Positroid Links and Braid varieties
Abstract
We study braid varieties and their relation to open positroid varieties. First, we construct a DGalgebra associated to certain braid words, possibly admitting negative crossings, show that its zeroth cohomology is an invariant under braid equivalence and positive Markov moves, and provide an explicit geometric model for its cohomology in terms of an affine variety and a set of locally nilpotent derivations. Second, we discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In particular, we prove that each open positroid stratum can be presented as the augmentation variety for different Legendrian fronts described in terms of either permutations, juggling patterns, cyclic rank matrices or Le diagrams. We also relate braid varieties to open Richardson varieties and brick manifolds, showing that the latter provide projective compactifications of braid varieties, with normal crossing divisors at infinity, and compatible stratifications. Finally, we state a conjecture on the existence and properties of cluster structures on braid varieties.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2105.13948
 Bibcode:
 2021arXiv210513948C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 Mathematics  Geometric Topology;
 Mathematics  Representation Theory;
 Mathematics  Symplectic Geometry;
 13F60;
 14M15;
 53D12;
 57K43
 EPrint:
 57 Pages