We use -Schur functions to get the minimal boundary of the -bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type 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 -Schur functions and elementary computations based on the Perron–Frobenius theorem.
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Mots clés : $k$-Schur functions, harmonic functions, random walks on alcoves
@article{ALCO_2021__4_2_241_0,
author = {Lecouvey, C\'edric and Tarrago, Pierre},
title = {Alcove random walks, $k${-Schur} functions and the minimal boundary of the $k$-bounded partition poset},
journal = {Algebraic Combinatorics},
pages = {241--272},
publisher = {MathOA foundation},
volume = {4},
number = {2},
year = {2021},
doi = {10.5802/alco.147},
language = {en},
url = {http://www.numdam.org/articles/10.5802/alco.147/}
}
TY - JOUR AU - Lecouvey, Cédric AU - Tarrago, Pierre TI - Alcove random walks, $k$-Schur functions and the minimal boundary of the $k$-bounded partition poset JO - Algebraic Combinatorics PY - 2021 SP - 241 EP - 272 VL - 4 IS - 2 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.147/ DO - 10.5802/alco.147 LA - en ID - ALCO_2021__4_2_241_0 ER -
%0 Journal Article %A Lecouvey, Cédric %A Tarrago, Pierre %T Alcove random walks, $k$-Schur functions and the minimal boundary of the $k$-bounded partition poset %J Algebraic Combinatorics %D 2021 %P 241-272 %V 4 %N 2 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.147/ %R 10.5802/alco.147 %G en %F ALCO_2021__4_2_241_0
Lecouvey, Cédric; Tarrago, Pierre. Alcove random walks, $k$-Schur functions and the minimal boundary of the $k$-bounded partition poset. Algebraic Combinatorics, Tome 4 (2021) no. 2, pp. 241-272. doi : 10.5802/alco.147. http://www.numdam.org/articles/10.5802/alco.147/
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