# Nonsymmetric Macdonald polynomials via integrable vertex models

@article{Borodin2020NonsymmetricMP,
title={Nonsymmetric Macdonald polynomials via integrable vertex models},
author={Alexei Borodin and Michael Wheeler},
journal={Transactions of the American Mathematical Society},
year={2020}
}
• Published 15 April 2019
• Mathematics
• Transactions of the American Mathematical Society
Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $\mu = (\mu_1,\dots,\mu_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik-Dunkl operators $Y_i$ for all $1 \leq i \leq n$, and are thus equal to nonsymmetric Macdonald polynomials $E_{\mu}$. Our partition functions have the combinatorial…

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## References

SHOWING 1-10 OF 46 REFERENCES
Coloured stochastic vertex models and their spectral theory
• Mathematics
• 2018
This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following:
Matrix product formula for Macdonald polynomials
• Mathematics
• 2015
We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik--Zamolodchikov equations, which arise by
Modified Macdonald Polynomials and Integrability
• Mathematics
Communications in Mathematical Physics
• 2020
We derive combinatorial formulae for the modified Macdonald polynomial $$H_{\lambda }(x;q,t)$$ H λ ( x ; q , t ) using coloured paths on a square lattice with quasi-cylindrical boundary conditions.
Non-symmetric Macdonald polynomials and Demazure-Lusztig operators
We extend the family non-symmetric Macdonald polynomials and define general-basement Macdonald polynomials. We show that these also satisfy a triangularity property with respect to the monomials
Dynamical stochastic higher spin vertex models
We introduce a new family of integrable stochastic processes, called dynamical stochastic higher spin vertex models, arising from fused representations of Felder’s elliptic quantum group E_{\tau ,
Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra
We define cylindric versions of skew Macdonald functions Pλ/μ(q, t) for the special cases q = 0 or t = 0. Fixing two integers n > 2 and k > 0 we shift the skew diagram λ/μ, viewed as a subset of the
Higher spin six vertex model and symmetric rational functions
• Mathematics
• 2016
We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and
Spin q–Whittaker polynomials
• Mathematics
• 2021
From multiline queues to Macdonald polynomials via the exclusion process
• Mathematics
American Journal of Mathematics
• 2022
abstract:Recently James Martin introduced {\it multiline queues}, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion process