Interactions between Representation Theory and Model Theory


* Jason Bell, Invariant hypersurfaces

Abstract: We study an extension of a theorem of Cantat, which says that if \phi: X\to X is a dominant rational self-map then the number of totally invariant hypersurfaces C (that is, hypersurfaces for which \phi^{-1}(C)=C) is finite unless \phi \circ f=\phi for some non-constant rational map f: X\to \mathbb{P}^1. We relate this to the question of the Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings and other algebras that ``come from geometry''.  This is joint work with Rahim Moosa and Adam Topaz.

* Georgia Benkart
  • Lecture I. McKay Quivers and Schur-Weyl Duality
The McKay Correspondence establishes a bijection between the finite subgroups G of SU_2 and the simply-laced affine Dynkin diagrams.  Such a subgroup must be one of the following: (a) a cyclic group C_n, (b) a binary dihedral group D_n, or (c) one of the 3 exceptional groups: the binary tetrahedral group T, binary octahedral group O, or binary icosahedral group I. McKay's observation was that the quivers determined by tensoring the simple modules of G = C_n,  D_n, T, O, I with the G-module  V= C^2 exactly correspond  to the Dynkin diagrams \hat{A}_{n-1}$, \hat{D}_{n+2}, \hat{E}_6, \hat{E}_7, \hat{E}_8, respectively. 

More generally, for any finite group G and any finite-dimensional G-module V,  the corresponding McKay quiver provides a way  to encode the rule for tensoring by the G-module V.  This first lecture will relate McKay quivers to another well-established representation theory concept,  Schur-Weyl duality, 
and will explain how combining these concepts gives information about  the tensor product module V^{\otimes k}, the centralizer algebra End_{G}(V^{\otimes k}), and the G-invariants in V^{\otimes k}.

  • Lecture II.  Diagram Algebras and Fundamental Theorems of Invariant Theory
Diagram algebras play an essential role in modeling the relations satisfied by the transfer matrices in the Potts model in statistical mechanics.
They also arise in the work of V. Jones (and others) on subfactors of operator algebras and on knot and link invariants.    This talk will focus on various
examples of diagram algebras as they relate to representations of groups and Schur-Weyl duality.    We will describe how diagram algebras provide realizations of centralizer algebras and how they can be used to give the fundamental theorems of invariant theory for tensor invariants.   

  • Lecture III.  Applications and Future Directions 
In addition to providing a wealth of information about representations, the McKay quiver -- Schur-Weyl duality approach can be used to count walks on graphs, study the dynamics of chip firing, and provide examples of Markov chains exhibiting new features. These applications have connections with the modular representation theory of finite groups and of finite-dimensional Hopf algebras such as quantum groups at roots of unity and restricted enveloping algebras of modular Lie algebras. Future applications point to connections with tensor categories and tensor invariants.

* Lorna Gregory, Interpretation functors

Abstract: Interpretation functors are a uniform additive version of the model theoretic notion of interpretation for module categories. Algebraically, they are simply additive functors I:Mod-R \rightarrow Mod-S between module categories which commute with direct limits and products.

In this talk, I will introduce interpretation functors from a model theoretic perspective and give an overview of results about their (non-)existence and their usefulness, with a particular focus on finite-dimensional algebras.

* Thorge Jensen, TBA

Abstract: TBA

* Moshe Kamensky, Tannakian categories over fields with operators

Abstract: The Tannakian formalism provides a complete description of the structure of categories of representations of linear group schemes. When the ground field is equipped with additional structure, such as a derivation or an automorphism, it is natural to consider linear groups with the same kind of structure. In this setting, the Tannakian structure is not sufficient to give a full description of the category of representations. I will provide an overview of the Tannakian formalism, and will explain how to adapt it to groups over fields with operators.

* Martina Lanini, Sheaves on the alcoves and modular representations

Abstract: I'll report on a joint project with Peter Fiebig. The aim of the project is to provide a new perspective on the problem of calculating irreducible characters of reductive algebraic groups in positive characteristics. Given a finite root system R and a field k we introduce an exact category C of sheaves on the partially ordered set of alcoves associated with R, and we show that the indecomposable projective objects in C encode the desired characters.
* Ben Martin, The First Kac-Weisfeiler Conjecture for representations of modular Lie algebras

Abstract: Let g be a finite-dimensional restricted Lie algebra over an algebraically closed field k of characteristic p> 0.  In 1971 Kac and Weisfeiler conjectured a formula giving the maximal dimension of a simple g-module.  This is known to hold when g is the Lie algebra of a reductive algebraic group over k and p>2, but the general case is still open.

Recently David Stewart, Lewis Topley and I proved that for fixed d, if g is a restricted subalgebra of $gl_d(k)$ and p is large enough then the Kac-Weisfeiler formula holds for g (Akaki Tikaradze has found an alternative proof).  I will explain some of the ideas behind the proof, which involves the Lefschetz Principle from first-order model theory.

* Rahim Moosa, Model theory and the Dixmier-Moeglin Equivalence
Abstract: About five years ago, a new application of the model theory of differentially closed fields arose. The target was the Dixmier-Moeglin equivalence problem (DME) in noncommutative affine algebras, as well as a variant for commutative Poisson algebras. It has become clear that the structure of D-varieties (i.e., finite-dimensional types in DCF), and D-groups, can be used to both prove the DME and produce counterexamples in various settings. There have been a handful of papers exploring and exploiting this connection. I will give an overview of this body of work.

* Anand Pillay, Groupoids, relative internality, and differential tangent bundles

Abstract: (Joint with Leo Jimenez) Jimenez studied relative internality in model theory, a typical example being a family of linear ODE's over some base (in DCF_0), associating to it a definable (Galois) groupoid, and introducing the notion of ``collapse" of the groupoid (a weakening of isotriviality).  We will work out some of these ideas in the context of differential tangent bundles of (finite dimensional) differential algebraic varieties.

* Mike Prest, Definability in Representation Theory

Abstract: I will give an introduction to various ideas, techniques and results from Model Theory which should enable using these in applications in representation theory.  I will begin with definable sets and the formulas that are used to describe them.  Realising types is a technique for obtaining elements with desirable properties and can be achieved using ultraproducts - a construction which also can yield structures with specified properties.  Calling on the Compactness Theorem, a corollary of {\L}os' Theorem on ultraproducts, is an alternative to directly using ultraproducts.  I will present, and illustrate the use of, other results about obtaining structures which are `nice' (small, large, saturated, homogeneous).  Choosing and changing the underlying formal language will also be discussed.

* Bea Schumann, A branching rule for the restriction of irreducible representations sl(2n,C) to sp(2n,C) via Littelmann paths

Abstract: Rules for decomposing the restriction of an irreducible representation into irreducible representations of a sub Lie algebra are called branching rules. Littelmann paths give a combinatorial tool to obtain such rules in many cases. However, there was no known branching rule in terms of Littelmann paths describing the restriction to a sub Lie algebra induced by a Dynkin diagram automorphism. In my talk I will explain how to do that for the case of the restriction from sl(2n,C) to sp(2n,C) based on joint work with Jacinta Torres.

* Sue Sierra, Enveloping algebras with just infinite GK-dimension

Abstract: Let W be the Witt algebra, the Lie algebra of derivations of the complex affine line.  We show that, although the universal enveloping algebra of W, U(W), has infinite GK-dimension, any proper quotient of it has polynomial growth:  that is, U(W) has _just_infinite_ GK-dimension.  We prove similar statements for the enveloping algebras of several related Lie algebras.  This is joint work with Natalia Iyudu.

In 2013 we proved with Walton that U(W) is neither left or right noetherian.  Our work with Iyudu provides supporting evidence for the conjecture that U(W) satisfies the ascending chain condition on two-sided ideals, and if time permits we will discuss other evidence in favour of this conjecture.
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Last updated 14 June, 2019.