Nikolaus 2004 The 2 modular characters of the Conway group Co1
Title:The 2 modular characters of the Conway group Co1
Author:Jon Thackray
Version:1
Date:20041209
Status:Draft

Contents

Section 1 : Purpose of the talk

Section 2 : Background

Section 3 : Demonstration of results

Section 4 : Some further thoughts

Section 5 : Status of results

Section 6 : The (partial) decomposition matrix for the principal block

Section 7 : References

Section 8 : Acknowledgements

Section 1 : Purpose of the talk

To describe present a set of results giving the 2 modular character table of Conway's first group, suitable for inclusion the the forthcoming Atlas of Brauer Characters Part 2.

Section 2 : Background

Conway's first group is described in [Conway]. Throughout this paper, it will be denoted Co1. The character table is given in [Conway et al] The representations used as a starting point are taken from the Birmingham Atlas of Finite Group Representations [Wilson].

At the point of starting on the problem, some small representations of Co1 were already known, as well a block of defect three containing three irreducibles. These are considered known, and will not be described in this talk. The talk will address only the principal block, in which we are seeking twenty six irreducibles.

At this stage the results are incomplete. The determination of the composition factors of one tensor condensation is ongoing. There are seven other results which are condensation only. There are a further three results which combine condensation and character theory to yield an existence proof (the character theory implies a decomposition which the condensation proves to be the most that can occur). The remaining fifteen irreducibles have all been constructed and their indicators computed, and for seven of these of indicator plus the orthogonal group has also been determined. Two further indicators are known, as two of the three ireducibles proved by condensation constitute a complex conjugate pair.

It is hoped that the final condensation conjecture will be known by around the middle of 2005.

Section 3 : Demonstration of results

Our first character ϕ1 is the trivial character 1. Co1 has a double cover 2.Co1 with a 24 dimensional real representation, which reduces modulo 2 to a 2 modular irreducible of the simple group. This is our second character ϕ2 Since this representation is real, its skew square 276 has a trivial submodule, and hence a composition factor 274, which can be seen to be irreducible by restriction to Co2 and to 3.Suz.2. This gives ϕ3

ϕ2ϕ3 decomposes as 2000 ⊕ 4576. This can be demonstrated theoretically using symmetrised power decomposition. But the meataxe proves the result anyway. This gives two new characters ϕ5 and ϕ6 The skew fourth of ϕ2 has composition factors 2.ϕ1 + 2.ϕ3 + 1496 + 8580. We thus have two further characters ϕ4 and ϕ7 ϕ2ϕ4 has composition factors 17952 and 17952 ¯¯¯¯¯ , giving ϕ8 and ϕ9. These characters have complex values on elements of orders 23 and 39, which can be computed by taking ranks of suitable sums of element powers.

The skew square of ϕ3 has composition factors 3.ϕ1 + ϕ2 + 2. ϕ3 + ϕ4 + ϕ7 + 26750. 26750 is ϕ10. The skew sixth of ϕ2 has composition factors 8.ϕ1 + 2.ϕ2 + 8.ϕ3 + 3.ϕ4 + 2.ϕ7 + 2.ϕ10 + 57200. 57200 is ϕ11.

ϕ2ϕ7 has composition factors ϕ6 + ϕ8 + ϕ9 + 165440. 165440 is ϕ12.

Analysis of the condensed permutation representation of Co2 on 3.Suz.2 yields two new characters ϕ13 and ϕ16. Uncondensing reveals the degree of ϕ13 to be 218800. ϕ16 can then be discovered to have degree 420256. This representation has not been constructed.

Using condensed tensor analysis, we find ϕ3ϕ4 has composition factors 2.ϕ1 + 2.ϕ2 + 4.ϕ3 + 2.ϕ7 + ϕ10 + 2.ϕ11 + 250448. 250448 is ϕ14. Also ϕ3ϕ5 has composition factors 2.ϕ2 + 4.ϕ6 + 3.ϕ8 + 3.ϕ9 + ϕ12 + 256496. 256496 is ϕ15. ϕ14 and ϕ15 are the last characters to have been constructed.

Using condensed tensor analysis, we find ϕ2ϕ11 has composition factors ϕ17 + ϕ18 + ϕ19. Character theory shows that at least this much decomposition occurs, hence each of thes is irreducible. Further, again from character theory, ϕ17 and ϕ18 are a complex conjugate pair. We obtain an irreducible subspace of the condensed tensor product, and analyse it using condensed generators of Co3 to determine the degree of ϕ17, and hence of the other two. These are the last characters currently known to exist. The remainder of these results are from tensor condensation only. They therefore assume that the condensation algebra is equal to the full Hecke algebra.

Decomposing the condensed tensor product ϕ3ϕ10 we obtain 26.ϕ1 + 14.ϕ2 + 28.ϕ3 + 6.ϕ4 + 4.ϕ5 + 4.ϕ6 + 6.ϕ7 + 3.ϕ8 + 3.ϕ9 + 7.ϕ10 + 8.ϕ11 + ϕ12 + 4.ϕ13 + 2.ϕ14 + 2.ϕ16 + ϕ20. From this we conjecture an irreducible ϕ20 of degree 4100096.

Decomposing the condensed tensor product ϕ4ϕ7 we obtain 28.ϕ1 + 12.ϕ2 + 26.ϕ3 + 6.ϕ4 + 3.ϕ5 + 2.ϕ6 + 6.ϕ7 + ϕ8 + ϕ9 + 7.ϕ10 + 6.ϕ11 + 4.ϕ13 + ϕ14 + 2.ϕ16 + ϕ20 + ϕ21. From this we conjecture an irreducible ϕ21 of degree 6120022.

Decomposing the condensed tensor product ϕ3ϕ11 we obtain 16.ϕ1 + 6.ϕ2 + 12.ϕ3 + 2.ϕ4 + 2.ϕ5 + 2.ϕ6 + 2.ϕ7 + 2.ϕ8 + 2.ϕ9 + 2.ϕ10 + 4.ϕ11 + 4.ϕ13 + 3.ϕ16 + ϕ21 + ϕ22. From this we conjecture an irreducible ϕ22 of degree 7025950.

Decomposing the condensed tensor product ϕ5ϕ7 we obtain 38.ϕ1 + 34.ϕ2 + 40.ϕ3 + 14.ϕ4 + 16.ϕ5 + 30.ϕ6 + 14.ϕ7 + 15.ϕ8 + 15.ϕ9 + 7.ϕ10 + 5.ϕ12 + 2.ϕ14 + 8.ϕ15 + 5.ϕ17 + 5.ϕ18 + 7.ϕ19 + ϕ25. From this we conjecture an irreducible ϕ25 of degree 4634432.

Decomposing the condensed tensor product ϕ4ϕ8 we obtain 44.ϕ1 + 36.ϕ2 + 50.ϕ3 + 16.ϕ4 + 13.ϕ5 + 24.ϕ6 + 18.ϕ7 + 14.ϕ8 + 13.ϕ9 + 11.ϕ10 + 4.ϕ11 + 2.ϕ12 + 2.ϕ14 + 6.ϕ15 + 3.ϕ17 + 3.ϕ18 + 4.ϕ19 + ϕ23 + 2.ϕ25. From this we conjecture an irreducible .

Decomposing the condensed tensor product ϕ4ϕ8 we obtain 44.ϕ1 + 36.ϕ2 + 50.ϕ3 + 16.ϕ4 + 13.ϕ5 + 24.ϕ6 + 18.ϕ7 + 14.ϕ8 + 13.ϕ9 + 11.ϕ10 + 4.ϕ11 + 2.ϕ12 + 2.ϕ14 + 6.ϕ15 + 3.ϕ17 + 3.ϕ18 + 4.ϕ19 + ϕ23 of degree 9144846.

Decomposing the condensed tensor product ϕ4ϕ11 we obtain 144.ϕ1 + 72.ϕ2 + 142.ϕ3 + 32.ϕ4 + 22.ϕ5 + 30.ϕ6 + 38.ϕ7 + 20.ϕ8 + 20.ϕ9 + 32.ϕ10 + 29.ϕ11 + 6.ϕ12 + 16.ϕ13 + 6.ϕ14 + 6.ϕ15 + 8.ϕ16 + 2.ϕ17 + 2.ϕ18 + 3.ϕ19 + 2.ϕ21 + 2.ϕ24. From this we conjecture an irreducible ϕ24 of degree 27621392.

The condensed tensor product ϕ5ϕ11 has not been complete created yet. The decomposition so far, consistent with character theory and tensor products is 72.ϕ1 + 49.ϕ2 + 82.ϕ3 + 24.ϕ4 + 17.ϕ5 + 27.ϕ6 + 30.ϕ7 + 15.ϕ8 + 15.ϕ9 + 19.ϕ10 + 8.ϕ11 + 5.ϕ12 + 4.ϕ14 + 4.ϕ15 + 4.ϕ17 + 4.ϕ18 + 7.ϕ19 + 4.ϕ23 + 3.ϕ25 + ϕ26 From this we conjecture an irreducible ϕ26 of degree ≤ 51741858.

Section 4 : Some further thoughts

The tensor condensations in the above section have resulted in some very large matrices, which take a long time to produce and are hard to reduce. For example, the condensed tensor product ϕ5ϕ11 has degree 470296 and size around 27.5Gb. Each matrix takes around 90 days to produce on a P4/3400, and a correspondingly long time to split. I wish to examine some possibilities for reducing the size of the matrices, and hence as a consequence the split time.

The degree of the condensed matrices is roughly equal to the uncondensed degree divided by the condensation subgroup order. So why don't we just pick a larger condensation subgroup? The answer is that the larger subgroups (of McL at least) are unfaithful, ie there is an irreducible representation which condeses to zero. In this case the culprit is 1496.

But, if we are merely interested in the multiplicities, and we had an alternative means to determine the missing multiplicities, we could then make use of a larger condensation subgroup. For example, the full Sylow 3 subgroup of McL is faithful on every representation except 1496, and would lead to condesned matrices of degree around 160000.

Condensation relies on looking at the subspace fixed by the condensation subgroup, under the action of a suitably smaller algebra. If M is an irreducible of the group algebra kG, and e the central idempotent formed from the sum over the elements of the condensation subgroup H, then our condensation space is Me acted on by ekGe. But Me is just the subspace in the direct sum decomposition of M as a kH module all of whose summands are isomorphic to the trivial representation.

So, what would happen if we chose a different central idempotent f? The first obvious consequence is that we no longer have control of the multiplicity of the trivial representation. But, if we also condensed using the original e, we could regain this information. In terms of our original example, we would gain a factor of nine, only to lose a factor of two on the matrix construction. But the reduction process is cubic in the degree, so here we would gain twenty seven divided by two.

However, if we choose an idempotent associated with a non-linear character, we could well lose any possible advantage due to higher multiplicity of the irreducible direct summand, coupled with large degree. So, initially I'd prefer to consider only non-trivial linear characters. A further complication is that condensing over GF(2), the trivial character is the only linear character. So, we might also have to consider moving to GF(4) (which for the Sylow 3 subgroup would give a number of non-trivial linear characters). We'd still get a factor of about two (or possibly better, since only the non-trivial character condensation needs to be done in the field extension) on overall space requirements, and a factor around six on reduction speed. With larger odd order subgroups greater savings might be made.

The next question that needs to be answered is how do we make these non-trivial character condensations? I think that for permutation condensations, we need to compute the relevant idempotent (I believe there are formulae from Schur for these). For tensor condensation, I think we need a permutation of the symmetry basis for the right hand component. Essentially, instead of taking the basis in the order of the duals of the irreducbles of the condensation subgroup, I believe we now need dual ⊗ λ, where λ is our non-trivial linear character. Note that in this case, the tensor product is irreducible (another advantage of sticking to linear characters).

Food for thought!

Section 5 : Status of results

Degree Proved Constructed Indicator Group sign Conjectured Incomplete
ϕ1 1 +
ϕ2 24 + +
ϕ3 274 + -
ϕ4 1496 + +
ϕ5 2000 + +
ϕ6 4576 + +
ϕ7 8580 + -
ϕ8 17952 o
ϕ9 17952 o
ϕ10 26750 + +
ϕ11 57200 +
ϕ12 165440 +
ϕ13 218800 +
ϕ14 250448 +
ϕ15 256496 +
ϕ16 420256
ϕ17 378016 o
ϕ18 378016 o
ϕ19 616768
ϕ20 4100096
ϕ21 6120022
ϕ22 7025950
ϕ23 9144846
ϕ24 27621392
ϕ25 4634432
ϕ26 ≤ 51741858
ϕ27 40370176 +
ϕ28 1507328000 +
ϕ29 313524224 +

Section 6 : The (partial) decomposition matrix for the principal block

ϕ1 ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ7 ϕ8 ϕ9 ϕ10 ϕ11 ϕ12 ϕ13 ϕ14 ϕ15 ϕ16 ϕ17 ϕ18 ϕ19 ϕ20 ϕ21 ϕ22 ϕ23 ϕ24 ϕ25 ϕ26 ϕ27 ϕ28 ϕ29
χ1 10000000000000000000000000000
χ2 20100000000000000000000000000
χ3 11100000000000000000000000000
χ4 10110000000000000000000000000
χ5 10100010000000000000000000000
χ6 22211110000000000000000000000
χ7 20200000010000000000000000000
χ8 21310010010000000000000000000
χ9 32311110010000000000000000000
χ10 62511111110000000000000000000
χ11 31310010011000000000000000000
χ12 41410010011010000000000000000
χ13 12310010011001000000000000000
χ14 1251132221131100000000000000000
χ15 64721121122001000000000000000
χ16 1241121121132110000000000000000
χ17 41400111110010001000000000000
χ18 41400111110010000100000000000
χ19 63621010022010010000000000000
χ20 84821121122010010000000000000
χ21 1051121131133021010000000000000
χ22 1971842232253120010000000000000
χ23 1681742342254121010000000000000
χ24 20142474885563102200000000000000
χ25 1191254331133101110010000000000
χ26 17101754452244111110010000000000
χ27 1171133442322011110110000000000
χ28 1171133443222011111010000000000
χ29 24142774785575122210000000000000
χ30 25162885785576122210000000000000
χ31 00000000000000000001000000000
χ32 392444127121388117233311110000000000
χ33 36203710710106696232221120000000000
χ34 1571332241134021010000100000000
χ35 24102153452255131020010100000000
χ36 28122863463376131020000010000000
χ37 3620359710116687142231120100000000
χ38 6743732215252212121810635523351000000000
χ39 9455992719312817172413766634461000000000
χ40 6331631591316991513273241110110000000
χ41 62326216101416881512363241120110000000
χ42 552954159121566139332221120101000000
χ43 613059159151788148342222230101000000
χ44 7444752216242111111812655443361100000000
χ45 6836681811171910101610353332230101000000
χ46 7844792115262313131911665434461100000000
χ47 8752902518302716162113666634461100000000
χ48 362038966105589142220000000100000
χ49 4824471071014771010162231110100100000
χ50 00000000000000000000000000100
χ51 1276912734233635191930228117674471210000000
χ52 5734621510151810101413253331110010110000
χ53 168941724630495027274126101210966691201000000
χ54 1701101865336585634344533912131276691110020000
χ55 90489221142125151521193104452220110110000
χ56 1116111727172934212127223126653320110110000
χ57 185115199563762593535483111111212677101111020000
χ58 192121208593964633737503410121313777101111020000
χ59 221135234654472704141563812151414988121211020000
χ60 210124220603966663838523410141212888111212020000
χ61 1408014837233842252534275127764440121120000
χ62 226132235644169703939563811161312988111312020000
χ63 215125226583964663939534110191312107791220120000
χ64 23213224263406270393957439181212106680212120000
χ65 2561442626946717541416145132114131288121311110000
χ66 00000000000000000000000000010
χ67 322186336865892985757786114281818151010131321230000
χ68 18010919248385558353543381116111186691121231000
χ69 2701532777253768145456450162215141388141322121000
χ70 256141263654674784646614114221313101010131313121000
χ71 00000000000000000000000000110
χ72 282161293755383875151684915231515121010141323131000
χ73 308179321856091955454755518231717131010161323131000
χ74 305178323825792965959765815271817141010131332141000
χ75 2761592907253788450506756152616151488121231231000
χ76 55832458015810317017096961379228403230242020292644040000
χ77 55832458015810317017096961379228403230242020293644040000
χ78 386225402106751181196969946822302221171414212433141000
χ79 4252494461188213013277771057524332524181515222433141000
χ80 3351943538764951036262816717311919161010141231341000
χ81 4162454361158112913076761027523322423181515222443151000
χ82 4282644581258813813881811088024302726181515232343161000
χ83 4572644721268813613978781107926342524201616252534141000
χ84 4382534581188212513477771078422362423201313191443251000
χ85 53632156215310817317098981319131383332222121323534151000
χ86 4512644751208613414084841108823412726211515202441351000
χ87 4972865191339214715390901219025422827221717232543251000
χ88 58434160816311317718010210214310332443432262020313644151000
χ89 57433359615711017217510010013910332463331262020303643251000
χ90 5142945401369715015895951279429443028221717242444252000
χ91 5223045501409915416499991289826442929231717231555272000
χ92 53831256514310315816710010013110129473130241818252544362000
χ93 00000000000000000000000000011
χ94 72843276920214122823113813818013039574342302727373766292000
χ95 72241775319413821522313113117513039604140322525363755372000
χ96 74243277920114122123013613618213739624341332525353765382000
χ97 73642377119513621122713413417913936644140332323312765482000
χ98 78446082521515023724514414419314242624544332727383766392000
χ99 00000000000000000000000000111
χ100 90853295225217427728316516522416049705351383232464967292000
χ101 900524945246168267279163163222164467251493830304139773102000

Section 7 : References

[Conway] J.H.Conway: A group of order 8315553613086720000. Bul LMS (1969) 1, 79.
[Conway et al] Conway, Curtis, Norton, Parker, Wilson - An atlas of finite groups
[Wilson] The Birmingham Atlas of Finite Group Representations

Section 8 : Acknowledgements

I'd like to thank Richard Parker, Rob Wilson, Gerhard Hiss (and LDFM), Juergen Mueller, Frank Lubeck and Klaus Lux for putting me up to all this sort of stuff.