Nous introduisons et étudions une généralisation inhomogène des polynômes spin de -Whittaker de [15]. Ce sont des polynômes symétriques, et nous prouvons une règle de branchement, des identités de Cauchy asymétriques duales et non duales, et une représentation intégrale pour ces polynômes. Nous prouvons une règle de branchement, des identités de Cauchy asymétriques, duales et non-duelles, et une représentation intégrale pour ces polynômes. Notre outil principal est une nouvelle famille d’équations de Yang–Baxter déformées.
We introduce and study an inhomogeneous generalization of the spin -Whittaker polynomials from [15]. These are symmetric polynomials, and we prove a branching rule, skew dual and non-dual Cauchy identities, and an integral representation for them. Our main tool is a novel family of deformed Yang–Baxter equations.
Accepté le :
Publié le :
Alexei Borodin 1 ; Sergei Korotkikh 1
![](/static/ptf/img/CC-BY%204.0.png)
@article{AFST_2024_6_33_1_1_0, author = {Alexei Borodin and Sergei Korotkikh}, title = {Inhomogeneous spin $q${-Whittaker} polynomials}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {1--68}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 33}, number = {1}, year = {2024}, doi = {10.5802/afst.1761}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1761/} }
TY - JOUR AU - Alexei Borodin AU - Sergei Korotkikh TI - Inhomogeneous spin $q$-Whittaker polynomials JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2024 SP - 1 EP - 68 VL - 33 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1761/ DO - 10.5802/afst.1761 LA - en ID - AFST_2024_6_33_1_1_0 ER -
%0 Journal Article %A Alexei Borodin %A Sergei Korotkikh %T Inhomogeneous spin $q$-Whittaker polynomials %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2024 %P 1-68 %V 33 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1761/ %R 10.5802/afst.1761 %G en %F AFST_2024_6_33_1_1_0
Alexei Borodin; Sergei Korotkikh. Inhomogeneous spin $q$-Whittaker polynomials. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 1, pp. 1-68. doi : 10.5802/afst.1761. https://afst.centre-mersenne.org/articles/10.5802/afst.1761/
[1] Current fluctuations of the stationary ASEP and six-vertex model, Duke Math. J., Volume 167 (2018) no. 2, pp. 269-384 | Zbl
[2] Phase transitions in the ASEP and stochastic six-vertex model, Ann. Probab., Volume 47 (2019) no. 2, pp. 613-689 | Zbl
[3] Random walk in beta-distributed random environment, Probab. Theory Relat. Fields, Volume 167 (2017) no. 3, pp. 1057-1116 | DOI | Zbl
[4] On a family of symmetric rational functions, Adv. Math., Volume 306 (2017), pp. 973-1018 | DOI | Zbl
[5] Symmetric elliptic functions, IRF models, and dynamic exclusion processes, J. Eur. Math. Soc., Volume 22 (2020) no. 5, pp. 1353-1421 | DOI | Zbl
[6] Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz, Commun. Math. Phys., Volume 339 (2015) no. 3, pp. 1167-1245 | DOI | Zbl
[7] Spectral theory for the -Boson particle system, Compos. Math., Volume 151 (2015) no. 1, pp. 1-67 | DOI | Zbl
[8] Shift-invariance for vertex models and polymers (2019) (https://arxiv.org/abs/1912.02957)
[9] Integrable probability: Stochastic vertex models and symmetric functions, Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School (Les Houches, 2015), Volume 104, Oxford University Press, 2017, pp. 26-131 | DOI | Zbl
[10] Higher spin six vertex model and symmetric rational functions, Sel. Math., New Ser., Volume 24 (2018) no. 2, pp. 751-874 | DOI | Zbl
[11] Inhomogeneous exponential jump model, Probab. Theory Relat. Fields, Volume 172 (2018) no. 1-2, pp. 323-385 | DOI | Zbl
[12] Coloured stochastic vertex models and their spectral theory (2018) (https://arxiv.org/abs/1808.01866)
[13] Nonsymmetric Macdonald polynomials via integrable vertex models (2019) (https://arxiv.org/abs/1904.06804, to appear in Trans. Am. Math. Soc.)
[14] Observables of coloured stochastic vertex models and their polymer limits, Probab. Math. Phys., Volume 1 (2020) no. 1, pp. 205-265 | DOI | Zbl
[15] Spin -Whittaker polynomials, Adv. Math., Volume 376 (2020), 107449, 51 pages | DOI | Zbl
[16] Construction of -matrices for symmetric tensor representations related to , J. Phys. A. Math. Theor., Volume 49 (2016) no. 49, 495204, 19 pages | Zbl
[17] Schur Polynomials and The Yang-Baxter Equation, Commun. Math. Phys., Volume 308 (2011) no. 2, pp. 281-301 | DOI | Zbl
[18] Double Grothendieck polynomials and colored lattice models (2020) (https://arxiv.org/abs/2007.04533)
[19] Observables of stochastic colored vertex models and local relation (2020) (https://arxiv.org/abs/2011.11426)
[20] Yang–Baxter random fields and stochastic vertex models (2019) (https://arxiv.org/abs/1905.06815)
[21] Matrix product formula for Macdonald polynomials, J. Phys. A. Math. Theor., Volume 48 (2015) no. 38, 384001, 25 pages | DOI | Zbl
[22] The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math., Volume 153 (1996) no. 1-3, pp. 123-143 | DOI | Zbl
[23] A new generalisation of Macdonald polynomials, Commun. Math. Phys., Volume 352 (2017) no. 2, pp. 773-804 | DOI | Zbl
[24] Modified Macdonald polynomials and integrability, Commun. Math. Phys., Volume 374 (2020) no. 3, pp. 1809-1876 | DOI | Zbl
[25] Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Commun. Math. Phys., Volume 318 (2013) no. 1, pp. 173-246 | DOI | Zbl
[26] Hidden diagonal integrability of -Hahn vertex model and Beta polymer model (2021) (https://arxiv.org/abs/2105.05058)
[27] Stochastic matrix for , Nucl. Phys., B, Volume 913 (2016), pp. 248-277 | DOI
[28] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Oxford Science Publications, 1995, x+475 pages | DOI | Zbl
[29] On the Yang-Baxter equation for the six-vertex model, Nucl. Phys., B, Volume 882 (2014), pp. 70-96 | DOI | Zbl
[30] Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice, J. Math. Phys., Volume 58 (2017) no. 9, 091703, 23 pages | Zbl
[31] Vertex models, TASEP and Grothendieck polynomials, J. Phys. A. Math. Theor., Volume 46 (2013) no. 35, 355201, 26 pages | Zbl
[32] Spin q-Whittaker polynomials and deformed quantum Toda (2020) (https://arxiv.org/abs/2003.14260)
[33] Refined Cauchy identity for spin Hall–Littlewood symmetric rational functions (2007) (https://arxiv.org/abs/2007.10886)
[34] A short proof of Macdonald’s conjecture for the root systems of type , Proc. Am. Math. Soc., Volume 102 (1988) no. 4, pp. 777-785 | Zbl
[35] Quantum inverse scattering method for the q-boson model and symmetric functions, Funct. Anal. Appl., Volume 40 (2006) no. 3, pp. 207-217 | DOI | Zbl
[36] Refined Cauchy/Littlewood identities and six-vertex model partition functions. III. Deformed bosons, Adv. Math., Volume 299 (2016), pp. 543-600 | DOI | Zbl
[37] Hall polynomials, inverse Kostka polynomials and puzzles, J. Comb. Theory, Ser. A, Volume 159 (2018), pp. 107-163 | DOI | Zbl
[38] Six-vertex, loop and tiling models: integrability and combinatorics (Habilitation thesis, http://www.lpthe.jussieu.fr/~pzinn/publi/hdr.pdf, https://arxiv.org/abs/0901.0665)
Cité par Sources :