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}
}
  • A. Borodin, M. Wheeler
  • 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… 
View PDF on arXiv
View With Semantic Reader
15 Citations
Highly Influential Citations
2
Background Citations
5
Methods Citations
5
Results Citations
1

Figures from this paper

15 Citations

Citation Type

Citation Type

Colored Vertex Models and Iwahori Whittaker Functions
We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group $G$…
Quasi-solvable lattice models for $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n+1}$ Demazure atoms and characters.
We construct colored lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able…
Double Grothendieck Polynomials and Colored Lattice Models
We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent…
Colored Fermionic Vertex Models and Symmetric Functions
In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra Uq (…
A Vertex Model for Supersymmetric LLT Polynomials
We describe a Yang-Baxter integrable vertex model, which can be realized as a degeneration of a vertex model introduced by Aggarwal, Borodin, and Wheeler. From this vertex model, we construct a…
Free field approach to the Macdonald process
The Macdonald process is a stochastic process on the collection of partitions that is a $(q,t)$-deformed generalization of the Schur process. In this paper, we approach the Macdonald process…
Schur Function Identities and The Yang-Baxter Equation
We construct a eight-vertex lattice model which includes two different types of freefermionic Boltzmann weights. We show our model admits a solution to the Yang-Baxter equation. Moreover we prove…
Bent Lattice Models
Exactly solvable lattice models have found numerous applications to the theory of symmetric functions among other areas of mathematics (See [1–7]). The terms solvable refers to the existence of a…
A Vertex Model for LLT Polynomials
We describe a novel Yang–Baxter integrable vertex model. From this vertex model we construct a certain class of partition functions that we show are essentially equal to the LLT polynomials of…
Interpolation Polynomials, Bar Monomials, and Their Positivity
We prove a conjecture of Knop–Sahi on the positivity of interpolation polynomials, which is an inhomogeneous generalization of Macdonald’s conjecture for Jack polynomials. We also formulate and…
...
...

References

SHOWING 1-10 OF 46 REFERENCES
Coloured stochastic vertex models and their spectral theory
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
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…
Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons
Modified Macdonald Polynomials and Integrability
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
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
From multiline queues to Macdonald polynomials via the exclusion process
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…
...
...