Masterseminar "Coxeter-Gruppen"

Im SS 2015 biete ich ein Masterseminar über Coxeter-Gruppen an.

Coxeter Gruppen liegen an der Schnittstelle von Kombinatorik, Algebra und Geometrie. Ihre Verbindungen zu Lie Algebren, algebraischen Gruppen, Hecke Algebren, sowie einer Vielzahl weiterer geometrischer und kombinatorischer Objekten aus der Darstellungstheorie, machen es möglich durch das Studium von Coxeter Gruppen Aussagen zu beweisen die in all diesen Bereichen Auswirkungen haben.


Das Seminar findet Mon 14:00-16:00 Uhr statt.  

Erster Termin: 20. April, EE.0135 (Cauerstraße 4)


Preprints

  1. Semi-infinite combinatorics in representation theory, Opens external link in new windowhttp://arxiv.org/abs/1505.01046

    Moment graphs coming from the geometry of flag varieties were used to provide new tools to attack representation theoretic questions involving  somehow the Bruhat order on the undelying Weyl group, such as multiplicity formulae where Kazhdan-Lusztig polynomials appear. In this work we discuss multiplicity formulae where Lusztig's semi-infinite order (and the semi-infinite analogues of Kazdhan-Lusztig polynomials) occurs and propose a moment graph approach to investigate them.  We motivate such an approach by considering the (not yet rigorously defined) geometric side of the story. We show that it is possible to compute stalks of the local intersection cohomology of the semi-infinite flag variety, and hence of spaces of quasi maps, by performing an algorithm due to Braden and MacPherson. This is mostly a survey paper about moment graph techniques in representation theory.

  2. with P. Fiebig, Filtered modules on moment graphs and periodic patterns, Opens external link in new windowarXiv:1504.01699

    We introduce and investigate the notion of group actions on moment graphs. We then consider the special case of an affine moment graph acted upon by the root lattice. Modules over the structure algebra of the quotient graph come hence naturally equipped with certain filtrations having a particularly nice behaviour. We then focus on a subcategory of the category of modules with such induced filtrations and show that it admits an exact structure. This allows us to talk about projectives. We prove that in our category there are enough projectives and that the multiplicities of Verma objects in them are given by Lusztig's periodic polynomials. In the last section we discuss the relation with the Andersen-Jantzen-Soergel category.


  3. with S. Griffeth, A. Gusenbauer, D. Juteau, Parabolic degeneration of rational Cherednik algebras, Opens external link in new windowarXiv:1502.08025

    We introduce the notion of parabolic degeneration of rational Cherednik algebras for complex reflection groups. We discuss two applications of our construction: a necessary condition for finite dimensionality of simple modules and a necessary condition for the existence of maps between standard objects. Both problems are still open in general. A consequence of (a weak version of) the finite dimensionality criterion is a new proof of a theorem of Berest-Etingof-Ginzburg classifying finite dimensional modules for the rational Cherednik algebra of the symmetric group. From the necessary condition for the existence of maps, it follows that category O for rational Cherednik algebras is a highest weight category with respect to an order which is much coarser than the one considered usually.



  4. with G. Cerulli Irelli, and P. Littelmann, Degenerate flag varieties and  Schubert varieties: a characteristic free approach, Opens external link in new window arXiv:1502.04590

    Evgeny Feigin's motivation to consider degenerate flag varieties was the study of a certain class of modules naturally arising from the representation theory of a simple Lie algebra. We show that in type A and C such modules are Demazure modules for Lie algebras of the same type, but doubled rank. Our constructions and proofs hold over the integers, so that as an application we generalise to any characteristic the result obtained in the previous paper with Cerulli Irelli.

 

Publications

 

  1. with G. Cerulli Irelli, Degenerate flag varieties of type A and C are Schubert varieties, Opens external link in new windowarXiv:1403.2889, International Mathematics Research Notices 2014, doi: 10.1093/imrn/rnu128

    Degenerate flag varieties have been introduced in 2010 by E.Feigin and since then several papers investigated their geometric and representation theoretic properties, showing their affinity with Schubert varieties. In this work we prove that in type A and C degenerate flags not only have a lot in common with Schubert varieties, but that they actually are isomorphic to Schubert varieties in an appropriate partial flag manifold. In the Appendix we show that the resolutions of type A degenerate flag varieties, defined by E.Feigin and Finkelberg, are in fact Bott-Samelson varieties.


  2. On the stable moment graph of an affine Kac-Moody algebra, Opens external link in new windowarXiv:1210.3218, Trans. Amer. Math. Soc., 367 (2015), 4111-4156

    367 (2015), 4111-4156

    367 (2015), 4111-4156

    367 (2015), 4111-4156


    In this paper, I introduce the stable moment graph of an affine Kac-Moody algebra g: a certain oriented graph with set of vertices in bijection with the alcoves in the fundamental chamber and with edges labelled by coroots of g. The study of indecomposable Braden-MacPherson sheaves on finite intervals (deep enough in the fundamental chamber) of the stable moment graph leads to a categorical analogue of a stabilisation property for affine Kazhdan-Lusztig polynomials proven by Lusztig here. Along the way, I introduce the notion of push-forward functor in the category of sheaves on a moment graph and prove that it is right adjoint to the pullback functor introduced in my previous paper 'Kazhdan-Lusztig combinatorics in the moment graph setting'.


  3. Categorification of a parabolic Hecke module via sheaves on moment graphs, Opens external link in new windowarXiv:1208.1492, Pac. J. Math. 271-2 (2014), 415-444.


    In this work, I generalise Fiebig's definition of the category of special modules of a Coxeter group to the parabolic setting and show that this provides a weak categorification of a parabolic Hecke module. I define here left translation functors, that turn out to be a very important tool in my paper on the stable moment graph. In the last section, I briefly discuss the relation of parabolic special modules with non-critical singular blocks of (an equivariant version of) category O for symmetrisable Kac-Moody algebras.


  4.  Moment graphs and Kazhdan-Lusztig polynomials,Opens external link in new window DMTCS Proceedings, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 491-502.

    This is an extended abstract for the FPSAC conference in 2012, that took place in Nagoya, Japan. It is a survey of the main results of my dissertation, focusing on the combinatorial side of the story. In particular, I recall some properties of Braden-MacPherson sheaves, that provide a categorical lifting of properties of Kazhdan-Lusztig polynomials. I also discuss the definition of category of k-moment graphs. 


  5. Kazhdan-Lusztig combinatorics in the moment graph setting, Opens external link in new windowarXiv:1103.2282, J. of Alg. 370 (2012), 152-170.

    In this paper, I introduce several strategies to interpret in terms of Braden-MacPherson sheaves certain elementary properties of Kazhdan-Lusztig polynomials. I define the pullback functor in the category of sheaves on a moment graph and prove that the pullback of an isomorphism maps indecomposable Braden-MacPherson sheaves to indecomposable Braden-MacPherson sheaves. This fact provides a trick that we use also in the proof of the main theorem of the paper on the stable moment graph. Since the arguments we provide work in any characteristic (under certain technical assumptions), by a theorem of Fiebig and Williamson, the result of this paper tell us that the stalks of indecomposable parity sheaves in positive characteristic behave very similarly to the ones of intersection cohomology complexes in characteristic zero, even in cases in which they are not perverse!