Seminare und Vorträge im WS 2020/2021

am Dienstag, 27. Oktober

Oberseminar

Daniel Kalmbach (Aachen), Branching polytopes and multiplicities We describe the branching of Lie algebras of classical type over \(A_{n-1}\). Therefore we label the highest weight vectors of the modules occurring in the decomposition of the restriction of a simple finite-dimensional module to \(A_{n-1}\) by lattice points of a polytope, which we can describe in both the string and Lusztig's parametrization. Intersecting this polytope with some hyperplanes allows us counting the multiplicities of the \(A_{n-1}\) modules.

15:00 online

am Dienstag, 03. November

Oberseminar

Valentin Rappel (Köln), Complex structures on Bott-Samelson manifolds using the path model: The path model and MV cycles of a complex algebraic group are known to be connected through the gallery model. We propose a direct connection using the differential-geometric Bott-Samelson manifold using a maximal compact subgroup K.The MV cycles can be constructed in the loop group of K directly via this approach without the Iwasawa decomposition. We further give different embedding of the affine Schubert varieties into the loop group of K. Along the way we prove symplecticness and under conditions on the path from which the Bott-Samelson manifold is constructed the image of the moment map.

16:00 online

am Dienstag, 10. November

Oberseminar

Alexander Yong ( University of Illionois at Urbana-Champaign), On combinatorial aspects of equivariant Schubert calculus I will overview certain combinatorial perspectives and problems about torus-equivariant Schubert calculus. This will be done by analogy with non-equivariant results, starting with the textbook case of Grassmannians. The talk is based on joint work with (subsets of) A. Adve, D. Anderson, E. Richmond, C. Robichaux, H. Thomas, and H. Yadav.

16:00 online

am Dienstag, 17. November

Oberseminar

Jonah Blasiak (Philadelphia), Demazure crystals and the Schur positivity of Catalan functions, Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety,are a rich class of symmetric functions which include k-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of \hat{sl}_n-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals. This is joint work with Jennifer Morse and Anna Pun.

16:00 online

am Dienstag, 24. November

Oberseminar

Sabino Di Trani (Firenze), Graded Multiplicities of Small Representations in the Exterior Algebra \(\Lambda \mathfrak{g}\) Let g be a simple Lie algebra over C. The structure as g representation of exterior algebra ∧g has been extensively studied during the past century. In particular many authors focused on the decomposition of ∧g into irreducible representations, formulating many conjectures and obtaining some elegant results. One of the most known theorems about exterior algebra asserts that the cohomology of a compact connected Lie group G is isomorphic as a graded vector space to the ring of G-invariants in ∧g, where g is the complexification of Lie(G). Finding Betti numbers of G then corresponds to identifying copies of the trivial representation in Λg. Reeder in '95 reduces this computation to the problem of finding copies of the trivial representation of the Weyl group of G in a suitable bi-graded algebra. As a generalization of this result, he conjectured that it is possible to compute the graded multiplicities in Λg of a special class of representations reducing to a similar 'Weyl group representation' problem. In the talk I will give a panoramic view of some known results and open conjectures about the isotypic components in the exterior algebra and their multiplicities, focusing on my work about Reeder's Conjecture for the Classical Lie algebras.

16:00 online

am Dienstag, 01. Dezember

Oberseminar

Evgeny Feigin (HSE Moscow), Global Demazure modules Abstract: The global Weyl modules form a well studied class of cyclic representations of the current Lie algebras. One of the key properties of these modules is that they are acted upon by polynomial algebras, the action is free and the spaces of generators areisomorphic to the level one affine Demazure modules (which coincide with the Weyl modules in the ADE case). We will drop the level one condition and give the definition of the arbitrary level global modules. We will describe the main properties of the global Demazure modules similar to that of the global Weyl modules. Time permitting, we will explain how the global Demazure modules naturally show up in the geometric representation theory. Joint with Ilya Dumanski and Michael Finkelberg.

10:30 online

am Dienstag, 01. Dezember

Oberseminar

Igor Makhlin (Skoltech), A family of semitoric degenerations, I will define a family of flat semitoric Gröbner degenerations for every Hibi variety. In particular, this construction can be applied to obtain families of semitoric degenerations for Grassmannians and flag varieties in types A and C. Each of the degenerations in question is the union of toric varieties associated with the parts of a (Gelfand-Kapranov-Zelevinsky) regular subdivision of the corresponding order polytope or Gelfand-Tsetlin polytope. This is a joint work with Evgeny Feigin: https://arxiv.org/abs/2008.13243

16:00 online

am Dienstag, 15.Dezember

Oberseminar

Peter Littelmann (Köln), Tensor products and geometric Satake, The geometric Satake correspondence can be regarded as a geometric construction of the rational representations of a complex connected reductive group G. In their study of this correspondence, Mirković and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in eachtensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.

16:00 online

am Dienstag, 19. Januar

Oberseminar

Martina Lanini und Alexander Pütz (Rome), Torus actions on quiver Grassmannians Abstract: In this talk, we will explain how to equip certain quiver Grassmannians with a torus action. In the case of nilpotent representations of the equioriented cycle, our torus actions are particularly nice behaved: their one skeleton admits an explicit combinatorial description and encodes the data needed to compute the equivariant cohomology, as well as a basis for it. As a very special case of our construction, we recover, respectively, the Bruhat graph of the type A flag variety, the moment graph description of its equivariant cohomology and the equivariant Schubert basis of the latter.

16:00 online

am Dienstag, 26. Januar

Oberseminar

Valentin Rappel (Bochum), Disputation

16:00 online

am Dienstag, 02. Februar

Oberseminar

Narasimha Chary (Bochum), Torus quotients of Schubert varieties. Abstract: In this talk, we consider the quotients of flag varieties and Schubert varieties for the action of a maximal torus. We compare the singular locus and semistable locus (w.r.t maximal torus action) of minuscule Schubert varieties. We study the smoothness of torus quotients of Schubert varieties in the Grassmannian. Finally, we consider the torus quotients of Schubert varieties for \(SL(n, \mathbb C)\) with respect to the highest root. This talk is based on a joint work with S.K. Pattanayak.

16:00 online

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