Research Seminar Representation Theory
Copyright: © Lehrstuhl B für MathematikThe research seminar is organized together with the chair of Algebra and Number Theory as a "Joint Algebra Seminar". The planned talks are listed on its website.
he speakers are invited guests or members of the team will present current research and developments. This could be a snapshot of 10 minutes or a regular seminar talk of 60 minutes.

 07.11.2022  George Balla (Aachen): The tropical symplectic Grassmannian

28.11.2022  Pablo Zadunaisky (MPI Bonn): Highest weight vectors and the Pieri rule
 Let L be any simple finite dimensional weight representation of gl(n,C). The tensor product of L with the natural representation V is a semisimple representation, and its decomposition is described by the Pieri rule. This purely combinatorial rule, along with some theory, guarantees the existence of highest weight vectors in the tensor product L (x) V. In this talk I will show that these highest weight vectors can be given explicitly, and this description is independent of L. If time allows it, I will discuss some variations of this idea: replacing V or L with more general representations, quantum versions, extension to other types ... This is joint work with Joanna Meinel (Bonn).

05.12.2022  Piotr Śniady (MPI Bonn): Museum of visual Asymptotic Representation Theory
 Museum of visual ART (Asymptotic Representation Theory) collects mathematical artifacts which can be visualized as beautiful images or animations related to large random combinatorial objects appearing in the representation theory. Our visit to the Museum will consist of two parts, focused around two exhibits. Slides, animations, homeworks, references and additional materials are available at psniady.impan.pl/fpsac.

12.12.2022  Daniel Corey (TU Berlin): Degenerations of flag varieties and subdivisions of generalized permutahedra
 We prove that the initial degenerations of the typeA flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of Fl_0(n)the open subvariety of the complete flag variety Fl(n) consisting of flags in general positionare smooth and irreducible when n ≤ 4. We also study the Chow quotient of Fl(n) by the diagonal torus of PGL(n), and show that, for n=4, this is a log crepant resolution of its log canonical model. This is based on joint work with Jorge Alberto Olarte.

19.12.2022  Daniel Orr (Virginia Tech): From quantum toroidal algebras to wreath Macdonald operators

Any simple finitedimensional Lie algebra has an associated affine Lie algebra, which is a central extension of its loop algebra. The corresponding quantized enveloping algebras are oneparameter deformations of these objects. By considering the double loop algebra, one has a notion of toroidal Lie algebra, and this admits a twoparameter deformation known as the quantum toroidal algebra.
In this talk, which is based on joint work with Mark Shimozono and Joshua Wen, I will explain how recent developments in the theory of quantum toroidal algebras (of type A) have enabled the discovery a new family of commuting difference operators. The significance of these operators is that their joint eigenfunctions are precisely Haiman's wreath Macdonald polynomials, a tantalizing twoparameter family of polynomials arising from the geometry of cyclic quiver Nakajima varieties. The new 'wreath Macdonald operators' provide a new characterization of wreath Macdonald polynomials, which were hitherto accessible only by indirect methods.
(Prior knowledge of Macdonald polynomials will not be assumed.)

 09.01.2023  Daniel Schäfer (Köln): Equivariant cohomology of degenerate Grassmannians

16.01.2023  Magdalena Boos (Bochum):
On degenerations in symmetric representation varieties
 The notion of a symmetric quiver was first introduced by Derksen and Weyman in 2002. Symmetric quiver representations are collected in socalled symmetric representation varieties which are acted on by reductive groups via change of basis. We motivate our interest in understanding orbit closure relations, i.e. symmetric degenerations of said actions. Our main result describes them explicitly in case the symmetric quiver is of finite representation type. This is joint work with G. Cerulli Irelli.

23.01.2023  Benjamin Klopsch (Düsseldorf): Finite axiomatizability of the rank and the dimension of compact \(p\)adic Lie groups
 Recently, Nies, Segal and Tent initiated an investigation into finite axiomatizability (in the sense of model theory) for profinite groups. Among several other types of groups, they consider in some detail the class of \(p\)adic analytic pro\(p\) groups. In joint work with Martina Conte, we have now answered some questions regarding the rank and the dimension of \(p\)adic analytic groups which arise naturally from the work of Nies, Segal and Tent. For instance, we prove that for every prime \(p\) and every positive integer \(r \) there is a sentence \(\sigma_{p,r} \) in the firstorder language of groups such that for every pro\(p\) group \(G\) the following are equivalent: (i) \(G\) is a model of \(\sigma_{p,r} \), (ii) \(G\) has rank \(r \). In my talk I will explain briefly the necessary background and motivation. Then I will discuss why the theorem and related results hold true. Time permitting I will present natural generalisations and indicate some open problems.

30.01.2023  Victoria Schleis (Tübingen):
Linear degenerate tropical flag varieties
 Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Their linear degenerations arise in representation theory as they describe quiver representations and their irreducible modules. As linear degenerations of flag varieties are difficult to analyze algebraically, we describe them in a combinatorial setting and further investigate their tropical counterparts. In this talk, I will introduce matroidal, polyhedral and tropical analoga and descriptions of linear degenerate flags and their varieties obtained in joint work with Alessio Borzì. To this end, we introduce and study morphisms of valuated matroids. Using techniques from matroid theory, polyhedral geometry and linear tropical geometry, we use the correspondences between the different descriptions to gain insight on the structure of linear degeneration. Further, we analyze the structure of linear degenerate flag varieties in all three settings, and provide some cover relations on the poset of degenerations. For small examples, we relate the observations on cover relations to the flat irreducible locus studied in representation theory.

 11.04.2022  Ievgen Makedonskyy (MPI Bonn): Reduced arc schemes over toric varieties
 25.04.2022  Jonathan Gruber (Lausanne)
 02.05.2022  Giulia Iezzi (Aachen)
 23.05.2022  Jacob Matherne (Bonn): KazhdanLusztig theory of matroids
 13.06.2022  Hans Franzen (Paderborn): Attractors of torus actions of certain GITquotients
 11.07.2022 10:30 Uhr Seminarraum 103 Pontdriesch 1012  Deniz Kus (Bochum): Parabolic versions of quantum groups
 13.07.2022 10:00 Uhr Seminarraum 103 Pontdriesch 1012  Shreepranav Enugandla (Indian Institute of Science): A general family of partial degenerations of finitedimensional simple Lie algebras

 16.07.2020 Irem Portakal (Magdeburg)
 09.07.2020 Kamal Saleh (Siegen)
 02.07.2020  Thomas Gerber (Lausanne)
 22.06.2020  George Balla (Aachen)
 18.06.2020  Darius Dramburg (Aachen)
 28.05.2020  Lara Bossinger (Oaxaca)

 25.06.2019  Hans Franzen (Bochum), GKSeminar
 17.06.2019  Moritz Weber (Saarbrücken), 10:3012:00 im Raum 405
 04.06.2019  Thorsten Heidersdorf (Bonn), 14:1515:45 im Raum 405
 21.05.2019  Siddhartha Sahi (Rutgers University), GKSeminar
 20.05.2019  Georg Loho (London)
 29.04.2019  Kai Wehrmaker (Aachen), Snapshot
 22.04.2019  Dominik Bernhardt (Aachen), Snapshot
 16.04.2019  Bart van Steirteghem (Erlangen, New York), GKSeminar

 29.01.2019  Narasimha Chary Bonata (MPI Bonn), GKSeminar
 28.01.2019  Verity Mackscheidt (Köln)
 14.01.2019  Thomas Gerber (Aachen)
 07.01.2019  Dominik Bernhardt ( Aachen), Snapshot
 17.12.2018  Xin Fang (Köln)
 11.12.2018  Ghislain Fourier (Aachen), GKSeminar
 26.11.2018  Lea Bittmann (Paris)
 19.11.2018  Deniz Kus (Bochum)
 13.11.2018  Rekha Biswal (MPI Bonn), GKSeminar
 12.11.2018  Markus Baumeister (Aachen), Snapshot
 05.11.2018  Ghislain Fourier (Aachen), Snapshot
 30.10.2018  Dirk Liebhold (Aachen)
 23.10.2018  Johannes Flake (Aachen), GKSeminar
 22.10.2018  Johannes Flake (Aachen), Snapshot
 16.10.2018  Thomas Gerber (Aachen), GKSeminar