Sugandha Maheshwary, Indian Institute of Technology Roorkee
Title: Cut groups and their generalisations
Abstract: In this talk, I will give an overview of finite cut groups, i.e., the groups which have only trivial central units in their integral group ring. I will discuss the developments on cut groups, made in recent years, in varied directions. The talk shall include a discussion and results on various classes of groups which generalise the notion of cut groups. These classes include extended cut groups, arbitrary (not necessarily finite) cut groups, semi-rational groups, uniformly semi-rational groups, quadratic groups etc.
12:00 20th May 2025 - Aula 1.5, Facultat de Ciències Matemàtiques
Pablo Andújar, Universitat de València
Title: Teoría de modelos y grupos pseudofinitos
Abstract: La primera mitad de este seminario será una introducción básica a la teoría de modelos como marco para el estudio de estructuras matemáticas y sus teorías. En la segunda mitad se presentarán resultados sobre grupos pseudofinitos, es decir grupos infinitos que poseen la teoría de los grupos finitos. Durante el seminario averiguaremos si hacer todo esto en una hora es posible
11:00 25th Mar 2025 - Aula 1.5, Facultat de Ciències Matemàtiques
Damiano Rossi, RPTU Kaiserslautern-Landau
Title: From Mckay's to Dade's conjecture
Abstract: The McKay conjecture in group representation theory states that the number of irreducible characters whose degree is not divisible by a prime p is determined by information encoded in the normaliser of a Sylow p-subgroup. But what happens when we consider characters whose degree is divisible by p? The answer is given by Dade's conjecture, a statement that provides a far-reaching generalisation of the p-local-global phenomenon described by the McKay conjecture. In this talk, I will introduce these conjectures and explain a strategy for their solution.
11:00 12th Mar 2025 - Aula 1.1, Facultat de Ciències Matemàtiques
Víctor Ortiz Sotomayor, Universitat Politècnica de València
Title: What do the frequencies of conjugacy class sizes tell us about the group structure?
Abstract: Within finite group theory, there are numerous results which endorse that the set of sizes of conjugacy classes of a group provides relevant information about its structure. However, there are some features that may not be inferred from this piece of information. For instance, the order of the group is unknown; less trivially, solubility and nilpotency cannot be determined. However, if the frequencies of the class sizes are also considered, then some of these properties can be read off from this multiset.
The aim of this contribution is to survey results within this framework, and to present recent progress in this research line from a local point of view.
12:00 24th Feb 2025 - Aula 1.5, Facultat de Ciències Matemàtiques
Steven (Meizheng) Fu, University of Auckland
Title: Elementary abelian subgroups and their local structure in classical groups
Abstract: Many open conjectures in the representation theory of finite groups, such as the Alperin Weight Conjecture and Dade’s conjecture, can be studied by reducing to quasi-simple groups. An important tool in such studies is p-radical subgroups and their local structure. In fact, radical subgroups play an important role in many areas of modular representation theory. For instance, defect groups of blocks are radical; the subgroup R of a weight (R,𝜑) is radical. A p-radical subgroup is a subgroup R of G such that R = Op(NG(R)), the largest normal p-subgroup of the normalizer. A subgroup of a finite group G is p-local if it is the normalizer of a nontrivial p-subgroup of G. Every p-radical subgroup R of G with Op (G) ≠ G is radical in some maximal-proper p-local subgroup M of G. And every M of G can be realized as the normalizer of an elementary abelian p-subgroup. Thus, to classify p-radical subgroups of a finite group G, one can first classify elementary abelian p-subgroups and find their local structure.
To achieve this, we first classify and obtain the local structure in a linear algebraic group G and then use a one-to-one correspondence derived from Lang-Steinberg theorem to transfer the results to finite groups of Lie type GF, the fixed point subgroup of G of a Steinberg endomorphism F. This approach was used successfully by A, D and L in [1] to classify and obtain local structure of the elementary abelian p-subgroups in finite exceptional groups of Lie type. I will report briefly on my work to solve this problem for classical groups.
References
[1] An, Jianbei, Heiko Dietrich, and Alastair J Litterick. “Elementary Abelian Subgroups: From Algebraic Groups to Finite Groups.” Trans. Amer. Math. Soc. 377 (2024): no. 12, 8335-8380.
12:00 3rd Feb 2025 - Aula 1.5, Facultat de Ciències Matemàtiques
Júlia Martínez-Marín, University of Bristol
Title: Rational points on K3 surfaces of degree 2
Abstract: K3 surfaces can be considered a 2-dimensional analogue of elliptic curves. However, their rational points (the solutions to the polynomial equations defining the surface) are not yet well understood. In this talk, we focus on K3 surfaces of degree 2, as their geometry allows us to use elliptic curves to produce infinitely many rational points.
12:00 8th Jan 2025 - Aula 1.5, Facultat de Ciències Matemàtiques
Iris Gilabert, Universitat Politècnica de València
Title: On a character correspondence associated to projectors
Abstract: A saturated formation F is a class of groups which is closed under quotients, such that G/(N ∩ M) ∈ F whenever G/N, G/M ∈ F, and such that G ∈ F if and only if G/Φ(G) ∈ F. For solvable groups G, we define F-projectors as the conjugacy class of subgroups H which verify that HN/N is F-maximal in G/N for every N ⊲ G. A classical example of saturated formation is that of nilpotent groups. The projectors of G are then its Carter subgroups.
For a solvable group G, Isaacs defined a correspondence between the linear characters of a Carter subgroup C of G and a certain subset of Irr(G) which he named the head characters of G. Navarro also defined, for a saturated formation F and an F-projector H of G, a canonical subset Irr_{F'} (G) of Irr(G) in correspondence with Lin(H). He called them the F'-characters of G. These again generalize the nilpotent case, as head characters correspond exactly to F'-characters when F is the formation of nilpotent groups.
In this talk, which is based on a joint work with Lucia Sanus and María José Felipe, we study several aspects of the behaviour of head characters, especially regarding normal subgroups of G, and we discuss how this could be generalized to F'-characters for other formations containing that of nilpotent groups.
12:00 28th Nov 2024 - Aula 1.5, Facultat de Ciències Matemàtiques
David Cabrera, Universitat de València
Title: Inducción de Artin y Brauer
Abstract: En 1946, Richard Brauer demostró que todo carácter de un grupo finito se puede expresar como combinación lineal entera de la inducción de caracteres lineales de un cierto tipo de subgrupos, los p-elementales. Una versión más débil de este teorema es el teorema de Emil Artin, que afirma la misma tesis pero con coeficientes racionales y subgrupos cíclicos. Una consecuencia importante del teorema de Brauer en teoría de números es que las funciones L de Artin L(ρ,s) son meromorfas.
El objetivo principal de este seminario es presentar los teoremas de inducción de Artin y Brauer tratándolos desde un punto puramente de teoría de caracteres. Se discutirá cómo en el teorema de Artin se puede obtener unicidad de los coeficientes y , además, presentaremos una generalización del teorema de Brauer, debida a Robert Boltje, que permite obtener una descomposición canónica.
12:00 14th Oct 2024 - Aula 1.5, Facultat de Ciències Matemàtiques
Gareth Tracy, University of Warwick
Title: Length parameters in finite groups and applications
Abstract: Studying the length of certain subnormal series’ in a finite group has had numerous applications over the years: from Hall & Higman’s early breakthroughs on the Burnside problems; to Babai’s study of the computational complexity of algorithms for finite permutation groups, and much more.
In this talk, I will speak about conjectures of Khukhro and Shumyatsky which associate some of these length parameters to properties of so-called weakly central elements of finite groups. I will also speak about applications to the study of maximal overgroups of elements in finite groups, and to Baer—Suzuki type theorems. Joint work with R. M. Guralnick.
12:00 1st Mar 2024 - Aula 1.5, Facultat de Ciències Matemàtiques
J. Miquel Martínez, University of Florence
Title: Degree divisibility in character correspondences
Abstract: We discuss situations in which it is possible to find character correspondences for the McKay and Alperin-McKay conjectures with good properties. In particular, we explore how these correspondences can interact with character degrees. This is joint work with Damiano Rossi.
12:00 8th Mar 2024 - Aula 1.5, Facultat de Ciències Matemàtiques
Juan Martínez, Universitat de València
Título: Covering the set of p-elements of a finite group
Abstract: There exists many papers studying the number of proper subgroups needed to cover a finite group. In this talk, we are interested in slightly different questions. How many proper subgroups do we need to cover the set of p-elements of a group? Do we need all Sylow p-subgroups to cover the set of p-elements of a group?
12:00 6th Mar 2024 - Aula 1.1, Facultat de Ciències Matemàtiques
Nariel Monteiro, University of California, Santa Cruz
Title: The Complex Stable Representations of the General Linear Group over Finite Local Rings.
Abstract: In this talk, we will give a survey of the representation theory of the general linear group over finite local rings. We will explain how Clifford's theory plays an important role in the construction of the irreducible characters of such groups. We will emphasize the construction of a class of irreducible representations called stable representations. The study of such a class of representations is motivated by constructions of strongly semisimple representations, introduced by the work of Hill.
11:00 11th Dec 2023 - Aula 1.5, Facultat de Ciències Matemàtiques
Charles Eaton, The University of Manchester
Title: Progress on Donovan's conjecture
Abstract: I will introduce Donovan's conjecture, explaining background and context, and giving a survey of progress. The conjecture regards Morita equivalence classes of blocks of group algebras for finite groups with respect to fields of prime characteristic. Essentially it is about the different ways we can build indecomposable modules from simple modules. While in some cases, such as tame blocks or those of finite type, general results can be proved, but for most cases we use the classification of finite simple groups.
12:00 1st Dec 2023 - Aula 1.5, Facultat de Ciències Matemàtiques
Robert Boltje, University of California, Santa Cruz
Title: An introduction to the canonical Brauer induction formula.
Abstract: Brauer's famous induction theorem states that
(1) one can write every character of a finite group as an integral linear combination of induced linear characters and
(2) one can do so by inducing from "elementary" subgroups only.
The first part of Brauer's induction theorem answered a question of Artin and had consequences for so-called Artin L-functions which are central objects in algebraic number theory, generalizing the Riemann zeta-function. Both parts of Brauer's theorem have become fundamental tools in character theory. In this series of two talks we introduce the concept of canonical Brauer induction, dating back to the late 1980s. It is an explicit and canonical version and provides an independent proof of aspect (1) of Brauer's theorem.
11:00 13th and 20th Nov 2023 - Aula 1.5, Facultat de Ciències Matemàtiques
Eugenio Giannelli, Università degli Studi di Firenze
Title: Sylow Branching trees
Abstract: Let G be a finite group, let p be a prime number and let P be a Sylow p-subgroup of G. The Sylow Branching Coefficients (SBCs) of G are the integers [χ, θ^G], where χ and θ are irreducible characters of G and P respectively. In this talk I will present some new results concerning SBCs for symmetric groups. More precisely, given any arbitrary character θ ∈ Irr(P), I will explain that the irreducible constituents of the induction θ^Sn can be read off quite easily from a combinatorial object (a tree or more generally a forest) naturally associated with θ. This talk is based on a recent joint work with Stacey Law.
12:00 17th Nov 2023 - Aula 1.5, Facultat de Ciències Matemàtiques
Britta Späth, Bergische Universität Wuppertal
Title: “On the McKay conjecture”
Abstract: J. McKay conjecture (from 1971) predicts that for any finite group G and prime l, the number of complex irreducible characters of G with a degree not divisible by l is controlled by the normaliser of a Sylow l-subgroup of G. By work of Isaacs, Malle and Navarro this conjecture was reduced to a statement on finite quasi-simple groups and their representation theory. I will report on recent progress on this problem, in joint work with Marc Cabanes.
17:00 4th Oct 2023 - Saló de Graus, Facultat de Ciències Matemàtiques
Taro Sakurai, Chiba University
Title: Divisibility of sizes in local-global correspondence of conjugacy classes
Abstract: Let p be a prime and let G be a finite group with a Sylow p-subgroup P. It is known for decades that there is a bijection from the set of conjugacy classes of G whose size is not divisible by p to that of N_G(P). This talk presents a refinement of this local-global correspondence by taking divisibility of sizes into account. We also propose a local-global conjectures on conjugacy classes of finite p-solvable groups whose size is not divisible by a prime p.
10:00 15th Sep 2023 - Aula 1.6, Facultat de Ciències Matemàtiques
Mandi Schaeffer Fry, University of Denver
Title: Character Tables and Brauer's Legacy
Abstract: Widely known as the founder of modular representation theory, Richard Brauer set the stage for the study of the so-called local-global conjectures in character theory. In this talk, I'll discuss some of these conjectures and their implications for other character-theoretic properties and lingering questions of Brauer.
13:00 23rd Mar 2023 - Saló de Graus, Facultat de Ciències Matemàtiques
Attila Maróti, Alfréd Rényi Insititute of Mathematics
Title: Bounds for the diameters of orbital graphs of affine groups
10:00 8th Feb 2023 - Facultat de Ciències Matemàtiques
Benjamin Sambale, LUH Hannover
Title: Fusion systems in representation theory
12:00 6th-9th Feb 2023 - Facultat de Ciències Matemàtiques