Tag - Representations of algebras

Rongwei Yang: Linear algebra in several variables

29th October, 2024

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.

Kai Meng Tan: Cores and core blocks of Ariki-Koike algebras

15th October, 2024

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

Kay Jin Lim: Integral basic algebras

1st October, 2024

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.

Sondre Kvamme: Higher torsion classes and silting complexes

20th June, 2024

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.

Aslak Bakke Buan: From exceptional to τ-exceptional sequences in module categories

30th May, 2024

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.

Karin Erdmann: The Hemmer-Nakano Theorem and relative dominant dimension

30th May, 2024

Let ℋq(d) be the Iwahori-Hecke algebra of the symmetric group where q is a primitive ℓ-th root of unity, and let A = Sq(n,d) be the q-Schur algebra. Hemmer and Nakano proved amongst others that for ℓ ≥ 4, the Schur functor gives an equivalence between the category of A-modules with Weyl filtration, and the category of ℋq(d)-modules with dual Specht filtration, and that certain extension groups get identified. This has been a surprise and has inspired further research. In this talk we discuss some extensions of this result.

Yuta Kimura: Classifying torsion classes of Noetherian algebras

28th February, 2024

Let R be a commutative Noetherian ring and A a Noetherian R-algebra. In this talk, we study classification of torsion classes, torsion free classes and Serre subcategories of mod-A. In the case where A = R, such subcategories were classified by Gabriel, Takahashi and Stanley-Wang by using prime ideals of R. If R is a field, then A is a finite-dimensional algebra, and there are many studies of such subcategories relating with tilting theory. For a Noetherian algebra case, localization of A at a prime ideal of R plays an important role. We see that classification can be reduced to finite dimensional algebras. If A is commutative, our results cover cases of commutative rings.

Markus Hunziker: Highest weight Harish-Chandra modules and classical invariant theory

26th February, 2024

In this mostly expository presentation, I will explain how certain combinatorial structures that arise in the representation theory of real reductive Lie groups can be used to solve several longstanding problems in classical invariant theory. Specifically, I will outline how to explicitly describe syzygies, Hilbert series, and linear bases of modules of covariants of several vectors and co-vectors.

Ziqing Xiang: Quantum Wreath Products and Their Representations

19th February, 2024

We introduce a new notion called the quantum wreath product, which produces an algebra B ≀Q H(d) from a given associative algebra B, a positive integer d, and a choice Q = (R, S, ρ, σ) of parameters. Important examples include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang’s wreath Hecke algebras, Rosso-Savage’s (affine) Frobenius Hecke algebras, Kleshchev-Muth’s affine zigzag algebras, and the Hu algebra that quantizes the wreath product Σm ≀ Σ2 between symmetric groups. We will discuss the bases of quantum wreath product algebras, and some of their representations.

Jon Carlson: Locally dualizable modules abound

5th February, 2024

The derived category of a commutative local noetherian ring and the module category of a modular group algebra are tensor triangulated categories. A dualizable object in such a category is one that has a dual that is compatible with the tensor structure. The question that we address in this paper is whether the subcategory dualizable objects in certain co-local subcategories is the idempotent closure of image of the compact objects under the local cohomology functor associated to the subcategory. In this lecture, I will try to explain what all of these words mean, why one might care about such a question and how we get a negative answer is certain cases.