Past Junior Algebra and Representation Theory Seminar

Spectral sequences are computational tools to find the (co-)homology of mathematical objects and are used across various fields. In this talk I will focus on the LHS spectral sequence, which we associate to an extension of groups to compute group cohomology. The first part of the talk will serve as introduction to both group cohomology and general spectral sequences, where I hope to provide and intuition and some reduced formalism. As main example, and core of this talk, we will look at the LHS spectral sequence associated to the group extension $(\mathbb{Z}/3\mathbb{Z})^3 \rightarrow S \rightarrow \mathbb{Z}/3\mathbb{Z}$, where $S$ is a Sylow-3-subgroup of $SL_2(\mathbb{Z}/9\mathbb{Z})$. In particular I will present arguments that all differentials on the $E^2$ page vanish.

In this talk I will discuss my extension of the Koszul duality theory of Beilinson, Ginzburg, and Soergel to the more general setting of quasi-abelian categories. In particular, I will define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders' embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of derived categories for certain categories of modules over Koszul monoids and their duals. The key examples of categories for which this theory works are the categories of complete bornological spaces and the categories of inductive limits of Banach spaces. These categories frequently appear in derived analytic geometry.

In this talk, I will talk about the category of coherent sheaves on the affine Flag variety of a simply-connected reductive group over $\mathbb{C}$. In particular, I'll explain how the convolution product naturally leads to a construction of the Iwahori-Hecke algebra, and present some combinatorics related to computing duals in this category.

Affine Hecke algebras and their representations play an important role in the representation theory of p-adic groups since they classify smooth representations generated by Iwahori-fixed vectors. The Deligne-Langlands correspondence, which was proved by Kazhdan and Lusztig, parametrises these representations by geometric data on the Langlands dual group. This talk is supposed to be a gentle introduction to this topic. I will also briefly talk about how this correspondence can be lifted to the derived level.

The Bruhat-Tits building is a mysterious combinatorial gadget that encodes key information about the structure and representation theory of a p-adic group. In this talk we will talk about apartments, buildings, and all the furnishings therein to hopefully demystify this beautiful subject.

Let $F$ be a finite field or a $p$-adic field. One method of constructing irreducible representations of $G = GL_2(F)$ is to consider spaces on which $G$ naturally acts and look at the representations arising from invariants of these spaces, such as the action of $G$ on cohomology groups. In this talk, I will discuss how this goes for abstract representations of $G$ (when $F$ is finite), and smooth representations of $G$ (when $F$ is $p$-adic). The first space is an affine algebraic variety, and the second a tower of rigid spaces. I will then mention some recent results about how this tower allows us to construct new interesting $p$-adic representations of $G$, before explaining how trying to adapt these methods leads naturally to considerations about certain geometric properties of these spaces.

The Specht modules are of fundamental importance to the representation theory of the symmetric group, and their 0th cohomology is understood through entirely combinatorial methods due to Gordon James. Over fields of odd characteristic, Hemmer proposed a similar combinatorial approach to calculating their 1st degree cohomology, or extensions by the trivial module. This combinatorial approach motivates the definition of universal $p$-ary designs, which we shall classify. We then explore the consequences of this classification to problem of determining extensions of Specht modules. In particular, we classify all extensions of Specht modules indexed by two-part partitions by the trivial module and shall see some far-reaching conditions on when the first cohomology of a Specht module is trivial.

Locally analytic representations of $p$-adic Lie groups are of interest in several branches of number theory, for example in the theory of automorphic forms and in the $p$-adic local Langlands program. To better understand these representations, Ardakov-Wadsley introduced a sheaf of infinite order differential operators $\overparen{\mathcal{D}}$ on smooth rigid analytic spaces, which resulted in several Beilinson-Bernstein style localisation theorems. In this talk, we discuss the current research on analogues of a Riemann-Hilbert correspondence for $\overparen{\mathcal{D}}$-modules, and what this has to do with complete convex bornological vector spaces.

Let $K$ be an algebraically closed field. Given three elements of some Lie algebra over $K$, we say that these elements form an $sl_2$-triple if they generate a subalgebra which is a homomorphic image of $sl_2(K).$ In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of $sl_2$-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic $p$. In particular, I will focus on the results in classical Lie algebras, which can be found as subsets of $gl_n(K)$.

The Krull dimension is an ideal-theoretic invariant of an algebra. It has an important meaning in algebraic geometry: the Krull dimension of a commutative algebra is equal to the dimension of the corresponding affine variety/scheme. In my talk I'll explain how this idea can be transformed into a tool for measuring non-commutative rings. I'll illustrate this with important examples and techniques, and describe what is known for Iwasawa algebras of compact $p$-adic Lie groups.

It is a well-known fact that Boolean rings, those rings in which $x^2 = x$ for all $x$, are necessarily commutative. There is a short and completely elementary proof of this. One may wonder what the situation is for rings in which $x^n = x$ for all $x$, where $n > 2$ is some positive integer. Jacobson and Herstein proved a very general theorem regarding these rings, and the proof follows a widely applicable strategy that can often be used to reduce questions about general rings to more manageable ones. We discuss this strategy, but will also focus on a different approach: can we also find ''elementary'' proofs of some special cases of the theorem? We treat a number of these explicit computations, among which a few new results.

The classification of finite simple groups shows that many (those of Lie type) are obtained as (projectivisations of) subgroups of some $GL_{n}(\mathbb{F}_{q})$.

Green first determined the character table of any $GL_{n}(\mathbb{F}_{q})$ in his influential 1955 paper, while others have since given more explicit constructions of certain `cuspidal' representations.

In this talk, I will introduce parabolic induction as a means of obtaining representations of $GL_{n}(\mathbb{F}_{q})$ from those of $GL_{m}(\mathbb{F}_{q})$ where $m<n$.

Finding the irreducible representations of any $GL_{n}(\mathbb{F}_{q})$ then becomes inductive on $n$ for fixed $q$, with the cuspidal representations serving as atoms for this process.

Harish-Chandra's philosophy of cusp forms reduces the problem to the following two steps:

•  Find the cuspidal representations of any $GL_{n}(\mathbb{F}_{q})$
•  Determine the irreducible components of any representation $\sigma_{1}\circ\dots\circ\sigma_{k}$ parabolically induced from cuspidals $\sigma_{i}$

The majority of my talk will then aim to address each of these points.

Let $G$ be a group acting transitively on a finite set $\Omega$. Then $G$ acts on $\Omega \times \Omega$ componentwise. Define the orbitals to be the orbits of $G$ on $\Omega \times \Omega$. The diagonal orbital is the orbital of the form $\Delta = \{(\alpha, \alpha) \mid \alpha \in \Omega \}$. The others are called non-diagonal orbitals. Let $\Gamma$ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set $\Omega$ and edge set $(\alpha,\beta) \in \Gamma$ with $\alpha, \beta \in \Omega$. If the action of $G$ on $\Omega$ is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding specific bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups. 

How to calculate the minimal number of summands of a nonzero polynomial in a given polynomial ideal? In this talk, we first discuss the roots of this question in computational algebra. Afterwards, we switch to the viewpoint of a commutative algebraist. In particular, we see that classical tools from this field, such as primary decomposition or the Castelnuovo–Mumford regularity, fail to provide a solution to this problem. Finally, we discuss a concrete example: A standard determinantal ideal generated by $t$-minors does not contain any polynomials with fewer than $t!/2$ terms.

Anabelian geometry asks how much we can say about a variety from its fundamental group. In 1997 Shinichi Mochizuki, using p-adic hodge theory, proved a fundamental anabelian result for the case of p-adic fields. In my talk I will discuss representation theoretical data which can be reconstructed from an absolute Galois group of a field, and also types of representations that cannot be constructed solely from a Galois group. I will also sketch how these types of ideas can potentially give many new results about p-adic Galois representations.

The Temperley-Lieb algebra is a diagrammatic algebra - defined on a basis of "string diagrams" with multiplication given by "joining the diagrams together".  It first arose as an algebra of operators in statistical mechanics but quickly found application in knot theory (where Jones used it to define his famed polynomial) and the representation theory of $sl_2$.  From the outset, the representation theory of the Temperley-Lieb algebra itself has been of interest from a physics viewpoint and in characteristic zero it is well understood.  In this talk we will explore the representation theory over mixed characteristic (i.e. over positive characteristic fields and specialised at a root of unity).  This gentle introduction will take the listener through the beautifully fractal-like structure of the algebras and their cell modules with plenty of examples.

Introduced by Fomin and Zelevinsky in 2002, cluster algebras have become ubiquitous in algebra, combinatorics and geometry. In this talk, I'll introduce the notion of a cluster algebra and present the approach of Kang-Kashiwara-Kim-Oh to categorify a large class of them arising from quantum groups. Time allowing, I will explain some recent developments related to the coherent Satake category.

For $G$ a finite group, $p$ a prime and $(K, \mathcal{O}_K, k)$ a $p$-modular system the group ring $\mathcal{O}_K G$ is an $\mathcal{O}_k$-order in the $K$-algebra $KG.$ Graduated $\mathcal{O}_K$-orders are a particularly nice class of $\mathcal{O}_K$-orders first introduced by Zassenhaus. In this talk will see that an $\mathcal{O}_K$-order $\Lambda$ in a split $K$-algebra $A$ is graduated if the decomposition numbers for the regular $A$-module are no greater than $1$. Furthermore will see that graduated orders can be described (not uniquely) by a tuple $n$ and a matrix $M$ called the exponant matrix. Finding a suitable $n$ and $M$ for a graduated order $\Lambda$ in the $K$-algebra $A$ provides a parameterisation of the $\Lambda$-lattices inside the regular $A$-module. Understanding the $\mathcal{O}_K G$-lattices inside representations of certain groups $G$ is of interest to those involved in the Langlands programme as well as of independent interest to algebraists.

Saturated fusion systems capture and abstract conjugacy in $p$-subgroups of finite groups and have recently found application in finite group theory, representation theory and algebraic topology. In this talk, we describe a situation in which we may identify a rank $2$ amalgam within $\mathcal{F}$ and, using some local group theoretic techniques, completely determine $\mathcal{F}$ up to isomorphism.

This is an elementary talk introducing the rational Cherednik algebra and its representations. Especially, we are interested in the case of complex reflection group. A tool called the Dunkl-Opdam subalgebra is used to decompose the standard modules into weight spaces and to construct the correspondence with the partitions of integers. If time allows, we might explore the concept of unitary representation and what condition a representation needs to satisfy to be qualified as one.

The well-known Schur-Weyl duality provides a link between the representation theories of the general linear group $GL_n$ and the symmetric group $S_r$ by studying tensor space $(\mathbb{C}^n)^{\otimes r}$ as a ${(GL_n,S_r)}$-bimodule. We will discuss a few variations of this idea which replace $GL_n$ with some other interesting algebraic object (e.g. O$_n$ or $S_n$) and $S_r$ with a so-called diagram algebra. If time permits, we will also briefly look at how this can be used to define Deligne's category which 'interpolates' Rep($S_t$) for any complex number $t \in \mathbb{C}$.

Buildings are geometric structures useful in understanding certain classes of groups. In a series of papers written during the 1980s, Ronan and Smith developed the theory of “presheaves on buildings”. By constructing a coefficient system consisting of kP-modules (where P is the stabiliser of a given simplex), and computing the sheaf homology, they proved several results relating the homology spaces with the irreducible G-modules. In this talk we discuss their methods as well as our implementation of the algorithms, which has allowed us to efficiently compute the irreducible representations of some groups of Lie type.

There is a connection between certain smooth representations of a reductive p-adic group and the representations of the Iwahori-Hecke algebra of this p-adic group. This Iwahori-Hecke algebra is a specialisation of a more general affine Hecke algebra. In this talk, we will discuss affine Hecke algebras and graded Hecke algebras. We will state a result from Lusztig (1989) that relates the representation theory of an affine Hecke algebra and a particular graded Hecke algebra and we will present a simple example of this relation.

The Drinfeld half-plane is a rigid analytic variety over a p-adic field. In this talk, I will give an overview of the geometric aspects of this space and describe its connection with representation theory.

One of the key objects in studying the Hilbert Scheme of points in the plane is a torus action of $(\mathbb{C}^*)^2$. The fixed points of this action correspond to monomial ideals in $\mathbb{C}[x,y]$, and this gives a connection between the geometry of Hilbert schemes and partition combinatorics. Using this connection, one can extract identities in partition combinatorics from algebro-geometric information and vice versa. I will give some examples of combinatorial identities where as yet the only proofs we have rely on the geometry of Hilbert schemes. If there is time, I will also sketch out a hope that such identities can also be seen by representations of appropriately chosen algebras.

One of the fundamental examples of geometric representation theory is the Springer correspondence which parameterises the irreducible representations of the Weyl group of a lie algebra in terms of nilpotent orbits of the lie algebra and irreducible representations of the equivariant fundamental group of said nilpotent orbits. If you don’t like geometry this may sound entirely mysterious. In this talk I will hopefully offer a gentle introduction to the subject and present a preprint by Lusztig (2020) which gives an entirely algebraic description of the springer correspondence.

A Real representation of a $C_2$-graded group $H < G$ ($H$ an index two subgroup) is a complex representation of $H$ with an action of the other coset $G \backslash H$ (“odd" elements) satisfying appropriate algebraic coherence conditions. In this talk I will present three such Real representation theories. In these, each odd element acts as an antilinear operator, a bilinear form or a sesquilinear form (equivalently a linear map to $V$ from the conjugate, the dual, or the conjugate dual of $V$) respectively. I will describe how these theories are related, how representations in each are classified, and how the first generalises the classical representation theory of $H$ over the real numbers - retaining much of its beauty and subtlety.

Let $k$ be a field and $A$ a $k$-algebra. The classical Quillen's Lemma states that if $A$ if is equipped with an exhaustive filtration such that the associated graded ring is commutative and finitely generated $k$-algebra then for any finitely generated $A$-module $M$, every element of the endomorphism ring of $M$ is algebraic over $k$. In particular, Quillen's Lemma may be applied to the enveloping algebra of a finite dimensional Lie algebra. I aim to present an affinoid version of Quillen's Lemma which strengthness a theorem proved by Ardakov and Wadsley. Depending on time, I will show how this leads to an (almost) classification of the primitive spectrum of the affinoid enveloping algebra of a semisimple Lie algebra.

The Schur functor and its inverses give an important connection between the representation theories of the symmetric group and the general linear group. Kleshchev and Nakano proved in 2001 that when the characteristic of the field is at least 5, the image of the Specht module under the inverse Schur functor is isomorphic to the dual Weyl module. In this talk I will address what happens in characteristics 2 and 3: in characteristic 3, the isomorphism holds, and I will give an elementary proof of this fact which covers also all characteristics other than 2; in characteristic 2, the isomorphism does not hold for all Specht modules, and I will classify those for which it does. Our approach is with Young tableaux, tabloids and Garnir relations.

Polynomial rings $R[X]$ are a fundamental construction in commutative algebra, under which Hilbert's basis theorem controls a finiteness property: being Noetherian. We will describe the picture for the non-commutative world; this leads us towards other interesting finiteness conditions.

Quiver varieties, as first studied by Grojnowski and Nakajima, form an interesting class of geometric objects, which can be constructed by an array of different techniques (GIT, symplectic and Hyperkaehler reduction). In this talk, we will explain how to construct these varieties, and how their homology gives rise to a categorification of the representations of Kac-Moody Lie algebras

This talk will introduce the notion of categorical rigidity and the automorphism class group of a category. We will proceed with calculations for several important categories, hopefully illuminating the inverse relationship between the automorphisms of a category and the extent to which the structure of its objects is determined categorically. To conclude, some discussion of what progress there is on currently open/unknown cases.

For a prime number p, we will consider completed group algebras, or Iwasawa algebras, of the form kG, for G a complete p-valued group of finite rank, k a field of characteristic p. Classifying the ideal structure of Iwasawa algebras has been an ongoing project within non-commutative algebra and representation theory, and we will discuss ideas related to this topic based on previous work and try to extend it. An important concept in studying ideals of group algebras is the notion of controlling ideals, where we say a closed subgroup H of G controls a right ideal I of kG if I is generated by a subset of kH. It was proved by Konstantin Ardakov in 2012 that for G nilpotent, every faithful prime ideal of kG is controlled by the centre of G, and it follows that the prime spectrum of kG can be realised as the disjoint union of commutative strata. I am hoping to extend this to a more general case, perhaps to when G is solvable. A key step in the proof is to consider a faithful prime ideal P in kG, and an automorphism of G, trivial mod centre, that fixes P. By considering the Mahler expansion of the automorphism, and approximating the coefficients, we can examine sequences of bounded k-linear functions of kG, and study their convergence. If we find that they converge to an appropriate quantized divided power, we can find proper open subgroups of G that control P. I have extended this notion to larger classes of automorphisms, not necessarily trivial mod centre, using which this proof can be replicated, and in some cases extended to when G is abelian-by-procyclic. I will give some examples, for G with small rank, for which these ideas yield results.

In recent years, Ardakov and Wadsley have been interested in extending the classical theory of Beilinson-Bernstein localisation to different contexts. The classical proof relies on fundamental geometric properties of the dual nilcone of a semisimple Lie algebra; in particular, finding a nice desingularisation of the nilcone and demonstrating that it is normal. I will attempt to explain the relationship between these properties and the proof, and discuss some areas of my own work, which focuses on proving analogues of these results in the case where the characteristic of the ground field K is bad.

In this talk, I wish to address the problem of evaluating an integral on an n-dimensional complex vector space whose n-form of integration has poles along a union of (affine) hyperplanes, following the work of Heckman and Opdam. Such situation arise often in the harmonic analysis of a reductive group and when that is the case, the singular hyperplane arrangement in question is dictated by the root system of the group. I will then try to explain how we can relate the intersection lattice of the hyperplane arrangement with nilpotent orbits of a complex Lie algebra related to the root system in question.

Abstract: Triangulations of surfaces serve as important examples for cluster theory, with the natural operation of “diagonal flips” encoding mutation in cluster algebras and categories. In this talk we will focus on the combinatorics of mutation on marked surfaces with infinitely many marked points, which have gained importance recently with the rising interest in cluster algebras and categories of infinite rank. In this setting, it is no longer possible to reach any triangulation from any other triangulation in finitely many steps. We introduce the notion of mutation along infinite admissible sequences and show that this induces a preorder on the set of triangulations of a fixed infinitely marked surface. Finally, in the example of the completed infinity-gon we define transfinite mutations and show that any triangulation of the completed infinity-gon can be reached from any other of its triangulations via a transfinite mutation. The content of this talk is joint work with Karin Baur.

Abstract: We define the Hochschild complex and cohomology of a monoid in an Ab-enriched monoidal category. Then we interpret some of the lower dimensional cohomology groups and discuss when the cohomology ring happens to be graded-commutative.

Firstly I will outline Dirac cohomology for graded Hecke algebras and the branching rules for the projective representations of $S_n$. Combining these notions with the Jucys-Murphy elements for $\tilde{S}_n$, that is the double cover of the symmetric group, I will go through a method to completely describe the spectrum data for the Jucys-Murphy elements for $\tilde{S}_n$. If time allows I will also explain how this spectrum data gives rise to a a concrete description for the matrices of the action of $\tilde{S}_n$.

We will introduce some ideas from geometric representation theory, including basic D-modules, (a realisation of) infinity categories, and the Ran space, and then go on to define chiral and factorisation algebras.
This talk has no prerequisites.

The computation of multidimensional persistent homology is one of the major open problems in topological data analysis. 

One can define r-dimensional persistent homology to be a functor from the poset category N^r, where N is the poset of natural numbers, to the category of modules over a commutative ring with identity. While 1-dimensional persistent homology is theoretically well-understood and has been successfully applied to many real-world problems, the theory of r-dimensional persistent homology is much harder, as it amounts to understanding representations of quivers of wild type. 

In this talk I will introduce persistent homology, give some motivation for how it is related to the study of data, and present recent results related to the classification of multidimensional persistent homology.