Infinite root systems in algebra and geometry & Enumeration in stochastic processes and polyhedral geometry

Speaker: Grant Barkley (Harvard)

Time: 3:00 PM – 4:00 PM

Title: Infinite root systems in algebra and geometry

Abstract:

We describe how biclosed sets of roots in a positive root system simultaneously give a natural completion of the weak order on a Coxeter group and a combinatorial skeleton for Calabi-Yau categories. In particular, for affine root systems, we show that biclosed sets form a profinite semidistributive lattice, strengthening the lattice property conjectured by Dyer. We also show that each torsion class in the category of modules for an affine type A preprojective algebra has an associated biclosed set of roots, which uniquely determines the spherical modules in the torsion class; in this way biclosed sets behave like stability conditions on the category. We apply biclosed sets to give a new proof that Bruhat intervals have EL-labelings and to prove an extension of the Gelfand-Serganova theorem on Coxeter matroids to infinite Coxeter groups. We also prove new cases of the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.

Speaker: Yuhan Jiang (Harvard)

 Time: 4:15 PM – 5:15 PM

Title: Enumeration in stochastic processes and polyhedral geometry

Abstract:

This dissertation explores the combinatorics of Markov chains and polyhedral geometry, with a focus on the asymmetric simple exclusion process (ASEP) and the Ehrhart theory of polytopes. The first part addresses the stationary distribution

of stochastic models, including the open ASEP, the Arndt-Heinzel-Rittenberg (AHR) model and the doubly ASEP (DASEP).We give a two-layer simple random walk interpretation for the open ASEP model, a tableaux formula for the AJR model, and show that the DASEP exhibits homomesy phenomenon. The second part of the dissertation studies the Ehrhart theory of positroid polytopes and alcoved polytopes. We present combinatorial formulas for the $h^*$-polynomials of positroid polytopes and alcoved polytopes.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/