Title: Brauer correspondent blocks with one simple module.
Author (s): Carolina Vallejo (ICMAT), Gabriel Navarro (Universitat de València) and Pam Huu Tiep (Rutgers University).
Source: Journal of Algebra.
Date of publication: In press.
Abstract: Finite groups model the symmetries that occur in nature. There are two classical ways of studying them; either by their actions on sets (the Theory of Permutation Groups) or by their actions on vector spaces (Representation Theory). By studying the action of a group G on a vector space V, what we do is study the representation of G as a sub-group of GL(V) that arises. The so-called Character Theory studies the trace of the representation. Characters were first defined by Frobenius in 1896. One year later, Burnside writes in the preface of his “Theory of groups of finite order” that in dealing purely with the theory of groups, no more concrete mode of representation should be used than is absolutely necessary. However, in 1911, in the preface of the second edition of his book he recognizes how useful the Representation Theory has been for the the Theory of groups of finite order and writes “the reason given in the original preface for omitting any account of it no longer holds good”. Not even Burnside was totally convinced from the beginning.
In this paper entitled Brauer correspondent blocks with one simple module, the researchers show how the p-local structure of a finite group G determines and is determined by the character theory of the principal p-block of G.
In this article, “Brauer correspondent blocks with one simple module”, we show that the p-local structure of a finite group G determines and is determined by the character theory of the principal p-block of G for any odd prime p. The case where p=2 is treated in [NV17] and [ST18].
The irreducible characters of a finite group G contain most of the relevant information about the actions of G on complex vector spaces. A character χ of G is a class function χ: G → C that codifies this information. For a prime number p that divides the order of G, the characters of G are partitioned into Brauer p-blocks, the principal block being the only one containing the character 1G that corresponds to the trivial action of G on C. The p-blocks appear when considering at the same time the ordinary and modular characters of G (the actions of G over complex vector spaces and spaces of positive characteristic p).
In the context of finite groups, the word «p-local» is related to the structure of the p-subgroups of G and its normalizers, the most paradigmatic case being that in which the p-subgroups are Sylow subgroups. A Sylow p-subgroup P is a subgroup of G of order pa, where pa is the greatest power of p that divides the order of G, and its normalizer NG(P) is the set of elements g ∈ G such that g−1Pg = P (namely those elements of G that act on P).
In the 1960s, Richard Brauer related the p-blocks of G with p-blocks of normalizers of p-subgroups of G. This is the well-known Brauer correspondence. An essential idea in Representation and Character Theory is to characterize the local structure of a group G by looking only at its global character theory. For odd primes p, we prove that the action of NG(P) on P by conjugation only generates inner automorphisms if, and only if, 1G is the only character of degree not divisible by p and p-rational contained in the principal p-block of G. The condition on NG(P) is equivalent to saying that NG(P) is p-nilpotent or that NG(P) decomposes as a direct product of P and a group of order not divisible by p. With regard to the conditions on the characters of the principal p-block, on the one hand, the condition on the degree of the characters is related to the McKay conjecture and, on the other hand, the condition of p-rationality has to do with a conjecture by Gabriel Navarro that refines the McKay conjecture by taking into account the values of the characters.
A character χ of G is associated with the action of G on a complex vector space. The degree of χ is the dimension of such a vector space. The McKay conjecture (for p = 2 it is now a theorem by [IMN07] and [MS16]) predicts that the number of characters of degree not divisible by p of G and of NG(P) is the same. John L. Alperin remarked that there must exist a bijection between these sets, which preserves the partitions of characters into p-blocks. These two conjectures are fundamental open problems in Group Representation Theory. Going one step further, Gabriel Navarro conjectured [Nav04] that there must also exist a bijection preserving the fields of values of the characters calculated over the p-adics (the field of values of χ over Qp is obtained by adjoining to Qp the values of χ). In particular, there must exist a bijection that preserves the number of p-rational characters (whatever they are, since I am not defining them) on both sides. This conjecture is attracting a great deal of interest from the community over recent years, see for example [Ruh17], [BN18], [NSV18]. It is worth mentioning that the Navarro conjecture implies our main result in “Brauer correspondent blocks with one simple module” so that our work supports this amazing conjecture.
One example of the surprising nature of the global-local conjectures in Representation Theory is that of the Monster group. The Monster group M is one of the 26 sporadic simple groups (and one of the key ingredients in the Monstruos Moonshine conjecture). It gets its name from its enormous size, since it is a group with 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 ≈ 8·1053 elements. Its character theory is very complex; nevertheless, according to the above-mentioned global-local conjectures, in order to study the irreducible actions of G on complex odd-dimensional vector spaces (or at least, certain aspects of them), it suffices to study the same type of actions of one of its Sylow 2-subgroups P (since in this case NM(P) = P). Such a group has 246 , so it is tiny compared to the Monster.
[BN18] O. Brunat, R. Nath. The Navarro Conjecture for the alternating groups. ArXiv:1803.01423.
[IMN07] I. M. Isaacs, G. Malle and G. Navarro. A reduction theorem for the McKay conjecture, Invent. Math. 170 (2007), 33–101.
[MS16] G. Malle, B. Späth. Characters of odd degree, Ann. of Math. (2) 184 3 (2016), 869–908.
[Nav04] G. Navarro. The McKay conjecture and Galois automorphisms, Ann. of Math. 160 (2004), 1129–1140.
[NTV18] G. Navarro, P. H. Tiep, C. Vallejo. Brauer correspondent blocks with one simple module. Trans. Amer. Math. Soc. to appear.
[NSV18] G. Navarro,B. Späth, C. Vallejo. A reduction theorem for the Galois-McKay conjecture. preprint
[NV17] G. Navarro, C. Vallejo. 2-Brauer correspondent blocks with one simple module. J. Of Algebra
[Ruh17] L. Ruhstorfer. The Navarro refinement of the McKay conjecture for finite groups of Lie type in defining characteristic. ArXiv:1703.09006.
[ST18] A. Schaeffer-Fry, J. Taylor. Principal 2-Blocks and Slow 2-subgroups. To appear in Bulletin of the London Mathematical Society.