Infinite root systems in algebra and geometry

Abstract

Given a root system, we study sets of roots, called biclosed sets, introduced by Matthew Dyer to generalize the Kazhdan-Lusztig conjecture. Biclosed sets generalize the notion of a set of positive roots. Most of their properties are standard for finite root systems and completely conjectural for infinite root systems. We prove several conjectures of Dyer about biclosed sets for affine root systems. In particular, we show that they form a lattice under containment order and coincide with the initial sections of reflection orders. Using those results, we apply biclosed sets to the study of torsion classes in Calabi-Yau categories. In particular, we show that biclosed sets give rise to generalized stability conditions on the representation category of an affine preprojective algebra and coincide with the restriction of torsion classes to the subcategory of spherical objects. In the case of type $\widetilde{A}$, biclosed sets case admit an explicit combinatorial model which can be interpreted (conjecturally) in terms of homological mirror symmetry; we use this model to parametrize the spherical objects of the representation category. We further use these result to give the first construction of Cambrian lattices for affine-type cluster algebras, giving a Coxeter-theoretic description of the exchange graph of the cluster algebra. We then turn to applications of biclosed sets in the Bruhat order on a Coxeter group, giving the first proof of EL-shellability not influenced by Hecke algebras, a generalization of the Gelfand-Serganova theorem on Coxeter matroids to infinite Coxeter groups, and a description of the faces of Bruhat interval polytopes. We also prove the broadest known case of the combinatorial invariance conjecture for Kazhdan--Lusztig $R$-polynomials in the symmetric group, and prove a related conjecture of Google DeepMind and Geordie Williamson in the case of lower intervals in the symmetric group.