Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 3, pp. 629-666.

This survey establishes some miscellaneous results on random Littelmann paths and generalized Pitman transform. We describe central probability distributions on Littelmann paths. Next we state a law of large numbers and a central limit theorem for the generalized Pitman transform. We then study harmonic functions on multiplicative graphs defined from the tensor powers of finite-dimensional Lie algebras representations. Finally, we explain there exists an inverse of the generalized Pitman transform defined almost surely on the set of infinite paths remaining in the Weyl chamber and how it can be computed.

Dans cet article de synthèse nous établissons des résultats complémentaires sur les chemins de Littelmann aléatoires et sur la transformée de Pitman généralisée. Nous décrivons les distributions de probabilité centrales sur les chemins de Littelmann. Ensuite nous donnons une loi des grands nombres et un théorème central limite pour la transformée de Pitman généralisée. Nous étudions alors les fonctions harmoniques sur les graphes multiplicatifs définis à partir des puissances tensorielles des représentations irréductibles des algèbres de Lie. Enfin, nous expliquons qu’il existe une transformée inverse de la transformée de Pitman généralisée définie presque sûrement sur les trajectoires infinies qui restent dans la chambre de Weyl et montrons comment elle peut être calculée.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1580
Classification: 05E05, 05E10, 60G50, 60J10, 60J22
Lecouvey, Cédric 1; Lesigne, Emmanuel 1; Peigné, Marc 1

1 Laboratoire de Mathématiques et Physique Théorique, UMR CNRS 7350, Université de Tours, UFR Sciences et Techniques, 37200 Tours, France
@article{AFST_2018_6_27_3_629_0,
     author = {Lecouvey, C\'edric and Lesigne, Emmanuel and Peign\'e, Marc},
     title = {Harmonic functions on multiplicative graphs and inverse {Pitman} transform on infinite random paths},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {629--666},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {3},
     year = {2018},
     doi = {10.5802/afst.1580},
     mrnumber = {3869076},
     zbl = {1400.05262},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1580/}
}
TY  - JOUR
AU  - Lecouvey, Cédric
AU  - Lesigne, Emmanuel
AU  - Peigné, Marc
TI  - Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
SP  - 629
EP  - 666
VL  - 27
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1580/
DO  - 10.5802/afst.1580
LA  - en
ID  - AFST_2018_6_27_3_629_0
ER  - 
%0 Journal Article
%A Lecouvey, Cédric
%A Lesigne, Emmanuel
%A Peigné, Marc
%T Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2018
%P 629-666
%V 27
%N 3
%I Université Paul Sabatier, Toulouse
%U http://www.numdam.org/articles/10.5802/afst.1580/
%R 10.5802/afst.1580
%G en
%F AFST_2018_6_27_3_629_0
Lecouvey, Cédric; Lesigne, Emmanuel; Peigné, Marc. Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 3, pp. 629-666. doi : 10.5802/afst.1580. http://www.numdam.org/articles/10.5802/afst.1580/

[1] Biane, Philippe; Bougerol, Philippe; O’Connell, Neil Littelmann paths and Brownian paths, Duke Math. J., Volume 130 (2005) no. 1, pp. 127-167 | DOI | MR | Zbl

[2] Bourbaki, Nicolas Groupes et algèbres de Lie, Chapitres 4,5 et 6, Actualités Scientifiques et Industrielles, 1337, Hermann & Cie, 1968, 288 pages

[3] Dynkin, Evgeniĭ Borisovich Markov Processes. Vol. I and II, Die Grundlehren der mathematischen Wissenschaften, 121/122, Springer, 1965, xii+365, viii+274 pages

[4] Kashiwara, Masaki On crystal bases, Representations of groups. Canadian Mathematical Society annual seminar, June 15-24, 1994 (CMS Conf. Proc.), Volume 16, American Mathematical Society (1995), pp. 155-197 | MR | Zbl

[5] Kerov, Sergei V. The boundary of Young lattice and random Young tableaux, Formal power series and algebraic combinatorics. Séries formelles et combinatoire algébrique 1994 (Series in Discrete Mathematics and Theoretical Computer Science), Volume 24, American Mathematical Society, 1996, pp. 133-158 | DOI | MR | Zbl

[6] Kerov, Sergei V. Asymptotic representation theory of the symmetric group and its applications in analysis, Translations of Mathematical Monographs, 219, American Mathematical Society, 2003, xv+201 pages | MR | Zbl

[7] Lecouvey, Cédric; Lesigne, Emmanuel; Peigné, Marc Random walks in Weyl chambers and crystals, Proc. Lond. Math. Soc., Volume 104 (2012) no. 2, pp. 323-358 | DOI | MR | Zbl

[8] Lecouvey, Cédric; Lesigne, Emmanuel; Peigné, Marc Conditioned one-way simple random walk and combinatorial representation theory, Sémin. Lothar. Comb., Volume 70 (2014) (B70b, 27 p.) | MR | Zbl

[9] Lecouvey, Cédric; Lesigne, Emmanuel; Peigné, Marc Conditioned random walks from Kac-Moody root systems, Trans. Am. Math. Soc., Volume 368 (2016) no. 5, pp. 3177-3210 | DOI | MR | Zbl

[10] Lenart, Cristian On the combinatorics of crystal graphs. I: Lusztig’s involution, Adv. Math., Volume 211 (2007) no. 1, pp. 204-243 | DOI | MR | Zbl

[11] Littelmann, Peter A Littlewood–Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math., Volume 116 (1994) no. 1-3, pp. 329-346 | DOI | MR | Zbl

[12] Littelmann, Peter Paths and root operators in representation theory, Ann. Math., Volume 142 (1995) no. 3, pp. 499-525 | DOI | MR | Zbl

[13] Littelmann, Peter The path model, the quantum Frobenius map and standard monomial theory, Algebraic groups and their representations (Mathematical and Physical Sciences), Volume 517, Kluwer Academic Publishers, 1998, pp. 175-212 | DOI | MR | Zbl

[14] O’Connell, Neil A path-transformation for random walks and the Robinson-Schensted correspondence, Trans. Am. Math. Soc., Volume 355 (2003) no. 9, pp. 3669-3697 | DOI | MR | Zbl

[15] Pitman, James W. One-dimensional Brownian motion and the three-dimensional Bessel process, Adv. Appl. Probab., Volume 7 (1975), pp. 511-526 | DOI | MR | Zbl

[16] Śniady, Piotr Robinson-Schensted-Knuth algorithm, jeu de taquin, and Kerov-Vershik measures on infinite tableaux, SIAM J. Discrete Math., Volume 28 (2014) no. 2, pp. 598-630 | DOI | MR | Zbl

Cited by Sources: