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

Source 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 rational expression for their drift. We also recover Rietsch’s parametrization 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 the Perron–Frobenius theorem.