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

Cédric Lecouvey (1) , Pierre Tarrago (2)

1 IDP - Institut Denis Poisson
2 CIMAT - Centro de Investigación en Matemáticas

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.

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]; Théorie des représentations [math.RT]

Cédric Lecouvey, Pierre Tarrago. Alcove random walks, k-Schur functions and the minimal boundary of the k-bounded partition poset. Algebraic Combinatorics, 2021, 4 (2), pp.241-272. ⟨10.5802/alco.147⟩. ⟨hal-01691407v2⟩