Publications

  1. with E. Feigin, A. Puetz, Totally nonnegative Grassmannians, Grassmann necklaces and quiver Grassmannians,

  2. with A. Puetz, Permutation actions on Quiver Grassmannians for the equioriented cycle via GKM-Theory,

  3. with A. Puetz, GKM-Theory for Torus Actions on Cyclic Quiver Grassmannians, arXiv:2008.13138

  4. with P. McNamara, Singularities of Schubert varieties within a right cell, arXiv:2003.08616

  5. with K. Zainoulline, A Riemann-Roch type Theorem for twisted fibrations of moment graphs, arXiv:2002.09936

  6. with K. Zainoulline, Twisted quadratic foldings of root systems, arXiv:1806.08962

  7. with A. Ram, The Steinberg-Lusztig tensor product theorem, Casselman-Shalika and LLT polynomials, arXiv:1804.03710, Representation Theory 23 (2019), 188-204.

  8. with L. Bossinger, Following Schubert varieties under Feigin's degeneration of the flag variety, arXiv:1802.04320

In type A, Feigin's degeneration of the flag variety (actually, a variation of it, appearing in a previous paper of mine with Cerulli Irelli) is a Groebner degeneration and induces a degeneration of the Schubert varieties therein. We study how the combinatorics of the symmetric group encodes some information about the induced degeneration of Schubert varieties, in particular about their (ir)reducibility.

  1. with R. Devyatov, K. Zainoulline, Oriented sheaves on double moment graphs, arXiv:1710.10275, Doc. Math. 24 (2019), 563-608.

  2. with E. Strickland, Cohomology of the flag variety under PBW degenerations, arxiv:1706.0707, to appear in Transf. Groups

We show that type A flag variety admits a family of flat degenerations (called PBW) whose cohomology surjects onto the cohomology of the original flag. Moreover, in the symplectic setting, the cohomology of Feigin's flat degeneration surjects onto the cohomology of the symplectic flag variety which has been degenerated. Central ingredient of our paper is the realisation of all these degenerations as Schubert varieties, which had been provided in my joint paper with Cerulli Irelli ("Degenerate flag varieties of type A and C are Schubert varieties") and with Cerulli Irelli and Littelmann ("Degenerate flag varieties and Schubert varieties: a characteristic free approach") for Feigin's degenerations and in a recent paper by Cerulli Irelli, Fang, Feigin, Fourier and Reineke for PBW degenerations.

  1. with L. Bossinger, X. Fang, G. Fourier, M. Hering, Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs, arxiv:1612.03838, to appear in Annals of Combinatorics

Inspired by recent work of Rietsch and Williams, we show that it is possible to use the combinatorics of Postnikov's plabic (planar bicolored) graphs to construct all toric degenerations of Gr(2,n) coming from maximal cones of the tropical Grassmannian. Our combinatorial definition makes sense for more general plabic graphs and we check that also for Gr(3,n) it gives rise to toric degenerations of it. Unluckily, in this case we do not recover all maximal cones of the corresponding tropical variety.

  1. with A. Ram, P. Sobaje, A Fock space model for decomposition numbers for quantum groups at roots of unity, arxiv:1612.03120, to appear in Kyoto J. of Math.

We generalise to any Lie type the GLn-construction due to Leclerc and Thibon, to get a combinatorial gadget that encodes information about representations of quantum groups at a root of unity.

  1. with P. Fiebig, The combinatorial category of Andersen, Jantzen and Soergel and filtered moment graph sheaves, Abh. Math. Semin. Univ. Hamburg 86-2 (2016), 203-212.

This paper is a survey on our work on filtered moment graph sheaves, a new moment graph theory which is able to control periodicity phenomena. As an application of our construction, we discuss a new approach towards the combinatorial category introduced by Andersen, Jantzen and Soergel in their work on Lusztig's conjecture on the irreducible highest weight characters of algebraic groups in positive characteristic.

  1. with P. Fiebig, Filtered moment graph sheaves, arXiv:1508.05579

We consider the notion of group action on a moment graph and of quotient graph. We construct a category of sheaves on the quotient graph, which come equipped with a suitable filtration. We define an extra structure on this category and hence classify its projective objects. We provide an algorithm to construct indecomposable projectives which is very similar to the one discovered by Braden and MacPherson to compute intersection cohomology.

  1. Semi-infinite combinatorics in representation theory, arxiv:1505.01046, to appear in EMS Ser. Congr. Rep., Eur Math. Soc.

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 underlying 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.

  1. with P. Fiebig, Sheaves on alcoves I: Projectivity and wall-crossing functors, arXiv:1504.01699v2

This is a completely revised version of our previous paper "Filtered modules on moment graphs and periodic patterns" in which we developed the notion of moment graph equipped with a group action and used it to construct an appropriate category whose aim was to study modular representations of algebraic groups. In this version, moment graph actions have disappeared. In fact, even moment graphs are not completely visible. What we propose now is a genuine category of sheaves on the set of alcoves, satisfying certain conditions. In this article, the first of a series, we describe the projective objects of this category in an intrinsic way, as well as obtaining them by iterated application of wall crossing functors.

  1. with S. Griffeth, A. Gusenbauer, D. Juteau, Parabolic degeneration of rational Cherednik algebras, arxiv:1502.08025, Selecta Math. 23 (2017), 2754-2705.

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.

  1. with G. Cerulli Irelli, and P. Littelmann, Degenerate flag varieties and Schubert varieties: a characteristic free approach, arXiv:1502.04590, Pac. J. Math. 284-2 (2016), 283-308.

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.

  1. with G. Cerulli Irelli, Degenerate flag varieties of type A and C are Schubert varieties, arXiv:1403.2889, Int. Math. Res. Not. 15 (2015), 6353-6374.

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.

  1. On the stable moment graph of an affine Kac-Moody algebra, arXiv:1210.3218, Trans. Amer. Math. Soc., 367 (2015), 4111-415.

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'.

  1. Categorification of a parabolic Hecke module via sheaves on moment graphs, arXiv: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.

  1. Moment graphs and Kazhdan-Lusztig polynomials, 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.

  1. Kazhdan-Lusztig combinatorics in the moment graph setting, arXiv: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!