In the last 15 years there has been an intense activity on problems concerning the representation theory of finite groups with respect to multiple primes. In this talk, we will introduce this topic and present a few recent results.
One of the central themes in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and those of its local subgroups. In particular, Sylow branching coefficients describe how an irreducible character of G decomposes upon restriction to a Sylow subgroup P of G, and have been shown to characterise group-theoretic properties such as the normality of P in G. We report on some new results on these coefficients in the case of symmetric groups.
Dade's conjecture predicts the vanishing of an alternating sum indexed by conjugacy classes of chains of p-subgroups. It is natural to ask whether this alternating sum arises from a (co)homological or topological construction and, if so, what property of this construction accounts for its vanishing. This perspective has been studied by several authors, including Boltje, Linckelmann, and Thévenaz.
In this talk, I will discuss the conjugacy-class version of Dade's conjecture and consider some natural refinement. I will show that the global sum can be decomposed into partial contributions indexed by local data, interpret each of them, via Bredon cohomology, as the reduced Euler characteristic of a naturally associated space and, ultimately, prove that all of these spaces are contractible. This provides a natural geometric explanation both for the shape of the alternating sum and for its vanishing.
Navarro's Galois—McKay conjecture is a refinement of the McKay conjecture that takes into account the action of specific Galois automorphisms. Similar to the McKay conjecture, the verification of this refinement has been reduced by Navarro—Späth—Vallejo to a condition on quasi-simple groups, the so called inductive Galois—McKay conditions. I will introduce these inductive conditions and present some recent progress on their verification for certain types of groups of Lie type.
TBD
I will discuss joint work with Lucas Ruhstorfer, in which we prove that a recent condition of Lyons–Martínez–Navarro–Tiep, regarding the field of values of extensions of characters in principal blocks, is satisfied for all finite simple groups. When combined with the results of L-M-N-T, this gives a new characterization of finite groups with a normal p-complement for a prime p and a new extension of a theorem of Thompson. This led us to results on the distribution of characters in unipotent blocks of disconnected reductive groups, which I will also discuss, along with implications for the blockwise Galois–McKay (also known as the Alperin–McKay–Navarro) conjecture.
Let G be a finite group and p be a prime number. If G is p-soluble, then it is easy to observe that the product of two p-regular conjugacy classes of G having coprime sizes is again a p-regular conjugacy class; this fact is crucial in the proof of several results concerning the so-called common divisor graph Γp(G) built on the set of p-regular conjugacy classes of a p-soluble finite group G (cf. [1]). The aim of this talk is to show to what extent the previous p-solubility assumption is necessary. In particular, we will discuss the implications for the existing literature on the common divisor graph Γp(G).
This is joint work with R.D. Camina, A. Maróti, E. Pacifici, C. Parker, K. Rekvényi, J. Saunders, G. Tracey, and M. van Beek.
References
[1] A. Beltrán, and M. J. Felipe: Conjugacy classes of p-regular elements in p-solvable groups, Groups St. Andrews 2005, London Math. Soc. Lecture Note Series 339 (2007) 224–229, Cambridge University Press.
[2] R.D. Camina, A. Maróti, E. Pacifici, C. Parker, K. Rekvényi, J. Saunders, V. Sotomayor, G. Tracey, and M. van Beek: Groups with conjugacy classes of coprime sizes, Bull. London Math. Soc. 58 (2026) e70320.
TBD
The Héthelyi-Külshammer conjecture asserts that whenever B is a p-block of a finite group G with nontrivial defect, the number 2√(p-1) is a lower bound for the number k(B) of irreducible characters belonging to B. In this talk, I will discuss the smallest possible subset of irreducible characters belonging to B for which the same bound should still apply, focusing on the case of principal blocks.
This is based on joint work with A. Maróti, J. M. Martínez, and M. Schaeffer Fry.
The character table of a finite group over the complex numbers encodes important arithmetic information about its representations and has deep connections with the structure of the group. It plays a fundamental role in several areas of finite group theory, including the classification of finite simple groups and classical results such as Burnside’s pmqn-theorem.
In this talk, I will discuss how character-theoretic information, especially the zeros of irreducible characters, influences the structure of finite groups. An element x ∈ G is called vanishing if there exists an irreducible complex character χ ∈ Irr(G) such that χ(x) = 0. I will survey known results on vanishing elements of prime-power order and explain their role in detecting structural properties of finite groups.
I will also present recent work proving that every non-abelian finite simple group, except SL2(4) and SL2(8), contains a vanishing element of prime-power order whose conjugacy class size is divisible by three distinct primes. As an application, we obtained a generalization of a result of Robati: if G is a nonsolvable finite group such that the conjugacy class size of every vanishing element of prime-power order has at most two distinct prime divisors, then G/Sol(G) is a direct product of copies of either SL2(4) or SL2(8), where Sol(G) denotes the largest normal solvable subgroup of G.
This is joint work with Dr. Rahul Kitture.
The contragredient map σ: g ↦ g-1 plays an important role in the study of real 2-blocks. In this talk, I introduce σ-blocks and extended lower defect groups, which provide a natural extension of classical lower defect group theory. As an application, I present a block-theoretic extension of a theorem of Gow and Murray, relating the number of quadratic principal indecomposable modules in a real 2-block to its strongly real extended lower defect groups.
Donovan's Conjecture states that for a fixed defect group, there are only finitely many blocks with this defect group up to Morita equivalence. Proving this conjecture for a given defect group is very hard! In this talk, I will try to explain the challenges we face when attempting to prove this conjecture. Recent progress in proving this conjecture has involved considering cases where it is possible to reduce the problem to quasisimple groups. This has been somewhat successful when the defect group is a 2-generated 2-group. We shall discuss previous work on these groups along with some generalisations of the result that I have been working on.
For a finite group G, let ρ(G) denote the set of all prime numbers dividing the degree of at least one irreducible character of G, and let σ(G) denote the maximal number of prime divisors of the degree of a single irreducible character of G. Huppert’s ρ-σ conjecture states that, for any group G, |ρ(G)| ≤ 3σ(G). A stronger version of this conjecture (first appearing in a 2023 survey by A. Moretó) claims that it is actually possible to find three irreducible characters ψ1, ψ2 , ψ3 ∈ Irr(G) such that every prime in ρ(G) divides the degree of at least one of them.
In this talk, I will first present an overview of the most recent developments on the conjecture, which is still not reduced to simple groups, highlighting the main problems one encounters when working towards a solution. In particular, one of those problems concerns extending characters to their inertia subgroups, a necessary step in order to argue by induction. To bypass this difficulty, at least partially, it may make sense to ask whether it is possible to formulate a variant of Huppert’s ρ-σ conjecture involving relative character degrees. Namely, if Z ◁ G and λ ∈ Irr(Z), is it possible to find a character χ ∈ Irr(G| λ) such that χ(1)/λ(1) is divisible by all the primes in a large subset of ρ(G/Z)? In the talk, I will propose a possible answer to this question, explaining why the selected large subset of primes makes sense in the context of Huppert’s ρ-σ conjecture.
The Ito-Michler theorem gives a character theoretic criterion for a finite group to possess a normal and abelian Sylow subgroup. We propose a new criterion in terms of decomposition numbers. This is joint work in progress with E. McDowell and G. Navarro.
In the 1950’s it was proven, such confirming a conjecture of Erdős, that if we have ℤ = ⨆ᵢ(ai + diℤ), a partition of the integers into arithmetic progressions, then at least two of the di’s are equal. Thinking of the integers just as a cyclic group, this lead Herzog and Schönheim to conjecture that if G = ⨆ᵢ giHi is a partition of a group G into cosets of subgroups H1, . . . , Hn, then two of these subgroups have the same index in G.
The conjecture has mostly been studied using combinatorial techniques and has found a positive answer for instance for groups having a Sylow tower, in particular for supersolvable groups. I will present a solution for symmetric and simple groups. This is based on an arithmetic observation and the vaste knowledge on the subgroup structure of symmetric and simple groups available in the literature.
This is joint work with Martino Garonzi.
The Alperin—McKay conjecture relates the number of height 0 characters in a block and in its Brauer correspondent. The proof of the conjecture has been reduced to the verification of certain conditions on (quasi-)simple groups. In this talk we will discuss the techniques used to verify these conditions for finite reductive groups of exceptional type.
The Brauer—Clifford group, introduced by Alexandre Turull, encodes Clifford theory over characters of normal subgroups, including fields of values and Schur indices. We explain how a generalization of projective representations determines an element of this group. Combined with earlier work of Ladisch, this shows that such a projective representation can be chosen to satisfy additional properties. This is joint work with F. Ladisch.
Many local–global problems in the representation theory of finite groups admit stronger versions that take Galois automorphisms into account. Such is the case for the Brauer Height Zero Conjecture. In this talk, we show that, in the principal p-block case of this conjecture, it suffices to consider the subset of characters fixed by a specific group of Galois automorphisms Ω. More concretely, a Sylow p-subgroup is abelian if, and only if, all irreducible characters in the principal block fixed by Ω are of p'-degree. This is joint work with A. Moretó and N. Rizo.
The study of the unit group V(ℤG) of an integral group ring has long been driven by fundamental questions, such as the Zassenhaus Conjecture, Subgroup Isomorphism Problem and the Prime Graph Question. We introduce a directed graph related to a group G, which we call the N-prime graph ΓN (G) of G and which is a refinement of the classical Gruenberg-Kegel graph. Generalizing some known results about the Gruenberg-Kegel graph, we proved that the group V(ℤG) of the units with augmentation 1 in the integral group ring ℤG has the same N-prime graph as G if G is a finite solvable group, and we reduced to almost simple groups the problem of whether ΓN (ℤG)=ΓN (G) holds for any finite group G. Finally, for G solvable we obtain some stronger results which give a contribution to the Subgroup Isomorphism Problem. More precisely, we prove that if V(ℤG) contains a Frobenius subgroup T with kernel of prime order and complement of prime power order, then G contains a subgroup isomorphic to T. This is a joint work with E. Pacifici and Á. del Río.
It is known that the Hochschild cohomology of a self-injective algebra is invariant under derived equivalence and stable equivalences of Morita type. In particular, the first-degree Hochschild cohomology HH1(B) of some p-blocks of a finite group algebra is preserved, as a Lie algebra, under these equivalences. We review some conjectures and results on this topic, and present recent research on determining the solvability and simplicity of HH1(B). This is based on a joint work with Markus Linckelmann.