|Title:||The 2 modular characters of the Harada Norton group|
|Id:||$Id: nikolaus.2003.html,v 1.1 2003/12/03 23:08:41 jon Exp $|
To describe present a set of results giving the 2 modular character table of the Harada Norton group, suitable for inclusion the the forthcoming Atlas of Brauer Characters Part 2.
The Harada Norton group is described in [Norton]. Throughout this paper, it will be denoted HN. 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 HN were already known, as well as one character of defect zero and a block of defect 4 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 seventeen irreducibles.
The starting point for the principal block is the 132 dimensional irreducible representation over GF(4) available from the Birmingham Atlas. Its Frobenius conjugate 132' is distinct and also irreducible. The skew square of 132 reduces as 2.1 + 3344 + 2650 + 2650'. 2650 and 2650' are a complex conjugate pair having irrationalities on elements of order 19 and 35. The problem of algebraic conjugacy is still open, at least as far as I am concerned. The character values for 2650 can be deduced by restriction of the representation to A12 and U3(8), or by direct computation.
132 tensored 132' reduces as 2.760 + 15904. This brings the total irreducible count to eight, and exhausts those easily obtainable. The indicators for all have been computed, and are + with the exception of 2650 and 2650' (which are complex and therefore o) and 132 and 132' which are -.
To make further progress we examine the tensor product of 132 with 760. We already have an ordinary character decomposition of 133 tensored 760 as 133' + 35112 + 65835. Using tensor condensation, we obtain a prediction of 2.1 + 2.132 + 1.132' + 31086 + 2 * 34352, suggesting that 35112 reduces as 760 + 34352, and 65835 reduces as 1 + 2.132 + 132' + 31086 + 34352. Uncondensing proves this prediction to be correct. 31086 and 34352 both require GF(4), and have Frobenius conjugates. All are real with indicator +. This brings our total to twelve.
We now need to dig deeper again, and look at the tonsor product of 132 and 2650. Tensor condensation reveals that this contains a pair of irreducibles of degree around 40000 and a further large dimension irreducible. Uncondensing and computing indicators reveals a complex conjugate pair of degree 43416. Computing the exact character of these irreducibles proved quite tricky, and was done in collaboration with Gerhard Hiss. The difficult part is to prove that 2650 and 2650' occur in the decomposition of the ordinary 69255 with equal decomposition number. This is done by computing explicitly the character value of 43416 on elements of order 19 (on which 69255 is real). Once this is done, computing the trace of 43416 on an element of order 25 distinguishes the two possibilities of either 69255 = 7.1 + 2.132 + 2.132' + 760 + 3344 + 4.(2650+2650') + 43416 or 69255 = 3.1 + 2.132 + 2.132' + 760 + 3344 + 2650 + 2650' + 15904 + 43416. The first possibility has character value -4 (and therefore trace 0 mod 2), whereas the second has character 1. Hence the mod 2 trace is sufficient to distinguish them. Computing this reveals 69255 = 3.1 + 2.132 + 2.132' + 760 + 3344 + 2650+2650' + 15904 + 43416, and therefore gives us the character of 43416.
A further GF(2) character 177286 also occurs in 132 tensor 2650. This can be uncondensed, constructed and proved to have indicator +. The final construcible (at present) irreducible occurs in 760 tensor 760. A laborious uncondensation can be used to create this. Its degree is 217130 and its indicator is also +.
This leaves one more irreducible to be found, which must therefore be constructible over GF(2) and be rational. Character theory demonstrates that it can be found in 2650 tensor 2650', as well as several other less symmetrical looking constructions of similar sizes. The full module here has dimension 7022500, which is well out of reach for current technology available to pure mathematicians. Even a condensation over an 11.5 subgroup, which was what I had been using, still has degree over 100000, over GF(4). Splitting such a module is quite a tricky prospect, given that of course the condensation card has already been played. In some sense, condensation has run out of steam.
Using the latest machine I could obtain, a P4/3066 with RAMBUS RD1066 memory, I began to construct and then split the condensed module. Producing the condensed generators, of which I needed four, took four days each. The information I had so far on its reduction indicated that 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 peak word was therefore undertaken to find a group algebra element with nullity one on the condensation of 217130, and hopefully zero on all others (apart from of course perhaps the one I was seeking, and the non-principal block irreducible). It turned out that the condensation of 217130 occurred twice, giving me considerable leverage in reducing the original unwieldy module. I followed this up by targetting the condensation of 177286, again using a peak word search, but this time without having to worry about the nullity on 217130, which was now out of the equation. The final part of the split was relatively routine, and suggests an irreducible of degree 1556136. Whilst condensation cannot in this case give an accurate answer (we know that if 1556136 exists it is irreducible, but cannot say if it exists), I have no reason to doubt that the generators I used for the condensation algebra give an exact mapping on irreducibles. Computing the indicator of this final irreducible is currently beyond me. The likes of Richard Parker and Rob Wilson have suggested methods for condensing indicators, but I have to confess to not understanding them anywhere near well enough to be able to implement them.
I finish with the decomposition matrix, as proved all bar the final irreducible, and suggested by the condensation.
I wish to thank Gerhard Hiss and Juergen Mueller for inviting me to Aachen in June 2002 where I began work on the problem.