1,720,991 research outputs found
Moment graphs and Kazhdan-Lusztig polynomials
Full text linkMotivated by a result of Fiebig (2007), we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs
Categorification of a parabolic Hecke module via sheaves on moment graphs
Full text linkWe investigate certain categories, associated by Fiebig with the geometric representation of a Coxeter system, via sheaves on Bruhat graphs. We modify Fiebig's definition of translation functors in order to extend it to the singular setting and use it to categorify a parabolic Hecke module. As an application we obtain a combinatorial description of indecomposable projective objects of (truncated) noncritical singular blocks of (a deformed version of) category O, using indecomposable special modules over the structure algebra of the corresponding Bruhat graph
On the stable moment graph of an affine Kac-Moody Algebra
No full textIn 1980 Lusztig proved a stabilisation property of the affine Kazhdan-Lusztig polynomials. In this paper we give a categorical version of such a result using the theory of sheaves on moment graphs. This leads us to associate with any Kac-Moody algebra its stable moment graph
Twisted quadratic foldings of root systems
No full textThe present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig's projection of the root system of type E8 onto the subring of icosians of the quaternion algebra, which gives the root system of type H-4.By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied
The combinatorial category of Andersen, Jantzen and Soergel and filtered moment graph sheaves
Full text linkWe give an overview on the series of articles (Fiebig and Lanini, Filtered moment graph sheaves, arXiv:1508.05579, 2015, Fiebig and Lanini, Periodic structures on affine moment graphs I: dualities and translation functors, arXiv: 1504.01699, 2015, Fiebig and Lanini, Periodic structures on affine moment graphs II: multiplicities and modular representations (in preparation)) that aims at introducing 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 modular algebraic groups
Permutation actions on Quiver Grassmannians for the equioriented cycle via GKM-theory
Full text linkIn our previous work, we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups and we investigate this permutation action via GKM techniques. In the case of (type A) flag varieties, or Schubert varieties therein, we recover Tymoczko's results on permutation representations
Sheaves on the alcoves and modular representations I
No full textWe consider the set of affine alcoves associated with a root system R as a topological space and define a certain category S of sheaves of Zk{mathcal{Z}}_k end{document}-modules on this space. Here Zk is the structure algebra of the root system over a field k. To any wall reection s we associate a wall crossing functor on S. In the companion article [FL] we prove that S encodes the simple rational characters of the connected, semisimple and simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number
The Steinberg-Lusztig tensor product theorem, Casselman-Shalika, and LLT polynomials
No full textIn this paper we establish a Steinberg-Lusztig tensor product theorem for abstract Fock space. This is a generalization of the type A result of Leclerc-Thibon and a Grothendieck group version of the Steinberg-Lusztig tensor product theorem for representations of quantum groups at roots of unity. Although the statement can be phrased in terms of parabolic affine Kazhdan-Lusztig polynomials and thus has geometric content, our proof is combinatorial, using the theory of crystals (Littelmann paths). We derive the Casselman-Shalika formula as a consequence of the Steinberg-Lusztig tensor product theorem for abstract Fock space.</p
Following Schubert varieties under Feigin’s degeneration of the flag variety
Full text linkWe study the effect of Feigin's flat degeneration of the type A flag variety on the defining ideals of its Schubert varieties. In particular, we describe two classes of Schubert varieties which stay irreducible under the degenerations and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some degenerate Schubert varieties (i.e. the vanishing sets of initial ideals of the ideals of Schubert varieties with respect to Feigin's Grobner degeneration) with Richardson varieties in higher rank partial flag varieties
A Riemann-Roch type theorem for twisted fibrations of moment graphs
No full textIn the present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and push-forwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch type theorem for the equivariant K-theory of some Kac-Moody flag varieties
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