Seminars in Algebraic Groups

Ian Tan: Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

30th September, 2024

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(ℂ) × SL2(ℂ) → SO4(ℂ), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

Artem Chernikov: Recognizing groups in Erdős geometry and model theory

15th March, 2024

Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory - variants of Hrushovski’s group configuration and of Zilber’s trichotomy principle - are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. I will overview some of the recent developments in this area, in particular explaining how all of this is not just about polynomials and works for definable sets in o-minimal structures.

Nate Harman: Oligomorphic and Linearly Oligomorphic groups

11th September, 2023

Oligomorphic groups are a class of groups arising in model theory. I will discuss where these groups come from, highlight some of their interesting properties, and explain why I (a non-model-theorist) am interested in them. Then I will introduce a new notion of linearly oligomorphic groups, and give new examples of infinite-dimensional algebraic groups.

Michael Wibmer: Expansive endomorphisms of profinite groups

25th May, 2023

Étale algebraic groups over a field k are equivalent to finite groups with a continuous action of the absolute Galois group of k. The difference version of this well-known result asserts that étale difference algebraic groups over a difference field k (i.e., a field equipped with an endomorphism) are equivalent to profinite groups equipped with an expansive endomorphism and a certain compatible difference Galois action. In any case, understanding the structure of expansive endomorphisms of profinite groups seems a worthwhile endeavour and that's what this talk is about.

Michael Wibmer: Difference algebraic groups

25th May, 2023

Difference algebraic groups are a generalization of algebraic groups. Instead of just algebraic equations, one allows difference algebraic equations as the defining equations. Here one can think of a difference equation as a discrete version of a differential equation. Besides their intrinsic beauty, one of the main motivations for studying difference algebraic groups is that they occur as Galois groups in certain Galois theories. This talk will be an introduction to difference algebraic groups.

Camila Sehnem: Equilibrium on Toeplitz extensions of higher-dimensional non-commutative tori

1st April, 2022

The C∗-algebra generated by the left-regular representation of ℕn twisted by a 2-cocycle is a Toeplitz extension of an n-dimensional non-commutative torus, on which each vector r ∈ [0,∞)n determines a one-parameter subgroup of the gauge action. I will report on joint work with Z. Afsar, J. Ramagge and M. Laca, in which we show that the equilibrium states of the resulting C∗-dynamical system are parametrized by tracial states of the non-commutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of r. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Our results generalize the phase transition on the Toeplitz non-commutative tori used as building blocks in work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims.

James Parkinson: Automorphisms and opposition in spherical buildings

4th March, 2022

The geometry of elements fixed by an automorphism of a spherical building is a rich and well-studied object, intimately connected to the theory of Galois descent in buildings. In recent years, a complementary theory has emerged investigating the geometry of elements mapped onto opposite elements by a given automorphism. In this talk we will give an overview of this theory.

William Graham: Geometry of generalized Springer fibres II

7th February, 2022

Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.

Sebastian Bischof: (Twin) Buildings and groups

4th February, 2022

Buildings have been introduced by Tits in order to study semisimple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of thick irreducible spherical buildings of rank at least 3. In particular, any such building comes from an RGD-system. The decisive tool in this classification is the Extension theorem for spherical buildings, i.e. a local isometry extends to the whole building.

Twin buildings were introduced by Ronan and Tits in the late 1980s. Their definition was motivated by the theory of Kac-Moody groups over fields. Each such group acts naturally on a pair of buildings and the action preserves an opposition relation between the chambers of the two buildings. This opposition relation shares many important properties with the opposition relation on the chambers of a spherical building. Thus, twin buildings appear to be natural generalizations of spherical buildings with infinite Weyl group. Since the notion of RGD-systems exists not only in the spherical case, one can ask whether any twin building (satisfying some further conditions) comes from an RGD-system. In 1992 Tits proves several results that are inspired by his strategy in the spherical case and he discusses several obstacles for obtaining a similar Extension theorem for twin buildings. In this talk I will speak about the history and developments of the Extension theorem for twin buildings.

Anne Thomas: A gallery model for affine flag varieties via chimney retractions

4th February, 2022

We provide a unified combinatorial framework to study orbits in affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat-Tits buildings, we relate these retractions and their effect on certain minimal galleries to double coset intersections in the corresponding affine flag variety.

William Graham: Geometry of generalized Springer fibres I

31st January, 2022

Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.

Uri Bader: Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity

6th June, 2021

Compact hyperbolic manifolds are very interesting geometric objects. Maybe surprisingly, they are also interesting from an algebraic point of view: They are completely determined by their fundamental groups (this is Mostow's Theorem), which is naturally a subgroup of the rational valued invertible matrices in some dimension, GLn(ℚ). When the fundamental group essentially consists of the integer points of some algebraic subgroup of GLn we say that the manifold is arithmetic. A question arises: is there a simple geometric criterion for arithmeticity of hyperbolic manifolds? Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen. In a recent work with Fisher, Miller and Stover we proved this conjecture. Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which Alex Furman and I have developed in recent years. In my talk I will survey the subject and focus on the relation between the geometric, algebraic and arithmetic concepts

Alexandr Zubkov: Harish-Chandra pairs and group superschemes

12th March, 2021

The purpose of my talk is to discuss the following results recently obtained in collaboration with A.Masuoka (Tsukuba University, Japan). First, we prove that a certain category of Harish-Chandra pairs is equivalent to the category of (not necessary affine) locally algebraic group superschemes. Using this fundamental equivalence we superize the famous Barsotti-Chevalley theorem and prove that the sheaf quotient of an algebraic group superscheme over its group super-subscheme is again a superscheme of finite type. I will also formulate some open problems whose solving would bring significant progress in the supergroup theory.

Harrison Chen: Coherent Springer theory and categorical Deligne-Langlands

23rd November, 2020

Kazhdan and Lusztig proved the Deligne-Langlands conjecture, a bijection between irreducible representations of unipotent principal block representations of a p-adic group with certain unipotent Langlands parameters in the Langlands dual group (plus the data of certain representations). We lift this bijection to a statement on the level of categories. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category equivalent to modules for the affine Hecke algebra (or specializing at q, unipotent principal block representations of a p-adic group). Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category.

Reuven Hodges: Coxeter combinatorics and spherical Schubert geometry

16th November, 2020

This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup. We will see that this conjecture extends and unifies previous sphericality results for Schubert varieties in G/B due to P. Karuppuchamy, J. Stembridge, P. Magyar–J. Weyman-A. Zelevinsky. In type A, the combinatorics of Demazure modules and their key polynomials, multiplicity freeness, and split-symmetry in algebraic combinatorics are employed to prove this conjecture for several classes of Schubert varieties.

Scott Larson: Small Resolutions of Closures of K-Orbits in Flag Varieties II

21st September, 2020

The geometry of closures of K-orbits in the flag variety governs key properties in representation theory of real reductive groups. For example, Kazhdan-Lusztig-Vogan polynomials and characteristic cycles of Harish-Chandra modules are of current interest. We recall how small resolutions have been used to compute these invariants, describe fibres of the resolutions from last week, and describe more small resolutions for the real reductive groups Sp2n(ℝ), U(p,q), and complex groups. Along the way we consider an application to Schubert varieties.