Flavors of Representation Theory

Speaker : Lara BOSSINGER (Universidad Nacional Autónoma de México,Mexico)

During these lectures I will cover the following topics: (1) Motivation: total positivity (2) Introduction to cluster algebras (3) cluster structures on partial flag varieties (4) realization of configuration spaces in quantum field theory as partial flag varieties (5) applications of cluster structures to scattering amplitudes

Speaker : Samuel LOPES (University of Porto,Portugal)

The goal of this course is to introduce the participants to some structural features and aspects of the representation theory of infinite-dimensional Lie algebras. As this is an introductory course, the focus will be on the following three types: (H) The (infinite) Heisenberg Lie algebra (also known in Physics as the oscillator algebra). Although just a central extension of a countable-dimensional abelian Lie algebra, hence in some sense the simplest nontrivial example in the course, the study of its representations paves the way for more complex examples. (V) The Virasoro Lie algebra. This Lie algebra is the central extension of the Witt Lie algebra of (complex) vector fields on the circle. These two Lie algebras are connected through Bosonic Fock space. Taking a dual point of view one obtains the fundamental Boson-Fermion correspondence and a class of infinite-dimensional Lie algebras appears naturally: (I) Lie algebras of infinite matrices. Some of these, such as gl(infinity) and sl(infinity) arise as direct limits of classical finite-dimensional reductive Lie algebras. We will try to discuss these objects and unveil some of the connections between them, as well as leverage on these examples to illustrate concepts which form the backbone of the representation theory of infinite-dimensional Lie algebras.

Speaker : Andrea SOLOTAR (University of Buenos Aires,Argentina)

The aim of this course is to describe recent progress on Han's conjecture. Recall that Han's conjecture states that if the global dimension of an algebra is infinite, then its Hochschild homology is infinite. This conjecture has been established for many important families of algebras. This course will serve as an introduction to homological methods/tools such as Hochschild (co)homology, global dimension, triangulated/derived category. These methods are fundamental in modern representation theory, for instance through categorification methods. We will also introduce more recent tools that have appeared in the study of Han's conjecture, and in particular the so-called tau-Hochschild (co)homology.

Speaker : Veronique BAZIER-MATTE ( Université Laval,Canada)

We present a connection between knot theory and cluster algebras via representation theory. More precisely, to every knot (or link) diagram K, we associate a cluster algebra A that contains a cluster x with the property that every cluster variable in x specializes to the Alexander polynomial of K. We call x the knot cluster of A. Furthermore, there exists a cluster automorphism of A of order two that maps the initial cluster to the cluster x. We realize this connection between knot theory and cluster algebras in two ways. On one hand, we construct indecomposable representations T(i) of the initial quiver Q of the cluster algebra A. Modulo the removal of 2-cycles, the quiver Q is the incidence quiver of the segments in K, and the representation T(i) of Q is built by taking successive boundaries of K cut open at the i-th segment. The relation to the Alexander polynomial stems from an isomorphism between the submodule lattice of T(i) and the lattice of Kauffman states of K relative to segment i. On the other hand, we identify the knot cluster x in A via a sequence of mutations that we construct from a sequence of bigon reductions and generalized Reidemeister III moves on the diagram K. On the level of diagrams, this sequence first reduces K to the Hopf link, then reflects the Hopf link to its mirror image, and finally rebuilds (the mirror image of) K by reversing the reduction. We show that every diagram of a prime link admits such a sequence. We further prove that the cluster variables in x have the same F-polynomials as the representations T(i). This establishes the important fact that our representations T(i) do indeed correspond to cluster variables in A. But it even establishes the much stronger result that these cluster variables are all compatible, in the sense that they form a cluster. We also prove that the representations T(i) have the following symmetry property. For all vertices i,j of Q, the dimension of T(i) at j is equal to the dimension of T(j) at i.

Speaker : Amit KUBER (Indian Institute of Technology Kanpur,India)

The Ziegler spectrum of an associative algebra A is a topological space whose underlying set consists of iso-classes of a significant class of A-modules, namely that of indecomposable algebraically compact modules, and whose topology reflects, via its Cantor-Bendixson analysis, the complexity of the module category. A module is said to be algebraically compact if it satisfies a solvability property for systems of A-linear equations akin to the finite intersection property in topology. A basis for closed sets for the Ziegler spectrum is given in terms of certain finitely presented functors on the category of finitely presented A-modules. Even though this topological space was first introduced in the model theory of modules, its surrounding ideas have been interesting and useful in representation theory for purely algebraic reasons using the language of functor categories. Recall that a finite-dimensional algebra is essentially (up to Morita equivalence) a bound quiver algebra. The course aims to provide a gentle introduction to the language of functor categories, and discuss the use of these methods in representation theory of bound quivers through key ideas like Serre localisation, Krull-Gabriel dimension, duality, Ziegler spectrum and definable categories.

Speaker : Pooja SINGLA (Indian Institute of Technology Kanpur,India)

This series of lectures explores the representation theory of groups, combining general theory with concrete examples from linear groups over finite and p-adic fields. We begin with the basic framework of representations, group algebras, irreducibility, and character theory. Key tools such as induction, restriction, Mackey theory, and Clifford theory are introduced, illustrated with examples from finite groups. The lectures then focus on GLn(Fq), explaining how its irreducible representations are constructed, including cuspidal and principal series representations and the use of parabolic induction. Finally, we transition to linear groups over p-adic fields, discussing smooth admissible representations, parabolic induction, and the Bernstein decomposition. Connections between the finite and p-adic settings will be highlighted. Throughout, explicit examples and computations illustrate the theory and its applications.

Speaker : Roozbeh HAZRAT (Western Sydney University,Australia)

Since the introduction of Leavitt path algebras associated to graphs, substantial effort has been devoted to understanding their irreducible representations. This has not only provided deeper insight into the structure of these algebras but also revealed application of their representation theory in symbolic dynamics. There have also been various generalisations of these algebras, notably those associated to higher rank graphs. In this series of lectures, we begin by introducing Kumjian-Pask algebras (a higher rank graph analogue of Leavitt path algebras) and study their (irreducible) representations in depth. We also highlight how these are related to important invariants in symbolic dynamics.

During these Jigsaw sessions in Week 1, students will be organised into groups and will present solutions to the exercises or problems assigned to them. These sessions will complement the tutorials and ensure that students actively put into practice what they have learned during the lecture series.

During the Jigsaw sessions in Week 2, students will continue working in groups, similar to the exercise carried out in Week 1. They will present solutions to a new set of exercises and problems assigned to their groups. These sessions, like before, will complement the tutorials and ensure students put into practice what they learned during the lecture series.