TheHarada Norton group is described in [Norton]. Throughout thispaper, it will be denoted HN. The character table is given in [Conwayet al] The representations used as a starting point are taken from theBirmingham Atlas of Finite Group Representations [Wilson].
At the point of starting on the problem, some small representations ofHN were already known, as well as one character of defect zero and ablock of defect 4 containing three irreducibles. These are consideredknown, and will not be described in this talk. The talk will addressonly the principal block, in which we are seeking seventeenirreducibles.
This leaves one more irreducible to be found, which must therefore beconstructible over GF(2) and be rational. Character theory demonstratesthat it can be found in 2650 tensor 2650', as well as several otherless symmetrical looking constructions of similar sizes. The fullmodule here has dimension 7022500, which is well out of reach forcurrent technology available to pure mathematicians. Even acondensation over an 11.5 subgroup, which was what I had been using,still has degree over 100000, over GF(4). Splitting such a module isquite a tricky prospect, given that of course the condensation cardhas already been played. In some sense, condensation has run out ofsteam.
Using the latest machine I could obtain, a P4/3066 with RAMBUS RD1066memory, I began to construct and then split the condensed module.Producing the condensed generators, of which I needed four, took fourdays each. The information I had so far on its reduction indicatedthat there would be only one or two factors 217130 in the full module,and also a non-principal block irreducible. A search for a near peakword was therefore undertaken to find a group algebra element withnullity one on the condensation of 217130, and hopefully zero on allothers (apart from of course perhaps the one I was seeking, and thenon-principal block irreducible). It turned out that the condensationof 217130 occurred twice, giving me considerable leverage in reducingthe original unwieldy module. I followed this up by targetting thecondensation of 177286, again using a peak word search, but this timewithout having to worry about the nullity on 217130, which was now outof the equation. The final part of the split was relatively routine,and suggests an irreducible of degree 1556136. Whilst condensationcannot in this case give an accurate answer (we know that if 1556136exists it is irreducible, but cannot say if it exists), I have noreason to doubt that the generators I used for the condensationalgebra give an exact mapping on irreducibles. Computing the indicatorof this final irreducible is currently beyond me. The likes of RichardParker and Rob Wilson have suggested methods for condensingindicators, but I have to confess to not understanding them anywherenear well enough to be able to implement them.
[Norton] S.P.Norton - F and other simple groups[Conway et al] Conway, Curtis, Norton, Parker, Wilson - An atlasof finite groups[Wilson] The Birmingham Atlas of Finite Group Representations
3a8082e126