Tag - Representations of algebraic groups

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.

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).

Nate Harman: Stability Phenomena in Representation Theory

28th August, 2023

For various natural sequences of groups, such as the general linear groups GLn or symmetric groups Sn, certain aspects of their representation theory act the same for all sufficiently large n. A classic example of this is Schur-Weyl duality, which gives a uniform description of degree d representations of GLn, provided n is at least d. I will discuss this and other examples of stability phenomena in representation theory, and how this sort of stabilization manifests itself in other areas of mathematics.

Daniel Nakano: On tilting modules and p-filtrations

26th April, 2023

In this talk, I will start by presenting historical background on the lifting of G-structures on modules for the corresponding Lie algebra. Early work was initiated in the 1960s with fundamental results by Curtis and Steinberg for simple modules. Later questions involving projective modules was studied by Humphreys-Verma, Ballard and Jantzen.

I will then present background material on two conjectures formulated by Donkin at MSRI in 1990. The first conjecture is Donkin's Tilting Module Conjecture (DTilt), and the second conjecture is Donkin's p-Filtration Conjecture (DFilt). Recent progress by Kildetoft-Nakano and Sobaje has shown that there are important connections between these conjectures. In particular, Jantzen's Question posed in 1980 on the existence of Weyl p-filtrations for Weyl modules for a reductive algebraic group constitutes a central part of the new developments.

I will describe how we produced infinite families of counterexamples to Jantzen’s Question and Donkin’s Tilting Module Conjecture. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type An or B2. Later, I will also present a complete answer to Donkin’s Tilting Module Conjecture for rank 2 groups.

Daniel Nakano: A Tribute to the Work of Cline, Parshall and Scott

17th April, 2023

For over 35 years, the triple author collaboration of Edward Cline, Brian Parshall and Leonard Scott was a leading force in the area of the representation theory of algebraic groups. The Cline-Parshall-Scott collaboration (also known as “CPS”) was the longest active triple authored collaboration in the history of mathematics. Many of the CPS results such as Mackey Imprimitivity and their foundational work on the induction functor were key to the early development of the subject.

In this talk I will survey three of their most important contributions,

   1.  the cohomology of finite groups of Lie type,
   2.  rational and generic cohomology, and
   3.  highest-weight categories.

In the process, I will also provide examples of how CPS’s work has inspired and influenced my research (with my collaborators) over the past 30 years.

Martha Precup: The cohomology of nilpotent Hessenberg varieties and the dot action representation

10th April, 2023

In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the combinatorics of chromatic symmetric functions to the geometry of Hessenberg varieties via a permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give a brief overview of that story and discuss how the dot action can be computed in all Lie types using the Betti numbers of certain nilpotent Hessenberg varieties. As an application, we obtain new geometric insight into certain linear relations satisfied by chromatic symmetric functions, known as the modular law.

Wilberd van der Kallen: A Friedlander-Suslin theorem over a noetherian base ring

29th March, 2023

We show that the celebrated Friedlander-Suslin theorem - on finite generation of cohomology of a finite group scheme G over a field - remains valid for a finite flat group scheme G over a commutative noetherian ring. In view of earlier work it suffices to put a uniform bound, depending on G only, on torsion in cohomology of G-modules.

Duc-Khanh Nguyen: A generalization of the Murnaghan-Nakayama rule for K–k-Schur and k-Schur functions

21st March, 2023

We introduce a generalization of K-k-Schur functions and k-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of K-k-Schur functions and k-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for k-Schur functions, and explains it as a degeneration of the rule for K-k-Schur functions. In particular, many other special cases promise to be detailed in the future.

Willem Adriaan De Graaf: Computing the first Galois cohomology set of a reductive algebraic group

6th March, 2023

In classification problems over the real field ℝ first Galois cohomology sets play an important role, as they often make it possible to classify the orbits of a real Lie group. In this talk we outline an algorithm to compute the first Galois cohomology set H1(G,ℝ) of a complex reductive algebraic group G defined over the real field ℝ. The algorithm is in a large part based on computations in the Lie algebra of G. This is joint work with Mikhail Borovoi.