117,296 research outputs found
The Hard Lefschetz theorem in positive characteristic for the flag varieties
No full textFor a given flag variety, we characterize the primes p for which there exists a weight l such that the hard Lefschetz theorem holds for multiplication by l on the cohomology of the flag variety with coefficients in an infinite field of characteristic p
The Néron-Severi Lie algebra of a Soergel module
No full textWe introduce the Néron-Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the Néron-Severi Lie algebra to provide an easy proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality
Bases of the intersection cohomology of Grassmannian Schubert varieties
No full textThe parabolic Kazhdan–Lusztig polynomials for Grassmannians can be computed by counting Dyck partitions. We “lift” this combinatorial formula to the corresponding category of singular Soergel bimodules to obtain bases of the Hom spaces between indecomposable objects. In particular, we describe bases of intersection cohomology of Schubert varieties in Grassmannians parametrized by Dyck partitions which extend (after dualizing) the classical Schubert basis of the ordinary cohomology
Singular Rouquier complexes
No full textWe generalize the construction of Rouquier complexes to the setting of one-sided singular Soergel bimodules. Singular Rouquier complexes are defined by takingminimal complexes of restricted Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier complexes: they are Δ-split, they satisfy a vanishing formula, and when Soergel’s conjecture holdsthey are perverse. As an application, we establish Hodge theory for singular Soergel bimodules
A combinatorial formula for the coefficient of q in Kazhdan-Lusztig polynomials
No full textWe propose a combinatorial interpretation of the coefficient of q in Kazhdan-Lusztig polynomials and we prove it for finite simply-laced Weyl groups
On the Induction of p-Cells
No full textWe study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see Jensen and Williamson (2017) and Jensen (2020)) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right p-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the p-canonical basis
Atoms and charge in type C2
No full textWe construct atomic decompositions for crystals of type C2 and use them to define a charge statistic, thus providing positive combinatorial formulas for the corresponding Kostka–Foulkes polynomials. Our methods include Kashiwara–Nakashima tableaux combinatorics as well as the combinatorics of string polytopes and twisted Bruhat graphs
On the affine Hecke category for SL3
No full textWe study the diagrammatic Hecke category associated with the affine Weyl group of type A~ 2 . More precisely we find a (surprisingly simple) basis in characteristic zero for the Hom spaces between indecomposable objects, that we call indecomposable double leaves
Pre-canonical bases on affine Hecke algebras
No full textFor any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases {N-i}1 <= i <= m+1 (where m is the height of the highest root) of the spherical Hecke algebra that interpolates between the standard basis N-1 and the canonical basis Nm+1. The expansion of Ni+1 in terms of the Ni is in many cases very simple and we conjecture that in type A it is positive
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