The Brauer characters of the sporadic simple Harada–Norton group and its automorphism group in characteristics 2 and 3

Abstract

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

Generators of Maximal Subgroups of Harada-Norton and some Linear Groups

“…The Harada-Norton group has been studied extensively in recent years and many papers are written on this group, here we mention a few [31][32][33][34][35][36][37][38][39]. Monomial modular representations and symmetric generation of the Harada-Norton group.…”

Broué's abelian defect group conjecture holds for the Harada–Norton sporadic simple group HN

“…Namely, relatively recently G. Hiss, J. Müller, F. Noeske and J.G. Thackray [17] have determined the 3-decomposition matrix of the group HN with defect group C 3 × C 3 , see 4.1. Our starting point for this work was actually to realize that the 3-decomposition matrix for the non-principal block of HN with an elementary abelian defect group of order 9 is exactly the same as that for the principal 3-block of the Higman-Sims simple group HS.…”

Matching simple modules of condensation algebras

“…The calculation of an exhaustive set of generators for the chosen local submodules is almost negligible: even for the larger composition factors it clocks in at most 3/10 of a second. However, in the case that we cannot find a match (and for which we know that a match does not exist, see [2]), the algorithm loops over as many generators for a local module as is given by the dimension of its simple head. For each of these vectors basis changes are conducted, thus giving us the output fail after about d × 3.5 s, where d is the dimension of the composition factor.…”

“…The idempotents e and e used are the trace idempotents of the subgroups 2 3 .2 2 .2 6 and 5 2 .5.5 2 , which are normal in the ninth, respectively tenth, maximal subgroup of G (see [2] for details). Condensation gives a 567-dimensional eF Ge module V e which possesses the composition factors 1 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 2 , 8, 8, 9, 9, 26 1 , 26 1 , 26 2 , 138, 138, 173 and a 438-dimension e F Ge -module V e with composition factors 1 , 3 , 3 , 7 , 7 , 8 1 , 8 1 , 8 1 , 8 2 , 8 2 , 8 3 , 8 3 , 8 3 , 8 4 , 16 , 16 , 16 , 16 , 89 , 96 , 96 .…”

“…In other words, the condensation algebras have the 'same' simple modules as their enveloping condensed algebras, but we allow them to have less glue. Such condensation algebras are a common sight in the Modular Atlas project [9], so that Algorithm 4 has immediate applications (see, for example, [2]). Section 4 is devoted to illustrating how the matching algorithms proposed in this paper fare when applied to practical examples.…”