C. Berg, N. Bergeron, H. Thomas, H. , and M. Zabrocki, Expansion of k-Schur functions for maximal rectangles within the affine nilCoxeter algebra, Journal of Combinatorics, vol.3, issue.3, pp.563-589, 2012.

A. Berenstein, S. Fomin, and A. Zelevinsky, Parametrization of canonical bases and totally positive matrices, Adv. in Math, vol.122, issue.1, pp.49-149, 1996.

S. Fomin and A. Zelevinsky, Total positivity: test and parametrizations, vol.22, pp.23-33, 2000.

S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, AMS Translation of Mathematical Monographs, vol.219, 2003.

M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Computational Commutative Algebra, vol.1, 2005.

T. Lam, Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc, vol.21, issue.1, pp.259-281, 2008.

T. Lam, Affine Stanley symmetric functions, American Journal of Mathematics, vol.128, issue.6, pp.1553-1586, 2006.

T. Lam, The shape of a random affine Weyl group element and random core partitions, Annals of Probability, vol.43, issue.4, pp.1643-1662, 2015.

T. Lam, A. Schilling, and M. Shimozono, Schubert polynomials for the affine Grassmannian of the symplectic group, Math. Zeit, vol.264, issue.4, pp.765-811, 2010.

T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono et al., , 2014.

L. Lapointe, A. Lascoux, and J. Morse, Tableaux atoms and a new Macdonald positivity conjecture, vol.116, pp.103-146, 2003.

C. Lecouvey, E. Lesigne, and M. Peigné, Conditioned random walks from Kac-Moody root systems, Transactions of the AMS, vol.368, issue.5, pp.3177-3210, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00833657

C. Lecouvey and P. Tarrago, Harmonic functions on multiplicative graphs and weight polytopes of representations

I. G. Macdonald, Symmetric functions, 1998.

J. Mittmann, Independence in algebraic complexity theory, Doctoral dissertation, 2013.

J. Morse and A. Schilling, Crystal approach to affine Schubert Calculus, IMRN, vol.8, pp.2239-2294, 2016.

N. O'-connell, A path-transformation for random walks and the Robinson-Schensted correspondence, Trans. Amer. Math. Soc, vol.355, pp.3669-3697, 2003.

N. Pon, Affine Stanley symmetric functions for classical types, Journal of Algebraic Combinatorics, vol.36, issue.4, pp.595-622, 2012.

A. Pandey, Algebraic Independence: Criteria and Structural Results over Diverse Fields, Doctoral dissertation, 2015.

K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, JAMS, vol.16, issue.2, pp.363-392, 2002.

K. Rietsch, A mirror construction for the totally nonnegative part of the Peterson variety, Nagoya Mathematical Journal, vol.183, pp.105-142, 2006.