# ALGEBRAIC COMBINATORICS

Refined Littlewood identity for spin Hall–Littlewood symmetric rational functions
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 37-51.

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions ${F}_{\lambda }$ are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials and can be viewed as partition functions in $\mathrm{𝔰𝔩}\left(2\right)$ higher spin six vertex models.

We obtain a refined Littlewood identity expressing a weighted sum of ${F}_{\lambda }$’s over all signatures $\lambda$ with even multiplicities as a certain Pfaffian. This Pfaffian can be derived as a partition function of the six vertex model in a triangle with suitably decorated domain wall boundary conditions. The proof is based on the Yang–Baxter equation.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.251
Classification: 05E05, 82B20, 81R50
Keywords: Littlewood identity, symmetric functions, six vertex model
Gavrilova, Svetlana 1

1 National Research University Higher School of Economics Faculty of Mathematics 6 Usacheva st. Moscow 119048 (Russia)
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Gavrilova, Svetlana. Refined Littlewood identity for spin Hall–Littlewood symmetric rational functions. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 37-51. doi : 10.5802/alco.251. https://alco.centre-mersenne.org/articles/10.5802/alco.251/

[1] Barraquand, Guillaume; Borodin, Alexei; Corwin, Ivan Half-space Macdonald processes, Forum Math. Pi, Volume 8 (2020), Paper no. e11, 150 pages | MR | Zbl

[2] Barraquand, Guillaume; Borodin, Alexei; Corwin, Ivan; Wheeler, Michael Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process, Duke Math. J., Volume 167 (2018) no. 13, pp. 2457-2529 | MR | Zbl

[3] Bisi, Elia; Zygouras, Nikos Point-to-line polymers and orthogonal Whittaker functions, Trans. Am. Math. Soc., Volume 371 (2019) no. 12, pp. 8339-8379 | DOI | MR | Zbl

[4] Borodin, Alexei On a family of symmetric rational functions, Adv. Math., Volume 306 (2017), pp. 973-1018 | DOI | MR | Zbl

[5] Borodin, Alexei Stochastic higher spin six vertex model and Macdonald measures, J. Math. Phys., Volume 59 (2018) no. 2, Paper no. 023301, 17 pages | MR | Zbl

[6] Borodin, Alexei; Corwin, Ivan; Petrov, Leonid; Sasamoto, Tomohiro Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz, Commun. Math. Phys., Volume 339 (2015) no. 3, pp. 1167-1245 | DOI | MR | Zbl

[7] Borodin, Alexei; Petrov, Leonid Higher spin six vertex model and symmetric rational functions, Sel. Math., New Ser., Volume 24 (2018) no. 2, pp. 751-874 | DOI | MR | Zbl

[8] Chen, Kailun; Ding, Xiangmao Stable spin Hall–Littlewood symmetric functions, combinatorial identities, and half-space Yang–Baxter random field, 2021 | arXiv

[9] Corwin, Ivan; Petrov, Leonid Stochastic higher spin vertex models on the line, Commun. Math. Phys., Volume 343 (2016) no. 2, pp. 651-700 | DOI | MR | Zbl

[10] Izergin, A. G. Partition function of a six-vertex model in a finite volume, Dokl. Akad. Nauk SSSR, Volume 297 (1987) no. 2, pp. 331-333 | MR

[11] Kirillov, A. N.; Noumi, M. $q$-difference raising operators for Macdonald polynomials and the integrality of transition coefficients, Algebraic methods and $q$-special functions, Providence, RI: American Mathematical Society, 1999, pp. 227-243 | DOI | Zbl

[12] Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993, xx+555 pages | DOI | MR

[13] Kupferberg, Greg Symmetry classes of alternating-sign matrices under one roof, Ann. Math. (2), Volume 156 (2002) no. 3, pp. 835-866 | DOI | MR | Zbl

[14] Macdonald, Ian Grant Symmetric functions and Hall polynomials, Oxford: Clarendon Press, 1995, x + 475 pages | Zbl

[15] Petrov, Leonid Refined Cauchy identity for spin Hall-Littlewood symmetric rational functions, J. Combin. Theory Ser. A, Volume 184 (2021), Paper no. 105519, 50 pages | MR | Zbl

[16] Povolotsky, A. M. On the integrability of zero-range chipping models with factorized steady states, J. Phys. A, Volume 46 (2013) no. 46, Paper no. 465205, 25 pages | DOI | MR | Zbl

[17] Rains, Eric; Warnaar, S. Ole Bounded Littlewood identities, Mem. Amer. Math. Soc., Volume 270 (2021) no. 1317, p. vii+115 | MR | Zbl

[18] Warnaar, S. Ole Rogers-Szegő polynomials and Hall-Littlewood symmetric functions, J. Algebra, Volume 303 (2006) no. 2, pp. 810-830 | DOI | Zbl

[19] Warnaar, S. Ole Bisymmetric functions, Macdonald polynomials and ${\mathrm{𝔰𝔩}}_{3}$ basic hypergeometric series, Compos. Math., Volume 144 (2008) no. 2, pp. 271-303 | DOI | MR | Zbl

[20] Wheeler, Michael; Zinn-Justin, Paul Refined Cauchy/Littlewood identities and six-vertex model partition functions. III. Deformed bosons, Adv. Math., Volume 299 (2016), pp. 543-600 | DOI | MR | Zbl

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