Research Seminar Representation Theory
The research seminar will take place on Mondays from 14:00 to 15:00 at the seminar room of the ART chair (room 407 in Pontdriesch 10–16).
The 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.
At the bottom of the page you find the slides of some talks.
- 07.11.2022 - George Balla (Aachen): The tropical symplectic Grassmannian
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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).
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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.
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12.12.2022 - Daniel Corey (TU Berlin): Degenerations of flag varieties and subdivisions of generalized permutahedra
- We prove that the initial degenerations of the type-A 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 position---are 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.
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19.12.2022 - Daniel Orr (Virginia Tech): From quantum toroidal algebras to wreath Macdonald operators
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Any simple finite-dimensional Lie algebra has an associated affine Lie algebra, which is a central extension of its loop algebra. The corresponding quantized enveloping algebras are one-parameter deformations of these objects. By considering the double loop algebra, one has a notion of toroidal Lie algebra, and this admits a two-parameter 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 two-parameter 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.)
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- 09.01.2023 - Daniel Schäfer (Köln): Equivariant cohomology of degenerate Grassmannians
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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 so-called 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.
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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 first-order 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.
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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.
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- 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): Kazhdan-Lusztig theory of matroids
- 13.06.2022 - Hans Franzen (Paderborn): Attractors of torus actions of certain GIT-quotients
- 11.07.2022 10:30 Uhr Seminarraum 103 Pontdriesch 10-12 - Deniz Kus (Bochum): Parabolic versions of quantum groups
- 13.07.2022 10:00 Uhr Seminarraum 103 Pontdriesch 10-12 - Shreepranav Enugandla (Indian Institute of Science): A general family of partial degenerations of finite-dimensional simple Lie algebras
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- 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)
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- 25.06.2019 - Hans Franzen (Bochum), GK-Seminar
- 17.06.2019 - Moritz Weber (Saarbrücken), 10:30-12:00 im Raum 405
- 04.06.2019 - Thorsten Heidersdorf (Bonn), 14:15-15:45 im Raum 405
- 21.05.2019 - Siddhartha Sahi (Rutgers University), GK-Seminar
- 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), GK-Seminar
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- 29.01.2019 - Narasimha Chary Bonata (MPI Bonn), GK-Seminar
- 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), GK-Seminar
- 26.11.2018 - Lea Bittmann (Paris)
- 19.11.2018 - Deniz Kus (Bochum)
- 13.11.2018 - Rekha Biswal (MPI Bonn), GK-Seminar
- 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), GK-Seminar
- 22.10.2018 - Johannes Flake (Aachen), Snapshot
- 16.10.2018 - Thomas Gerber (Aachen), GK-Seminar