The heart fan is a new convex-geometric invariant of an abelian category which captures interesting aspects of the related homological algebra. I will review the construction and some of its key properties, illustrating them through examples. In particular, I will explain how the heart fan can be viewed as a 'universal phase diagram' for Bridgeland stability conditions with the given heart.
Describing the conjugacy classes and/or irreducible characters of the unitriangular group over a finite field is known to be an impossibly difficult problem. Superclasses and supercharacters have been introduced (under the names of "basic varieties" and "basic characters") as an attempt to approximate conjugacy classes and irreducible characters using a cruder version of Kirillov's method of coadjoint orbits. In the past thirty years, these notions have been recognized in several areas (seemingly unrelated to representation theory): exponential sums in number theory, random walks in probability and statistics, association schemes in algebraic combinatorics... In this talk, we will describe and illustrate the main ideas and recent developments of the standard supercharacter theory of adjoint groups of radical rings. We will explore the close relation to Schur rings, and extend a well-known factorization of supercharacters of unitriangular groups which explains the alternative definition as basic characters.
Many results about schemes can be generalized to the non-commutative setting of stable ∞-categories. For bounded weighted categories, things are even better: one can formulate and prove a natural analogue of the theorem of Dundas-Goodwillie-McCarthy which is one of the fundamental tools in studying algebraic K-theory. However, in algebraic geometry bounded weighted categories do not show up very often: for instance, the ∞-category of perfect complexes over a scheme X only admits a reasonable weight structure when X is affine. We introduce a new notion of a c-category which is designed to cover a diverse class of geometric examples, including all quasi-compact quasi-separated schemes, yet allowing for all the weighted arguments to work out in this setting. Our main result shows that a c-category can be resolved in finitely many steps by categories of perfect complexes over connective ring spectra. This allows us to prove an analogue of the DGM theorem for c-categories, as well as the vanishing of their Hochschild homology below a certain degree. We also show that either of the following admits a structure of a c-category:
1. the derived category of any exact (∞-)category that has finite Ext-dimension;
2. the subcategory of compact objects in any weakly approximable stable ∞-category in the sense of Neeman.
In particular, using the computation of the Hochschild homology for c-categories mentioned before we obtain that the category of module-spectra over the ring C*(𝕊2) of cochains over the 2-sphere is not weakly approximable.
By an ultra classical result, the tensor product of a simple representation of 𝔤𝔩n(ℂ) and its defining representation decomposes as a direct sum of simple representations without multiplicities. This means that for each highest weight, the space of highest weight vectors is 1-dimensional. We will give an explicit construction of these highest weight vectors, and show that they arise from the action of certain elements in the enveloping algebra of 𝔤𝔩n(ℂ) + 𝔤𝔩n(ℂ) on the tensor product. These elements are independent of the simple representation we started with, and in fact produce highest weight vectors in several other contexts.
Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.
Given a graded-commutative ring acting centrally on a triangulated category, the main result of this talk shows that if the cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
Many mathematical and scientific problems concern systems of linear operators (A1,...,An). Spectral theory is expected to provide a mechanism for studying their properties, just like the case for an individual operator. However, defining a spectrum for non-commuting operator systems has been a difficult task. The challenge stems from an inherent problem in finite dimension: is there an analogue of eigenvalues in several variables? Or equivalently, is there a suitable notion of joint characteristic polynomial for multiple matrices A1,...,An? A positive answer to this question seems to have emerged in recent years.
Definition. Given square matrices A1,...,An of equal size, their characteristic polynomial is defined as
QA(z):=det(z0I + z1A1 + ⋯ + znAn), z=(z0,...,zn) ∈ ℂn+1.
Hence, a multivariable analogue of the set of eigenvalues is the eigensurface (or eigenvariety) Z(QA):={z ∈ ℂn+1 ∣ QA(z) = 0}. This talk will review some applications of this idea to problems involving projection matrices and finite-dimensional complex algebras. The talk is self-contained and friendly to graduate students.
Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.
This talk will consist of two parts. In the first part, we will see how certain results (such as the Nakayama 'Conjecture') for the symmetric groups and Iwahori-Hecke algebras of type A can be generalised to Ariki-Koike algebras using the map from the set of multipartitions to that of (single) partitions first defined by Uglov. In the second part, we look at Fayers's core blocks, and see how these blocks may be classified using the notation of moving vectors first introduced by Yanbo Li and Xiangyu Qi. If time allows, we will discuss Scopes equivalences between these blocks arising as a consequence of this classification
We will discuss joint work with Victor Ginzburg that proves a conjecture of Nadler on the existence of a quantization, or non-commutative deformation, of the Knop-Ngô morphism, a morphism of group schemes used in particular by Ngô in his proof of the fundamental lemma in the Langlands programme. We will first explain the representation-theoretic background, give an extended example of this morphism for the group GLn(ℂ), and then present a precise statement of our theorem.
Time permitting, we will also discuss how the tools used to construct this quantization can also be used to prove conjectures of Ben-Zvi and Gunningham, which predict a relationship between the quantization of the Knop-Ngô morphism and the parabolic induction functor, as well as an "exactness" conjecture of Braverman and Kazhdan in the D-module setting.
Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of Σn character sums over a new set called Ev(λ). When investigating the alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of Σn characters over the set Ev(λ) equals 0.
Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. In the recent preprint with David J. Benson, we initiate the study by focusing on the integral basic algebras. That is, we consider a p-modular system (K,𝒪,k) and an 𝒪-algebra A where both the algebras K⊗𝒪A and k⊗𝒪A are basic. When the algebra satisfies the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. As a case study, we focus on the descent algebras of Coxeter groups. They have been extensively studied since the introduction by Louis Solomon in 1976. We investigate their invariants as mentioned previously, their Ext quivers and representation type. The classification of the representation type in the p = 0 case has previously achieved by Manfred Schocker. In a recent preprint, together with Karin Erdmann, we complete the classification in the p > 0 case.
The theory of symmetric polynomials plays a key role in Representation Theory, Schubert Calculus, and Algebraic Combinatorics. Fundamental rules like the Pieri, Murnaghan-Nakayama, and Littlewood-Richardson rules describe the decomposition of products of Schubert classes into Schubert classes. We focus on the decomposition of polynomial representatives of Schubert classes in homology and K-homology of the affine Grassmannian of SLn, as well as quantum Schubert classes in quantum cohomology and K-cohomology of the full flag manifold of type A. Specifically, we explore how to use the Peterson isomorphism to connect formulas between homology and quantum cohomology, and between K-homology and quantum K-cohomology, extending techniques from the work of Lam-Shimozono on Schubert classes.
Draisma recently proved that finite length polynomial representations of the infinite general linear group GL are topologically GL-noetherian, i.e., the descending chain condition holds for GL-stable closed subsets. The scheme-theoretic variant of this theorem is a major open problem in the area. I will briefly outline the rich history of this problem and provide a negative answer in characteristic 2.
Higher Auslander-Reiten theory was introduced by Iyama in 2007 as a generalization of classical Auslander-Reiten theory. The main objects of study in the theory are d-cluster tilting subcategories of module categories. It turns out that many notions in algebra and representation theory have generalizations to higher Auslander-Reiten theory. In particular, in 2016 Jørgensen introduced a generalization of torsion classes, called higher torsion classes.
In this talk, I will recall the definition of higher torsion classes. I will then explain how functorially finite d-torsion classes give rise to (d+1)-term silting complexes, and hence to derived equivalences. The construction is analogous to the construction of 2-term silting complexes due to Adachi-Iyama-Reiten in 2014. I will illustrate the constructions and results on higher Nakayama algebras of type An.
Carlson's connectedness theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. In this talk, we will discuss a generalization, where it is proved that the Balmer support for an arbitrary monoidal triangulated category satisfies the analogous property. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.
Let G be a simple, simply connected algebraic group defined over a field of positive characteristic p, Gr be its rth Frobenius kernel, and, for q = pr, G(q) denotes the group of rational points over a field with q elements.
Motivated by work of Curtis and Steinberg, who showed that the simple Gr- and the simple G(q)-modules can be lifted to G, Humphreys and Verma conjectured that the projective covers of the simple Gr-modules also afford a G-module structure. Donkin later refined this conjecture by suggesting that these Gr-projectives lift uniquely to G in the form of tilting modules.
Ballard and Jantzen verified Donkin’s Tilting Module Conjecture for primes that are roughly twice the Coxeter number of the underlying root system or larger. But it was shown by Nakano and his collaborators that the conjecture fails in general. Counterexamples exist for all root systems with the exception of types B2, where the conjecture holds, and type A, where the conjecture remains completely open.
In this talk we delve into the rich history of these and closely related conjectures and report on their current status.
The Virasoro algebra is the central extension of derivations on Laurent polynomials. It plays an important role in mathematical physics and is itself a nice case study of an infinite-dimensional Lie algebra with triangular decomposition. I’ll give an overview of several families of representations of the Virasoro algebra and some connections between them.
The quest to find a character formula for the simple modules of a reductive algebraic group in positive characteristic took an unexpected turn roughly a decade ago when Williamson found a large number of counterexamples to the Lusztig Conjecture. Since then, the path to the simple characters has gone through the characters of the indecomposable tilting modules, thanks to the work of Riche and Williamson. However, the combinatorics required for determining all tilting characters are quite complicated, and the vast majority of these characters are not necessary to determine the simple characters. This talk is based on our pursuit of a more simplistic model in terms of what we’ve called the 'Steinberg quotient' of special tilting characters.
Exceptional sequences and their mutations were first considered in triangulated categories by the Moscow school of algebraic geometers. In the early nineties, Crawley-Boevey and Ringel studied exceptional sequences for module categories of hereditary algebras. We first recall their definitions and their main results, and then proceed to discuss a natural generalization to all (not necessarily hereditary) finite-dimensional algebras. This is the theory of τ-exceptional sequences, which was developed in joint work with Marsh, motivated by τ-tilting theory, by Adachi-Iyama-Reiten, by Jasso's reduction techniques for such modules and corresponding torsion pairs, and by the introduction of signed exceptional sequences by Igusa-Todorov.
The interplay between theories for τ-rigid modules, torsion pairs, and wide subcategories is central to our discussions.
