Tag - Representation theory

Dan Nakano: Realizing Rings of Regular Functions via the Cohomology of Quantum Groups

2nd November, 2023

Let G be a complex reductive group and be a parabolic subgroup of G. In this talk, the presenter will address questions involving the realization of the G-module of the global sections of the (twisted) cotangent bundle over the flag variety G/P via the cohomology of the small quantum group.

Our main results generalize the important computation of the cohomology ring for the small quantum group by Ginzburg and Kumar and provide a generalization of well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case and the parabolic setting. As an application, we answer the question (first posed by Friedlander and Parshall for Frobenius kernels) about the realization of coordinate rings of Richardson orbit closures for complex semisimple groups via quantum group cohomology. Formulas will be provided that relate the multiplicities of simple G-modules in the global sections with the dimensions of extension groups over the large quantum group.

Scott Balchin: A jaunt through the tensor-triangular geometry of rational G-spectra for G profinite or compact Lie

26th October, 2023

In this talk, I will report on joint work with Barnes-Barthel and Barthel-Greenlees which analyses the category of rational G-equivariant spectra for G a profinite group or compact Lie group respectively. In particular, I will focus on a series of results regarding the Balmer spectra of these categories, and how the topology of these topological spaces informs structural results regarding the category.

Emily Norton: Decomposition numbers for unipotent blocks with small 𝔰𝔩2-weight in finite classical groups

24th October, 2023

There are many familiar module categories admitting a categorical action of a Lie algebra. The combinatorial shadow of such an action often yields answers to module-theoretic questions, for instance via crystals. In proving a conjecture of Gerber, Hiss, and Jacon, it was shown by Dudas, Varagnolo, and Vasserot that the category of unipotent representations of a finite classical group has such a categorical action. In this talk I will explain how we can use the categorical action to deduce closed formulas for certain families of decomposition numbers of these groups.

Leovigildo Alonso Tarrio: Derivators in additive context

20th October, 2023

By a theorem of Cisinksi, every combinatorial model category defines a strong derivator. For a Grothendieck category A, there are several combinatorial model structures defined on A, thus its derived category is the base of a strong derivator. In this talk, we present an alternative path to this result assuming further that A has enough projective objects. This approach has the benefit of simplicity (and less prerequisites) and gives a very explicit description of homotopy Kan extensions, in particular homotopy limits and colimits. We will present these results. Further, as an application, we will show how to extend the description of local cohomology via Koszul complexes from closed subsets to arbitrary systems of supports, i.e. stable for specialization subsets. Time permitting, we will discuss how this point of view applies to the co/homology of groups.

Vanessa Miemietz: Higher representation theory

12th October, 2023

I will try to motivate the development of a subject called finitary 2-representation theory and explain some techniques and results on the example of Soergel bimodules of finite Coxeter type.

Benjamin Sambale: Groups of p-central type

10th October, 2023

A finite group G with centre Z is of central type if there exists an irreducible character χ such that χ(1)2=|G:Z|. Howlett–Isaacs have shown that such groups are soluble. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p≠5. I have shown that there are no exceptions for p=5. Moreover, I give some applications to p-blocks with a unique Brauer character.

Bill Graham: The lookup conjecture and rational smoothness in type Ã2

9th October, 2023

Carrell and Peterson proved a test for rational smoothness of Schubert varieties at torus-fixed points, which depends on the number of torus-fixed curves through such points. The lookup conjecture of Boe and Graham is a conjectural simplification of the Carrell-Peterson criterion for rational smoothness. In this talk I will survey previous work by Boe-Graham and Graham-Li on the lookup conjecture, and describe recent work with Brian Boe. We identify the locus of rationally smooth points of a Schubert variety of type Ã2, and complete the proof of the lookup conjecture of Boe and Graham in type Ã2. We also identify the locus of smooth points (which is different from the rationally smooth locus).

Geordie Williamson: A Panoramic View of Modular Representation Theory

6th October, 2023

I will try to give a glimpse of exciting developments in representation theory over the last two decades. A central focus will be on the representations of symmetric groups over the complex numbers and fields of positive characteristic. Over the complex numbers our understanding is very good, however the case of positive characteristic fields has turned out to be more complicated than (I suspect) the pioneers would have ever imagined. Remarkably, there appears to be a way forward which combines ideas which emerged in the Langlands programme with techniques from mod p algebraic topology (Smith theory).

Paul Balmer: The geometry of permutation modules

5th October, 2023

In joint work with Martin Gallauer, we study the tensor-triangular geometry of the derived category of permutation modules over a finite group, and more generally over profinite groups. Martin and I have already spoken on this topic in various venues. So I’ll try to comment on aspects that were not highlighted so far, like the construction of the modular fixed points (or Brauer quotients), the Koszul objects and the reduction to elementary abelian groups. If time permits, I’ll say a few words about the profinite case, which is still partially work-in-progress.