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.
Revised:
Accepted:
Published online:
Keywords: $k$-Schur functions, harmonic functions, random walks on alcoves
Lecouvey, Cédric 1; Tarrago, Pierre 2
CC-BY 4.0
@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 = {https://alco.centre-mersenne.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 - https://alco.centre-mersenne.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 https://alco.centre-mersenne.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, Volume 4 (2021) no. 2, pp. 241-272. doi : 10.5802/alco.147. https://alco.centre-mersenne.org/articles/10.5802/alco.147/
[1] Parametrizations of canonical bases and totally positive matrices, Adv. Math., Volume 122 (1996) no. 1, pp. 49-149 | DOI | MR
[2] Expansion of -Schur functions for maximal rectangles within the affine nilCoxeter algebra, J. Comb., Volume 3 (2012) no. 3, pp. 563-589 | DOI | MR | Zbl
[3] Total positivity: tests and parametrizations, Math. Intelligencer, Volume 22 (2000) no. 1, pp. 23-33 | DOI | MR | Zbl
[4] Asymptotic representation theory of the symmetric group and its applications in analysis, Translations of Mathematical Monographs, 219, American Mathematical Society, Providence, RI, 2003, xvi+201 pages | DOI | MR | Zbl
[5] Computational commutative algebra. 2, Springer-Verlag, Berlin, 2005, x+586 pages | MR | Zbl
[6] Affine Stanley symmetric functions, Amer. J. Math., Volume 128 (2006) no. 6, pp. 1553-1586 | MR | Zbl
[7] Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc., Volume 21 (2008) no. 1, pp. 259-281 | DOI | MR | Zbl
[8] The shape of a random affine Weyl group element and random core partitions, Ann. Probab., Volume 43 (2015) no. 4, pp. 1643-1662 | DOI | MR | Zbl
[9] -Schur functions and affine Schubert calculus, Fields Institute Monographs, 33, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2014, viii+219 pages | MR | Zbl
[10] Schubert polynomials for the affine Grassmannian of the symplectic group, Math. Z., Volume 264 (2010) no. 4, pp. 765-811 | DOI | MR | Zbl
[11] Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J., Volume 116 (2003) no. 1, pp. 103-146 | DOI | MR | Zbl
[12] Conditioned random walks from Kac–Moody root systems, Trans. Amer. Math. Soc., Volume 368 (2016) no. 5, pp. 3177-3210 | DOI | MR | Zbl
[13] Harmonic functions on multiplicative graphs and weight polytopes of representations (To appear in Annales de l’Institut Fourier, https://arxiv.org/abs/1609.00138) | Zbl
[14] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages | MR | Zbl
[15] Independence in algebraic complexity theory, Ph. D. Thesis, Universitäts-und Landesbibliothek Bonn (2013)
[16] Crystal approach to affine Schubert calculus, Int. Math. Res. Not. IMRN (2016) no. 8, pp. 2239-2294 | DOI | MR | Zbl
[17] A path-transformation for random walks and the Robinson–Schensted correspondence, Trans. Amer. Math. Soc., Volume 355 (2003) no. 9, pp. 3669-3697 | DOI | MR | Zbl
[18] Algebraic Independence: Criteria and Structural Results over Diverse Fields, Masters thesis, Indian Institute of Technology Kanpur (2015)
[19] Affine Stanley symmetric functions for classical types, J. Algebraic Combin., Volume 36 (2012) no. 4, pp. 595-622 | DOI | MR | Zbl
[20] Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc., Volume 16 (2003) no. 2, pp. 363-392 | DOI | MR | Zbl
[21] A mirror construction for the totally nonnegative part of the Peterson variety, Nagoya Math. J., Volume 183 (2006), pp. 105-142 | DOI | MR | Zbl
Cited by Sources: