Maud De Visscher (City University, London)
Title -- Representations of centraliser algebras
(Here are the Slides (pdf))
Abstract: In this series of lecture, we will consider centraliser algebras of the action of classical groups (and their quantum analogues) on tensor spaces. The most studied and best understood is the Temperley-Lieb algebra and we will start by reviewing its representation theory.
We will then turn to the Brauer, walled Brauer and partition algebra and explain how their representation theory can be studied in a uniform way by generalising the techniques used for the Temperley-Lieb algebra.
We will show how their decomposition matrices can all be described by certain parabolic Kazhdan-Lusztig polynomials. These, in turn, have very nice combinatorial description in terms of oriented generalised Temperley-Lieb algebras.
Cédric Lecouvey (Université de Tours)
Title -- The Kostka polynomials and their generalizations
Abstract: The topic of this mini-course will focused on the notion of Kostka polynomial, a natural quantization of the Kostka numbers. These polynomials appear in many different contexts related to representation theory of symmetric groups (Green functions), Lie algebras (graded multiplicities), algebraic combinatorics (crystal graphs) and quantum groups (energy function and one-dimensional sums). After recalling their definition and certain of the main properties of the Kostka polynomials in type A, we shall introduce more recent results on their generalizations to the other root systems.
Here is a directory where you can find slides of the lectures and additional contents:
https://plmbox.math.cnrs.fr/d/481f97d97e7b469590b6/
Paul Purdon Martin (University of Leeds)
Title -- Statistical Mechanics, Representation Theory and diagram categories
Abstract: In this mini course we are interested in how computation in statistical mechanics, and representation theory, fit together. Each one informs and provides assistance for, and a certain path through, the other. A short course must lean heavily on examples, and this is where the diagram categories come in. For us, a diagram category is a subcategory of the partition category (of Potts model transfer matrix algebras), or a deformation thereof. So this can include Brauer categories, blob categories, Temperley-Lieb, and so on.
Evgeny Mukhin (Indiana University)
Title -- Gaudin models
Abstract: Lecture 1. Gaudin models and Bethe ansatz.
I will define the Gaudin models and discuss a few basic properties. Then I will explain the Bethe ansatz - the method invented by physicists to look for the eigenvalues and eigenvectors of the Gaudin Hamiltonians. I will discuss the power and limitations of the Bethe ansatz method.
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Lecture 2. The gl(m)-gl(n) duality of Gaudin models.
I will briefly discuss the classical Schur-Weyl and gl(n)-gl(m) dualities and then prove the corresponding statement for the Gaudin models. The proof is reduced to a generalization of the famous Schur formula in linear algebra to the case of particular matrices with non-commuting entries. Time permitting, I will explain the relation to the property of bispectrality in the KP hierarchy.
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Lecture 3. The Gaudin models and the Schubert Calculus.
I will use the Bethe ansatz to explain that the algebra of gl(n) Gaudin Hamiltonians acting in a module coincides with the corresponding scheme theoretic intersection of Schubert varieties in a Grassmannian. This can be thought of as a version of Geometric Langlands duality. I will discuss the implications of this duality for both geometry and integrable systems.
