Alcove random walks, k-Schur functions and the minimal boundary of the k-bounded partition poset

TL;DR

This paper uses k-Schur functions to analyze the minimal boundary of the k-bounded partition poset, describing central random walks and deriving explicit polynomial drift expressions, connecting combinatorics with affine Grassmannian structures.

Contribution

It introduces a novel approach to characterize the minimal boundary of the k-bounded partition poset using k-Schur functions, linking combinatorics with affine Grassmannian analysis.

Findings

Explicit polynomial expression for the drift of random walks.

Recovery of Rietsch's parametrization of totally nonnegative matrices.

Homeomorphisms made explicit via combinatorics and Perron-Frobenius theorem.

Abstract

We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.