Abstracts

We discuss how ideas from knot theory and categorical Lie theory can be used to understand the representation theory of Hecke algebras of complex reflection groups. The talk will focus mostly on the combinatorics and diagrammatics underlying these structures.

A Noetherian ring $S$ whose simple modules have the property that their finitely generated essential extensions are Artinian is said to satisfy property $(\diamond)$. In 1958 Matlis proved that commutative Noetherian rings satisfy this property. In this talk we will discuss $(\diamond)$ for skew polynomial rings $S=R[\theta; \alpha]$ where $R$ is a commutative Noetherian ring and $\alpha$ is an automorphism of $R$. When such a skew polynomial ring S satisfies $(\diamond)$ turns out to be a surprisingly subtle question, not completely settled, and which leads naturally to other fundamental representation-theoretic questions concerning these rings. A complete characterization is given when $R$ is an affine algebra over a field $K$ and $\alpha$ is a $K$-automorphism. This is joint work with Ken Brown and Jerzy Matczuk.

In this talk, I will show a description of the irreducible smooth representations of the unitriangular groups and upper triangular groups over a non-archimedean field (for example the $p$-adic field), and related groups (the algebra groups and the unit group of split basic algebras). In particular, I will discuss the admissibility and unitarisability of the irreducible smooth representations. In the end, if time permits I will give some examples.

In this talk, I will discuss techniques to study irreducible representations of various quantum algebras at roots of unity, focussing in particular on the example of quantum determinantal varieties. This is based on joint works with Jason Bell, Samuel Lopes and Alexandra Rogers.

In the construction of the irreducible projective characters of symmetric groups, I. Schur introduced the so called Schur $Q$-functions $Q_{\lambda}$, indexed by strict partitions. They can be scaled to define Schur $P$-functions $P_{\lambda} = 2^{-\ell(\lambda)} Q_{\lambda}$ where $\lambda$ has $\ell(\lambda)$ parts. Both are special cases of Hall-Littlewood polynomials, and are bases for the subring of the symmetric functions over $\mathbb{Q}$ generated by the odd degree power sums. We have then the linear expansions of the product $P_{\mu}P_{\nu} = \sum\limits_{\lambda} f_{\mu\nu}^{\lambda} P_{\lambda}$ and of the skew Schur $Q$-functions $Q_{\lambda/\mu} = \sum\limits_{\nu} f^{\lambda}_{\mu\nu} Q_{\nu}$, where the structure constants $f_{\mu\nu}^{\lambda}$ are non negative integers, called the shifted Littlewood-Richardson (LR) coefficients. While the symmetry $f_{\mu\nu}^{\lambda} = f_{\nu\mu}^{\lambda}$ is an immediate consequence of the $P$-product, it is  a natural problem to exhibit this and other shifted LR symmetries in the combinatorial objects that they do enumerate.

Gillespie, Levinson and Purbhoo (2017) introduced a doubled type A crystal structure on shifted tableaux. This crystal graph, denoted $\mathcal{B} (\lambda/\mu, n)$, has vertices the skew shifted tableaux of shape $\lambda/\mu$ on the alphabet $ [n]' =\{1' < 1 < \ldots < n' < n\}$, and the double edges,  prime and unprimed-coloured, corresponding to the action of the primed and unprimed lowering and raising operators which commute with the shifted jeu de taquin.

One can define on $\mathcal{B} (\lambda/\mu, n)$ a shifted analogue of  crystal reflection operator $\sigma_i$ that coincides, after rectification, with the restriction of the shifted Schützenberger involution to the alphabet $\{i' < i < (i+1)' < i+1\}$, $i\in [n]$. Unlike type A crystals, the operators $\sigma_i$ do not define an action of the symmetric group $\mathfrak{S}_n$ on $\mathcal{B} (\lambda/\mu, n)$, as the braid relations do not hold. However, the restrictions of the Schützenberger involution to alphabets of the form $\{p' < p < \ldots < q' < q\}\subseteq [n]'$, where the crystal reflection operators are included, define an action of the cactus group $J_n$ on this crystal. In this talk we will present some recent results on this action as well as the symmetries on the shifted LR coefficients it yields.

Gentle algebras are a class of algebras of tame representation type, meaning it is often possible to get a global understanding of their representation theory. In this talk, we will show how to encode the module category of a gentle algebra using combinatorics of a surface. This is joint work with Karin Baur (Graz/Leeds).

Let $G$ be a connected, simply connected nilpotent Lie group. For $p,q\in [1,+\infty]$, the $L^p-L^q$ analogue of Morgan’s theorem was proved only for two step nilpotent Lie groups. In order to study this problem in larger subclasses, we formulate and prove a version of $L^p-L^q$ Morgan’s theorem on nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. An analogue of Cowling-Price Theorem is also provided in the same context. Representation theory and a localized Plancherel formula play an important role in the proof.

The modular representation theory of the general linear Lie algebra over a field of positive characteristic is a subject which has been studied intensively for around seventy years. The simple modules for the enveloping algebra all factor through certain finite dimensional quotients, known as reduced enveloping algebras. In 2002 Premet demonstrated that these reduced enveloping algebras are actually matrix algebras over certain endomorphism rings, which are now known as restricted finite W-algebras. Hence these finite dimensional algebras are the fundamental unit of currency when trying to understand $\operatorname{gl}_n$-modules. In this talk I will explain a joint work with Simon Goodwin, in which we provide a presentation for the restricted finite W-algebras, by showing that they are isomorphic to quotients of a truncated shifted Yangian.

A famous theorem of Auslander states that a finite dimensional algebra is of finite representation type if and only if every module is additively equivalent to a finite dimensional one. This establishes a correlation between quantity (of indecomposable finite dimensional modules) and size (of indecomposable modules).

We will discuss the ocurrence of an analogous correlation for torsion pairs. Indeed, for a finite dimensional algebra $A$ we prove that the category of $A$-modules has finitely many torsion pairs if and only if every torsion pair is generated by a finite dimensional module. Time allowing, we will also mention an analogous result for the derived category of $A$. This is based on joint work with L. Angeleri Hügel and F. Marks and on ongoing joint work with L. Angeleri Hügel and D. Pauksztello.

According to Drozd’s trichotomy theorem for every finite dimensional algebra $A$ there are three possibilities 

• there is only finitely many indecomposable module over $A$, in which case $A$ is said to have finite representation type; 
• for every natural number d there are only finitely many one-parameter families of indecomposable $A$-modules of dimension $d$.  In this case $A$ is said to have tame representation type; 
• the representation theory of $A$ is as complicated as the representation theory of $k[x,y]$, in which case $A$ is said to have wild representation type. 

The Borel-Schur algebras $S(B,n,r)$ is a family of finite-dimensional algebras depending on two natural numbers $n$ and $r$. Their representation theory is closely related to the representation theory of Schur algebras $S(n,r)$,  representation theory of the general linear groups $\operatorname{GL}_n$ in defining characteristic,  and representation theory of symmetric group. In this talk I will describe the representation types of the algebras $S(B,n,r)$ for every $n$ and $r$. 

This is joint work with Karin Erdmann and Ana Paula Santana.