Past Junior Algebra and Representation Theory Seminar

A monoid S is said to be weakly right coherent if every finitely generated right ideal of S is finitely presented as a right S-act. It is known that S is weakly right coherent if and only if it satisfies the following conditions: S is right ideal Howson, meaning that the intersection of any two finitely generated right ideals of S is finitely generated; and the right annihilator congruences r(a)={(u,v) in S x S | au=av} for each a in S are finitely generated as right congruences.

This talk will introduce basic semigroup theoretic concepts as is necessary before briefly surveying some important coherency-related results. Closure properties of the classes of monoids satisfying each of the above properties will be shared, with details explored for a specific construction. Time permitting, connections with axiomatisation will be discussed.

This talk will in part be based on a paper written with coauthors Craig Miller and Victoria Gould, preprint available at: arXiv:2411.03947.

The category PAb of profinite abelian groups is an abelian category with many nice properties, which allows us to do most of standard homological algebra. The category PAb naturally embeds into the category TAb of topological abelian groups, but TAb is not abelian, nor does it have a satisfactory theory of tensor products. On the other hand, PAb also naturally embeds into the category CondAb of "condensed abelian groups", which is an abelian category with nice properties. We will show that the embedding of profinite modules into condensed modules (actually, into "solid modules") preserves usual homological notions such Ext and Tor, so that the condensed world might be a better place to study profinite modules than the topological world.

Hilbert’s fourteenth problem is concerned with whether invariant rings under algebraic group actions are finitely generated. A number of examples have been constructed since the mid-20th century which demonstrate that this is not always the case. However such examples by their nature are difficult to construct, and we know little about their underlying structure. This talk aims to provide an introduction to the topic of Hilbert’s fourteenth problem, as well as the finite generation ideal - a key tool used to further understand these counterexamples. We focus particularly on the example constructed by Daigle and Freudenberg at the turn of the 21st century, and describe the work undertaken to compute the finite generation ideal of this example. 

There is a well-known relationship between finite W-algebras and Yangians. The work of Rogoucy and Sorba on the "rectangular case" in type A eventually led Brundan and Kleshchev to introduce shifted Yangians, which surject onto the finite W-algebras for general linear Lie algebras. Thus, these W-algebras can be realised as truncated shifted Yangians. In parallel, the work of Ragoucy and then Brown showed that truncated twisted Yangians are isomorphic to the finite W-algebra associated to a rectangular nilpotent element in a Lie algebra of type B, C or D. For many years there has been a hope that this relationship can be extended to other nilpotent elements.

I will report on a joint work with Lewis Topley in which we introduced the shifted twisted Yangians, following the work of Lu-Wang-Zhang, and described Poisson isomorphisms between their truncated semiclassical degenerations and the functions Slodowy slices associated with even nilpotent elements in classical simple Lie algebras( which can be viewed as semiclassical W-algebras). I will also mention a work in progress with Lu-Peng-Topley-Wang which deals with the quantum analogue of our theorem.

I will also recall what Poisson algebras and (filtered) quantizations are and give a brief intro to Slodowy slices, finite W-algebras and Yangians so that the talk should be quite accessible.

Bernstein–Gelfand–Gelfand (BGG) resolutions and the Grothendieck–Cousin complex both play central roles in modern algebraic geometry and representation theory. The BGG approach provides elegant, combinatorial resolutions for important classes of modules especially those arising in Lie theory; while Grothendieck–Cousin complexes furnish a powerful framework for computing local cohomology via filtrations by support. In this talk, we will give an overview of these two constructions and illustrate how they arise from the same categorical consideration.

We begin with the Fontaine--Wintenberger isomorphism, which gives an example of an extension of Q_p and of F_p((t)) with isomorphic absolute Galois groups. We explain how by trying to lift maps on mod p reductions one encounters Witt vectors. Next, by trying to apply the theory of Witt vectors to the two extensions, we encounter the idea of tilting. Perfectoid fields are then defined more-or-less so that tilting may be reversed. We indicate the proof of the tilting correspondence for perfectoid fields following the Witt vectors approach, classifying the untilts of a given characteristic p perfectoid field along the way. To end, we touch upon the Fargues--Fontaine curve and the geometrization of l-adic local Langlands as motivation for globalizing the tilting correspondence to perfectoid spaces.

The Junior Algebra and Representation Theory Seminar will kick-off the start of Hilary Term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

In ’95 Lusztig gave a local Langlands correspondence for unramified representations of inner to split adjoint groups combining many deep results from type theory and geometric representation theory. In this talk I will present a gentle reformulation of his construction revealing some interesting new structures, and with a view toward proving functoriality results in this framework. 

A non-nilpotent graph Γ(G) of a finite group G has elements of G as vertices, with x and y joined by an edge iff a subgroup generated by these two elements is non-nilpotent. During the talk we will prove several (often unrelated) properties of this construction; for instance, any simple graph can be found as an induced subgraph of Γ(G) for some (solvable) group G. The talk is based on my article "A few remarks on the theory of non-nilpotent graphs" (May 2023).

After recalling how Hecke algebras occur in the representation theory of reductive groups, we will introduce affine Hecke algebras through a combinatorial object called a root datum. Through a worked example we will construct a filtration on the affine Hecke algebra from which we obtain the graded Hecke algebra. This has a role analogous to the Lie algebra of an algebraic group.

We will discuss star operations on these rings, with a view towards the classical problem of studying unitary representations of reductive groups.

The Bruhat-Tits building is a crucial combinatorial tool in the study of reductive p-adic groups and their representation theory. Given a p-adic group, its Bruhat-Tits building is a simplicial complex upon which it acts with remarkable properties. In this talk I will give an introduction to the Bruhat-Tits building by sketching its definition and going over some of its basic properties. I will then show the usefulness of the Bruhat-Tits by determining the maximal compact subgroups of a p-adic group up to conjugacy by using the Bruhat-Tits building.

The Junior Algebra and Representation Theory Seminar will kick-off the start of the academic year with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Let F be a p-adic field. In this talk I'll study the Om(F)-distinction of some specific principal series representations  of Glm(F). The main goal is to give a computing method to see if those representations are distinguished or not so we can also explicitly find a non zero  Om(F)-equivariant linear form. This linear form will be given by the integral of the representation's matrix coefficient over Om(F).
 

After explaining on what specific principal series representations I'm working and why I need those specificities, I'll explain the different steps to compute the integral of my representation's matrix coefficient over Om(F). I'll explicitly give the obtained result for the case m=3. After that I'll explain an asymptotic result we can obtain when we can't compute the integral explicitly.

It is well-known that the set of irreducible (finite-dimensional) representations of a semisimiple complex Lie algebra g can be indexed by the dominant weights. The Borel-Weil theorem asserts that they can be seen geometrically as the global sections of line bundles over the flag variety. The Borel-Weil-Bott theorem computes the higher sheaf cohomology groups. There are several ways to prove the Borel-Weil-Bott theorem, which we will discuss. The classical idea is to study how the Casimir operator acts on the sheaf of sections of line bundles. Instead of this, the geometric idea is trying to compute the Doubeault cohomology, transferring the sheaf cohomology to the Lie algebra cohomology. The algebraic idea is to realize that the sheaf cohomology group can be computed by the derived functor of the induction, by using the Peter-Weyl the Borel-Weil theorem can be shown immediately.

Choose your favourite connected graph $\Delta$ and shade a subset $J$ of its vertices. The intersection arrangement associated to the data $(\Delta, J)$ is a collection of real hyperplanes in dimension $|Jc|$, first defined by Iyama and Wemyss. This construction involves taking the classical Coxeter arrangement coming from $\Delta$ and then setting all variables indexed by $J$ to be zero. It turns out that for many choices of $J$ the chambers of the intersection arrangement admit a nice combinatorial description, along with a wall crossing rule to pass between them. I will start by making all this precise before discussing my work to classify tilings of the hyperbolic plane arising as intersection arrangements. This has applications to local notions of stability conditions using the tilting theory of contracted preprojective algebras.

The Hecke category first rose to prominence through the proof of the Kazhdan-Lusztig conjecture. Since then, the Hecke category has proven to be a fundamental object in representation theory with many interesting applications to modular representation theory, quantum groups, knot theory, categorification and diagrammatic algebra. This talk aims to give a gentle introduction to the Hecke category. We will first discuss the geometric incarnation of the Hecke category and how it was used to prove the Kazhdan-Lusztig conjecture. Then, we move on to a more modern approach due to Soergel and Elias-Williamson which is purely algebraic, and we will discuss some recent advances in representation theory based on this approach.

Kashiwara’s theory of crystal bases provides a powerful tool for studying representations of quantum groups. Crystal bases retain much of the structural information of their corresponding representations, whilst being far more straightforward and ‘stripped-back’ objects (coloured digraphs). Their combinatorial description often enables us to obtain concrete realizations which shed light on the representations, and moreover turn challenging questions in representation theory into far more tractable problems.

After reviewing the construction and basic theory regarding quantum groups, I will introduce and motivate crystal bases as ‘nice q=0 bases’ for their representations. I shall then present (in both finite and affine types) the construction of Young wall models in the important case of highest weight representations. Time permitting, I will finish by discussing some applications across algebra and geometry.

In this talk I will discuss the representation theory of truncated current Lie algebras in prime characteristic. I will first give an introduction to modular representation theory for general restricted Lie algebras and introduce the Kac-Weisfeiler conjectures. Then I will introduce a family of Lie algebras known as truncated current Lie algebras, and discuss their representation theory and its relationship with the representation theory of reductive Lie algebras in positive characteristic.

The orbit method is a fundamental tool to study a finite dimensional solvable Lie algebra g. It relates the annihilators of simple U(g)-module to the coadjoint orbits of the adjoint group on g^* . In my talk, I will extend this story to the Witt algebra – a simple (non-solvable) infinite dimensional Lie algebra which is important in physics and representation theory. I will construct an induced module from an element of W^* and show that its annihilator is a primitive ideal. I will also construct an algebra homomorphism that allows one to relate the orbit method for W to that of a finite dimensional solvable algebra.

Associated to a complex admissible representation of a p-adic group is an invariant known is the "canonical dimension". It is closely related to the more well-studied invariant called the "wavefront set". The advantage of the canonical dimension over the wavefront set is that it allows for a completely different approach in computing it compared to the known computational methods for the wavefront set. In this talk we illustrate this point by finding a lower bound for the canonical dimension of any depth-zero supercuspidal representation, which depends only on the group and so is independent of the representation itself. To compute this lower bound, we consider the geometry of the associated Bruhat-Tits building.

The Junior Algebra and Representation Theory Seminar will kick-off the start of Trinity term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

String topology is an umbrella under which lives a family of algebraic structures on the homology of the (compact-open) loop space of a closed smooth manifold, M. Of great interest are the string product and coproduct, in view of the failure of the latter to be a homotopy invariant. We will discuss some existing algebraic and geometric perspectives on these operations, and give some examples that probe the extent to which the string coproduct fails to be a homotopy invariant. We will sketch an alternative point of view on string topology as the study of the derived bornological smooth loop stack and explain why this is a promising model for the observed phenomena of string topology.

Questions in algebra, while deep and interesting, can be incredibly difficult. Thankfully, when studying the representation theory of the symmetric groups, one can often take algebraic properties and results and write them in the language of combinatorics; where one has a wide variety of tools and techniques to use. In this talk, we will look at the specific example of the submodule structure of 2-part Specht modules in characteristic 2, and answer which hook Specht modules are uniserial in characteristic 2. We will not need to assume the Riemann hypothesis for this talk.