ALGEBRAIC COMBINATORICS

no. 2
Alcove random walks, k-Schur functions and the minimal boundary of the k-bounded partition poset
Lecouvey, Cédric1; Tarrago, Pierre2
1 Institut Denis Poisson (UMR CNRS 7013) Université de Tours Parc de Grandmont 37200 Tours, France
2 Laboratoire de Probabilités, Statistique et Modélisation (UMR CNRS 8001) Sorbonne Université 75005 Paris, France
Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 241-272.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.147
Classification: 05E05, 05E81, 31C20
Keywords: $k$-Schur functions, harmonic functions, random walks on alcoves
Lecouvey, Cédric 1; Tarrago, Pierre 2

1 Institut Denis Poisson (UMR CNRS 7013) Université de Tours Parc de Grandmont 37200 Tours, France
2 Laboratoire de Probabilités, Statistique et Modélisation (UMR CNRS 8001) Sorbonne Université 75005 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Lecouvey, Cédric; Tarrago, Pierre. Alcove random walks, $k$-Schur functions and the minimal boundary of the $k$-bounded partition poset. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 241-272. doi : 10.5802/alco.147. https://alco.centre-mersenne.org/articles/10.5802/alco.147/

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