6,872 research outputs found
Singular link Floer homology
We define a grid presentation for singular links i.e. links with a finite
number of rigid transverse double points. Then we use it to generalize link
Floer homology to singular links. Besides the consistency of its definition, we
prove that this homology is acyclic under some conditions which naturally make
its Euler characteristic vanish.Comment: 29 pages, many figure
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
Bijections and symmetries for the factorizations of the long cycle
We study the factorizations of the permutation into factors
of given cycle types. Using representation theory, Jackson obtained for each
an elegant formula for counting these factorizations according to the
number of cycles of each factor. In the cases Schaeffer and Vassilieva
gave a combinatorial proof of Jackson's formula, and Morales and Vassilieva
obtained more refined formulas exhibiting a surprising symmetry property. These
counting results are indicative of a rich combinatorial theory which has
remained elusive to this point, and it is the goal of this article to establish
a series of bijections which unveil some of the combinatorial properties of the
factorizations of into factors for all . We thereby obtain
refinements of Jackson's formulas which extend the cases treated by
Morales and Vassilieva. Our bijections are described in terms of
"constellations", which are graphs embedded in surfaces encoding the transitive
factorizations of permutations
When is a Schubert variety Gorenstein?
A (normal) variety is Gorenstein if it is Cohen-Macualay and its canonical
sheaf is a line bundle. This property, which measures the ``pathology'' of the
singularities of a variety, is thus stronger than Cohen-Macualayness, but is
also weaker than smoothness. We determine which Schubert varieties are
Gorenstein in terms of a combinatorial characterization using generalized
pattern avoidance conditions. We also give an explicit description as a line
bundle of the canonical sheaf of a Gorenstein Schubert variety.Comment: 15 pages, geometric characterization of Gorensteinness added; final
version to appear in Adv. Mat
- …