1.3k citations · 3k across the 62 of their papers we have counts for
66 papers
Rank recursion for -Whittaker and Macdonald operators
Philippe Di Francesco, Hieu Trung Vu
In this paper, we introduce and prove a set of rank recurrence relations for -Whittaker and Macdonald operators. We also derive an explicit expression for the -th power of th…
Arctic curves of the T-system with Slanted Initial Data
Philippe Di Francesco, Hieu Trung Vu
We study the T-system of type , also known as the octahedron recurrence/equation, viewed as a 2+1-dimensional discrete evolution equation. Generalizing the study of [P. D…
Cluster structure on genus 2 spherical DAHA: seven-colored flower
Semeon Arthamonov, Leonid Chekhov, Philippe Di Francesco +4
We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutat…
Ruijsenaars wavefunctions as modular group matrix coefficients
Philippe Di Francesco, Rinat Kedem, Sergey Khoroshkin +2
We give a description of the Hallnäs--Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $S\in SL(2,\mathbb{Z…
Duality and Macdonald difference operators
Philippe Di Francesco, Rinat Kedem
This note summarizes certain properties common to Macdonald, Koornwinder and Arthamonov-Shakirov -difference operators, relating to the duality or bi-spectrality properties of t…
Arctic curves of the 20V model on a triangle
Philippe Di Francesco
We apply the Tangent Method of Colomo and Sportiello to predict the arctic curves of the Twenty Vertex model with specific domain wall boundary conditions on a triangle, in the Dis…
Exponents for Hamiltonian paths on random bicubic maps and KPZ
Philippe Di Francesco, Bertrand Duplantier, Olivier Golinelli +1
We evaluate the configuration exponents of various ensembles of Hamiltonian paths drawn on random planar bicubic maps. These exponents are estimated from the extrapolations of exac…
Macdonald Duality and the proof of the Quantum Q-system conjecture
Philippe Di Francesco, Rinat Kedem
The -symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution…
OPE for XXX
Philippe Di Francesco, Fedor Smirnov
We explain a new method for finding the correlation functions for the XXX model which is based on the concepts of Operator Product Expansion of Quantum Field Theory on one hand and…
Arctic Curves in path models from the Tangent Method
Philippe Di Francesco, Matthew F. Lapa
Recently, Colomo and Sportiello introduced a powerful method, known as the \emph{Tangent Method}, for computing the arctic curve in statistical models which have a (non- or weakly-…
(t,q) Q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators
Philippe Di Francesco, Rinat Kedem
We introduce difference operators on the space of symmetric functions which are a natural generalization of the -Macdonald operators. In the limit, they satisfy…
Systematic procedure for analyzing cumulants at any order
Philippe Di Francesco, Maxime Guilbaud, Matthew Luzum +1
We present a systematic procedure for analyzing cumulants to arbitrary order in the context of heavy-ion collisions. It generalizes and improves existing procedures in many respect…
Quantum Q systems: From cluster algebras to quantum current algebras
Philippe Di Francesco, Rinat Kedem
In this paper, we recall our renormalized quantum Q-system associated with representations of the Lie algebra , and show that it can be viewed as a quotient of the quantum cur…
Difference equations for graded characters from quantum cluster algebra
Philippe Di Francesco, Rinat Kedem
We introduce a new set of -difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebr…
A path model for Whittaker vectors
P. Di Francesco, R. Kedem, B. Turmunkh
In this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-…
Bessenrodt-Stanley polynomials and the octahedron recurrence
Philippe Di Francesco
We show that a family of multivariate polynomials recently introduced by Bessenrodt and Stanley can be expressed as solution of the octahedron recurrence with suitable initial data…
The Non-Commutative -system and its positive Laurent property
P. Di Francesco
We define a non-commutative version of the T-system, which underlies frieze patterns of the integer plane. This system has discrete conserved quantities and has a particular…
Arctic curves of the octahedron equation
P. Di Francesco, R. Soto-Garrido
We study the octahedron relation (also known as the -system), obeyed in particular by the partition function for dimer coverings of the Aztec Diamond graph. For a s…
T-systems, networks and dimers
P. Di Francesco
We study the solutions of the T-system for type A, also known as the octahedron equation, viewed as a 2+1-dimensional discrete evolution equation. These may be expressed entirely i…
A doubly-refined enumeration of alternating sign matrices and descending plane partitions
Roger E. Behrend, Philippe Di Francesco, Paul Zinn-Justin
It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and desc…
Quantum cluster algebras and fusion products
Philippe Di Francesco, Rinat Kedem
-systems are recursion relations satisfied by the characters of the restrictions of special finite-dimensional modules of quantum affine algebras. They can also be viewed as mut…
On the weighted enumeration of alternating sign matrices and descending plane partitions
Roger E. Behrend, Philippe Di Francesco, Paul Zinn-Justin
We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m an…
The solution of the quantum T-system for arbitrary boundary
Philippe Di Francesco, Rinat Kedem
We solve the quantum version of the -system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cl…
Discrete integrable systems, positivity, and continued fraction rearrangements
Philippe Di Francesco
In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the - and -systems based on $A_r…