Schedule for: 19w5046 - Groups and Geometries

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We will discuss recent joint work of Nick Katz and the speaker on local systems on the affine line in characteristic p, whose monodromy groups are certain finite (almost quasi)simple groups.

We will describe efforts to classify Tits quadrangles. The focus of our work has been the Tits quadrangles that arise from exceptional groups. We will discuss applications of our results to the study of exceptional groups of relative rank 1. This is joint work with Bernhard Mühlherr.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

There is an analogy between the locally ringed spaces of algebraic geometry and the objects (variously known as localities or linking systems) that provide "classifying spaces" for fusion systems. We shall develop this analogy, and pose some questions regarding the sorts of groups (finite groups at least, then "finite-dimensional" groups, and then what?) that can serve as the analogs of affine schemes. There will be an application to a problem raised by Aschbacher at the Banff conference from six years ago.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Motivated by questions in model theory, Greg Cherlin introduced the idea of “relational complexity”, a statistic connected to finite permutation groups. He also stated a conjecture classifying those permutation groups with minimal relational complexity. We report on recent progress towards a proof of this conjecture. We also make some remarks about permutation groups with large relational complexity, and we explain how this statistic relates to others in the literature, notably base-size.

Let W be a Coxeter group. We provide a precise description of the conjugacy classes in W, yielding an analogue of Matsumoto's theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of W, thereby proving a conjecture of A. Cohen from 1994.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for p any prime number. A fusion system F on a p-group S is a category where the objects are all the subgroups of S and the morphisms are certain injective morphisms that “behave as conjugations” (in particular all restrictions of conjugation maps induced by elements of S are morphisms in F). Given a finite group G, it is always possible to define the saturated fusion system realized by G on one of its Sylow p-subgroups S: this is the category where the morphisms are the restrictions of conjugation maps induced by the elements of G. However, not all saturated fusion systems can be realized in this way. When this is the case, we say that the fusion system is exotic. The understanding of the behavior of exotic fusion systems (in particular at odd primes) is still an important open problem. In this talk we aim to present a new approach to the study of exotic fusion systems at odd primes. First, we will introduce the notion of pearls: essential subgroups that are either elementary abelian of order p^2 or non-abelian of order p^3. Then we will show the connections between pearls and exotic fusion systems and we will present new results concerning the classification of saturated fusion systems containing pearls.

I will discuss the current state of the GLS project, mentioning in particular uniqueness theorems, the generic case, and the bicharacteristic case.

I'll discuss the status of the program to classify odd 2-fusion systems. Briefly: a finite number of cases remain to be treated.

In our earlier study of low rank geometries related to exceptional groups, it became clear that structurable algebras play an important role. The natural question arose to what extent it would be possible to recover those geometries directly from the structurable algebras and their associated Tits-Kantor-Koecher Lie algebra. It turns out that the notion of an inner ideal is essential. We have been able to recover many geometries of rank one and two directly from the algebras in a surprisingly direct fashion. This is related to the geometries of extremal elements studied intensively by Arjeh Cohen and his collaborators, but our approach allows for many more geometries.

Let G=G(q) be a finite reductive group with Weyl group W. A long time ago Malle noticed that the process by which one calculates the unipotent character degrees of G from the Iwahori-Hecke algebra of W works just as well if W is replaced by a *complex* reflection group (with certain properties). A "spets" is the mysterious object which replaces G in this situation, named after the Greek island Spetses on which these observations were first made. Now suppose l is prime and (l,q)=1. I will argue via the theory of l-compact groups in algebraic topology that we know the l-fusion system of a spets G(q). Moreover, I claim this observation can be combined with results of Cabanes--Enguehard in Deligne—Lusztig theory to provide degrees of characters in the principal l-block of G. We thus have all the ingredients necessary to formulate exotic counting conjectures inspired by those of Alperin, Dade, Robinson and others. I will explain the proof of such a conjecture when l=2.

Let G be an algebraic group over an algebraically closed field. A subset S of G topologically generates G if the subgroup generated by S is dense in G, with respect to the Zariski topology. In this talk, I will report on recent work with Spencer Gerhardt and Bob Guralnick on the topological generation of simple algebraic groups by elements in specified conjugacy classes. I will explain some of the main ideas and I will present new results for exceptional algebraic groups. I will also describe an application concerning the random generation of finite simple groups of Lie type.

How can local properties involving conjugation give information about the whole group? Baer's Theorem and Glauberman's Z*-Theorem give two famous examples for how this question can be answered.In the Z_p^*-project we study isolated elements of prime order in finite groups and develop techniques in the direction of a Z_p^*-Theorem.

Aschbacher announced a program to revisit the classification of finite simple groups via fusion systems. For that purpose, he developed a local theory of fusion systems in analogy to the local theory of finite groups. Chermak introduced localities, which are group-like algebraic structures associated to fusion systems. We outline how one can revisit and extend the local theory of fusion systems via a local theory of localities, and we indicate how one might be able to overcome some of the technical problems in Aschbacher’s program by working with localities rather than with fusion systems. The first part of this work is joint with Chermak, the second part is joint with Grazian.

We present a new approach to computing the normaliser of a primitive group G in an arbitrary subgroup H of S_n. Our method runs in quasipolynomial time O(2^{\log^3 n}), whereas the previous best known algorithm was simply exponential. The main tools used are Babai’s recent breakthrough on the graph isomorphism problem, and results on the base size, the order, and the minimal number of generators of a primitive group.

For any smooth connected group G over an arbitrary field k, the irreducible modules are in 1-1 correspondence with those of the pseudo-reductive quotient G/R_{u,k}(G) where R_{u,k}(G) is the k-defined unipotent radical of G. If k is imperfect, a pseudo-reductive group may not be reductive. That means that over the algebraic closure of k, one sees more unipotent radical. If G has a split maximal torus, much of the theory of split reductive groups carries over and we give dimension formulae for irreducible G-modules which reduce the study to the split reductive case and commutative pseudo-reductive case.

In order to determine the Weierstrass points of a compact connected Riemann surface of genius at least 3 the knowledge of the permutation groups (G, \Omega) with fixity at most 4 is needed, i.e. a non-trivial element in G fixes at most 4 points in \Omega. I will discuss the classification of these groups by focusing on the case that (G, \Omega) is of fixity 4. This is a joint project with Kay Magaard and Rebecca Waldecker.

The MSS-project splits into two parts. The E-uniqueness part and the non-E-uniqueness part. The E-uniqueness part leads to groups with a large p-subgroup. In this talk I will report on those results on groups with a large p-subgroup, which have been obtained so far. Moreover I will sketch a strategy which could end up with a full classification at least for the prime 2. Here the Baumann characteristic will come into the picture.

The crux of a card trick performed with a deck of cards usually depends on understanding how shuffles of the deck change the order of the cards. By understanding which permutations are possible, one knows if a given card may be brought into a certain position. The mathematics of shuffling a deck of 2n cards with two “perfect shuffles” was studied thoroughly by Diaconis, Graham and Kantor in 1983. I will report on our efforts to understand a generalisation of this problem, with a so-called ``many handed dealer'' shuffling kn cards by cutting into k piles with n cards in each pile and using k! possible shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, as long as k \ne 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers, and also initiate a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles.

In this talk I report on a some work that has been done to understand the action of automorphisms on spherical buildings regarding their behaviour wrt the opposition relation on flags. It turns out that this behaviour can be captured by a decorated Coxeter diagram. These diagrams remarkably resemble the Tits indices, in particular those corresponding to a Galois group of order 2.

A classical theorem of Brauer asserts that every finite-dimensional non-modular representation \rho of a finite group G defined over a field K, whose character takes values in a subfield k, descends to k, provided that k has suitable roots of unity. If k does not contain these roots of unity, it is natural to ask how far \rho is from being definable over k. The classical answer is given by the Schur index of \rho, which is the smallest degree of a finite field extension l/k such that \rho can be defined over l. In this talk, based on joint work with Nikita Karpenko, Julia Pevtsova and Dave Benson, I will discuss another invariant, the essential dimension of \rho, which measures "how far" \rho is from being definable over k in a different way by using transcendental, rather than algebraic field extensions. I will also talk about recent results of Federico Scavia on essential dimension of representations of algebras.

There has been recent interest in the connections between automorphisms of fusion systems, of their associated linking systems, and of the finite groups realizing them (when such groups exist). Comparison between the automorphism groups of fusion systems and their associated linking systems has importance for various group-like constructions in fusion systems. I plan to discuss work with Glauberman describing the group of rigid automorphisms of a linking system, i.e. those which are the identity on the fusion system. Then I plan to discuss a cohomological obstruction theory, similar to the Broto-Levi-Oliver obstruction theory for the existence and uniqueness of linking systems, which is setup to begin to provide a general framework for understanding when it is possible to define a unique centralizer of a fusion subsystem.

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We provide a characterization of a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular many of the exceptional Lie geometries occur. In fact, our approach unifies and extends several earlier characterizations of (exceptional) Lie geometries arising from spherical Tits-buildings.

The talk revolves around the following question: given a finite dimensional algebra, what is the smallest ring of coefficients over which the algebra is defined? In general, this is a rather intractable issue but one can say quite a lot in the case of blocks of modular group algebras. I will give an introduction to the question, describe its relevance to modular representation theory, and present some answers.

A classical result states that the real Lie algebra so(n) admits a presentation via the embedded so(3) subalgebras along an A_{n-1} diagram, similar to the Curtis-Tits theorem and Phan's theorems. String theorists are interested in studying the corresponding (infinite-dimensional) real Lie algebra k for the diagram E_{10}. In my talk I will discuss a natural generalization of the classical 1/2-spin representation of so(n) to this Lie algebra k, I will exhibit a Cartan-Bott-type periodicity for the images of this representation along the E_n series, and I will present a general machinery how to extend this 1/2-spin representation to higher spin representations. It will turn out that all these extended higher spin representations are controlled by a concise Weyl-group based formula.

We report on the proof of Robinson's conjecture on heights of characters for all odd primes. The proof relies on the classification of finite simple groups. This is joint work with Feng, Li, Liu and Zhang.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.