_{3}  − . 2 1 5
760^{−}  k1′_{4}  − . . 2 11
3 344^{+}  k9′_{1}  − 2 1 6 11
3 344^{−}  k9′_{2}  − . 3 . 4
9 139^{+}  k8′_{1} c7′_{1} 2 . 5 8
9 139^{−}  k8′_{2} c7′_{2} . 1 2 5
17 556  − c32′ 1 . . 4
24 528  − c16′_{1} . . 2 10
63 536 k52′ c16′_{2} . 2 3 9
137 236^{+} k50′_{1} c40′_{1} . . 2 4
137 236^{−} k50′_{2} c40′_{2} . 1 . .
147 061^{+} k89′ c33′ . . 1 2
147 061^{−}  ….
255 037^{+} k173′_{1} c89′_{1} 1 . 3 5
255 037^{−} k173′_{2} c89′_{2} … 1
679 404 k276′ c192′ … 2
783 696^{+}  ….
783 696^{−} k322′ c272′ … 1
993 848  ….

Table 18. Condensation results for p = 3 for HN.2

is easily achieved. Thus the matching between the simple modules of the condensed algebras D_{1} and D_{2} can be derived from the matching obtained in Section 3.2. We moreover see that (S↑^{G.2})e_{i}↓_{D_{i}} is simple for all non-invariant simple F G-modules occurring, and S^{±}e_{i}↓_{D_{i}} is simple for all invariant simple F G-modules occurring. Furthermore, we can conclude from the condensation results that S^{+}e_{i}↓_{D_{i}} and S^{−}e_{i}↓_{D_{i}} are non-isomorphic. The matching of a condensed module to a member of a pair of extensions follows from the characters of the considered modules and through the subsequent analysis of the columns of Table 18 as follows.

The representations needed to condense the modules of Table 18 are again obtained by restricting the 4 371-dimensional representation of the Baby Monster group B to G.2:

4 371↓_{G.2} = 1^{−} + 266 + 760^{−} + 3 344^{−},

from which we immediately obtain 760^{+} and 3 344^{+}.

▶ To identify the condensed modules belonging to 760^{+} and 3 344^{+}, we condense them individually to reveal that

760^{+}e_{1} ↦ k1′_{3}, 3 344^{+}e_{1} ↦ k9′_{1};

to distinguish the composition factors found here from earlier ones notationally we add a dash.

▶ From the character of 760^{+} ⊗ 760^{+} it is clear that the extension 255 037^{+} is a constituent: The restriction to B_{0} of 760^{+}⊗760^{+} is the sum χ′_{1}+χ′_{6}+χ′_{8}+χ′_{11}+χ′_{20}. By the previous section and the above we know that χ′_{20} has as its constituents 3 344^{+} and extensions of 255 037, and 9 139. As χ′_{20} takes the value 384 on the class 2C, and 3 344^{+} and 9 139^{±} take the values 36 and ±51, respectively, the consituent of χ′_{20} of degree 255 037 also assumes a positive value on 2C, i.e. it is 255 037^{+}.

Hence for the component in B_{0}, Table 18 leaves the two possibilities

1^{+} + 266 + 2 × 3 344^{+} + 17 556 + 2 × 9 139^{±} + 255 037^{+},

i.e., either 9 139^{+} or 9 139^{−} is a constituent. Assuming 9 139^{−} is, it turns out that the character χ′_{20} cannot be written as a linear combination of the known Brauer characters with non-negative integer coefficients. Hence, 9 139^{+} is a constituent, from which we obtain 255 037^{+} explicitly. Furthermore, we identify the condensed modules

9 139^{+}e_{1} ↦ k8′_{1}, 255 037^{+}e_{1} ↦ k173′_{1}.

▶ Each of the three modules 760^{+} ⊗ 3 344^{+}, 1_{4.HS.2}^{G.2} and 3 344^{+} ⊗ 3 344^{+} only introduces one new composition factor which is an extension of a simple F G-module. Hence we may easily deduce their corresponding Brauer characters 137 236^{−}, 147 061^{+}, and 783 969^{−}, where

137 236^{−}e_{1} ↦ k50′_{2}, 147 061^{+}e_{1} ↦ k89′, 783 969^{−}e_{1} ↦ k322′.

The decomposition matrix for the principal 3-block of G.2 is given in Table 19.

GERHARD HISS, JÜRGEN MÜLLER, FELIX NOESKE, AND JON THACKRAY 27

χ  χ(1) Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5} Φ_{6} Φ_{7} Φ_{8} Φ_{9} Φ_{10} Φ_{11} Φ_{12} Φ_{13} Φ_{14} Φ_{15} Φ_{16} Φ_{17} Φ_{18} Φ_{19} Φ_{20} Φ_{21} Φ_{22}
1  1  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
2  1  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
3  266  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
4  760  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
5  760  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
6  3 344  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
7  3 344  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
8  17 556  .  .  .  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .
11  9 405  .  .  1  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .
12  9 405  .  .  1  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .
15  70 224  .  .  .  .  .  1  1  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .
16  131 670  1  1  .  1  1  .  .  .  .  1  2  1  .  .  .  .  .  .  .  .  .  .
18  214 016  1  .  .  .  1  1  .  1  .  .  .  1  1  .  .  .  .  .  .  .  .  .
19  214 016  .  1  .  1  .  .  1  .  1  .  .  1  .  1  .  .  .  .  .  .  .  .
20  267 520  .  .  .  .  .  1  .  1  .  .  .  .  .  .  .  .  1  .  .  .  .  .
21  267 520  .  .  .  .  .  .  1  .  1  .  .  .  .  .  .  .  .  1  .  .  .  .
24  365 750  1  .  1  1  .  1  .  1  1  .  1  1  .  .  .  .  1  .  .  .  .  .
25  365 750  .  1  1  .  1  .  1  1  1  .  1  1  .  .  .  .  .  1  .  .  .  .
26  749 056  .  .  .  2  2  .  .  .  .  1  2  .  .  .  .  .  .  .  1  .  .  .
29  653 125  1  .  .  .  1  2  .  1  1  .  1  1  1  .  1  .  1  .  .  .  .  .
30  653 125  .  1  .  1  .  .  2  1  1  .  1  1  .  1  .  1  .  1  .  .  .  .
31 1 312 500  1  1  .  .  .  .  .  .  .  .  1  .  .  .  1  1  .  .  .  .  .  1
32 1 436 400  .  .  1  2  2  2  2  2  2  1  2  2  .  .  .  .  1  1  1  .  .  .
33 1 053 360  .  .  .  3  .  .  .  .  .  1  2  1  .  1  .  .  .  .  .  1  .  .
34 1 053 360  .  .  .  .  3  .  .  .  .  1  2  1  1  .  .  .  .  .  .  .  1  .
40 1 361 920  .  .  .  1  2  2  1  1  1  .  2  1  1  .  1  .  1  .  1  .  .  .
41 1 361 920  .  .  .  2  1  1  2  1  1  .  2  1  .  1  .  1  .  1  1  .  .  .
42 2 723 840  .  .  .  2  2  1  1  1  1  .  2  .  .  .  1  1  .  .  2  .  .  1
49 2 375 000  1  1  1  2  1  2  1  2  1  .  2  1  .  .  1  1  1  .  1  .  .  1
50 2 375 000  1  1  1  1  2  1  2  1  2  .  2  1  .  .  1  1  .  1  1  .  .  1
51 2 407 680  1  1  1  .  1  1  .  .  2  .  2  2  1  .  2  .  .  .  .  1  .  1
52 2 407 680  1  1  1  1  .  .  1  2  .  .  2  2  .  1  .  2  .  .  .  .  1  1
53 2 661 120  1  1  1  2  2  1  1  2  2  .  3  1  .  .  1  1  1  1  1  .  .  1
54 2 661 120  1  1  1  2  2  1  1  2  2  .  3  1  .  .  1  1  1  1  1  .  .  1
59 3 200 000  .  1  1  2  1  1  1  .  2  .  3  1  .  1  2  1  .  .  1  1  .  1
60 3 200 000  1  .  1  1  2  1  1  2  .  .  3  1  1  .  1  2  .  .  1  .  1  1
61 3 424 256  1  1  1  2  3  2  .  1  2  1  4  2  1  .  2  .  1  .  1  1  .  1
62 3 424 256  1  1  1  3  2  .  2  2  1  1  4  2  .  1  .  2  .  1  1  .  1  1
65 4 156 250  .  1  .  4  3  2  2  1  1  1  5  2  1  1  2  1  .  .  2  1  .  1
66 4 156 250  1  .  .  3  4  2  2  1  1  1  5  2  1  1  1  2  .  .  2  .  1  1
77 5 878 125  1  1  1  4  5  3  2  1  2  1  6  2  1  .  3  1  .  .  3  1  .  2
78 5 878 125  1  1  1  5  4  2  3  2  1  1  6  2  .  1  1  3  .  .  3  .  1  2

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Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

E-mail address: G.H.:  Gerhard.Hiss@math.rwth-aachen.de
E-mail address: J.M.:  Juergen.Mueller@math.rwth-aachen.de
E-mail address: F.N.:  Felix.Noeske@math.rwth-aachen.de
E-mail address: J.T.:  jgt@pobox.com