Matrix product formula for Macdonald polynomials

Abstract

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 considering representations of the Zamolodchikov--Faddeev and Yang--Baxter algebras in terms of $t$-deformed bosonic operators. These solutions form a basis of the ring of polynomials in $n$ variables, whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at $q=1$.

27 pages; typos corrected, references added and some better conventions adopted in v2

References in corpus (6)

• Matrix representation of the stationary measure for the multispecies TASEP
• Two-Species Asymmetric Simple Exclusion Process with Open Boundaries
• On polynomials interpolating between the stationary state of a O(n) model and a Q.H.E. ground state
• Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2
• Tetrahedron Equation and Quantum $R$ Matrices for modular double of $U_q(D_{n+1}^{(2)}),U_q(A_{2n}^{(2)})$ and $U_q(C_n^{(1)})$
• A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule

Cited by in corpus (24)

• Multispecies TASEP and the tetrahedron equation
• Exact solution to integrable open multi-species SSEP and macroscopic fluctuation theory
• Integrable Floquet dynamics, generalized exclusion processes and "fused" matrix ansatz
• Matrix product solution to a 2-species TASEP with open integrable boundaries
• A note on $q$-oscillator realizations of $U_q(gl(M|N))$ for Baxter $Q$-operators
• Integrable stochastic dualities and the deformed Knizhnik-Zamolodchikov equation
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• Matrix product solution to multi-species ASEP with open boundaries
• Modified Macdonald polynomials and the multispecies zero range process: I
• Limiting current distribution for a two species asymmetric exclusion process
• Multiline queues with spectral parameters
• Modified Macdonald polynomials and the multispecies zero range process: II
• Transition probability and total crossing events in the multi-species asymmetric exclusion process
• Stochastic symplectic ice
• A $q$-boson representation of Zamolodchikov-Faddeev algebra for stochastic $R$ matrix of $U_q(A_n^{(1)})$
• Matrix product construction for Koornwinder polynomials and fluctuations of the current in the open ASEP
• Tetrahedron equation and Schur functions
• New formulas for Macdonald polynomials via the multispecies exclusion and zero range processes
• Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion
• Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials
• Quasi-solvable lattice models for ${\mathrm{Sp}}_{2n}$ and ${\mathrm{SO}}_{2n+1}$ Demazure atoms and characters
• Correlations in the multispecies PASEP on a ring
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday
• Steady state representations for the harmonic process