arXiv subjects

Loïc Poulain d'Andecy

Publications and source records attributed to Loïc Poulain d'Andecy.

16 recordsLinked to original sources

arXiv2026

Integrable multi-species SSEP with reactive particle species

We investigate integrable exclusion Markov processes constructed from set-theoretical solutions of the Yang-Baxter Equation (YBE) that generalise the multi-species Symmetric Simple Exclusion Process (SSEP). We first introduce the process called the $ (p,1) $-SSEP that is analogous to the multi-species SSEP but with an extra particle species qualified as reactive. Reactive species are able to evaporate and condense by pairs or to transform by pairs depending on the interpretation. We provide a full combinatorial study of the sectors in the periodic case. We prove the integrability of the process in the periodic case and in the open case for two types of boundaries that we introduce using Baxterisations of solutions of the reflection equation. Next we move on to the process called $ (p,q) $-SSEP, i.e. the multi-species SSEP with an arbitrary number of reactive particle species. We prove its integrability in the periodic case and, for the open case, we introduce integrable boundaries generalising the ones considered for the $ (p,1) $-SSEP. Finally, we define physical quantities relevant to the models and we compute them in the non-equilibrium stationary state of the open $ (p,q) $-SSEP for one type of integrable boundaries.

math-ph↗

arXiv2026

Polynomials of the Askey Scheme as Clebsch-Gordan Coefficients

Given a semi-simple algebra equipped with a coproduct, the Clebsch-Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the Clebsch-Gordan coefficients of the Lie algebra $\mathfrak{sl}_2$ are given in terms of the dual Hahn polynomials. Taking the reversed point of view, we show that any finite dimensional family of polynomials belonging to the Askey scheme can be interpreted as Clebsch-Gordan coefficients of an algebra. The Hahn polynomials are thus associated to the oscillator algebra with the Krawtchouk polynomials treated through a limit. The dual Hahn polynomials and Racah polynomials are seen to be associated to $\mathfrak{sl}_2$ with a more general coproduct than the standard one. The $q$-Hahn polynomials are interpreted as Clebsch-Gordan coefficients of a $q$-deformation of the oscillator algebra and the $q$-Racah polynomials are seen to be connected in this way to ${\rm U}_q(\mathfrak{sl}_2)$ with a generalized coproduct.

math.QA↗

arXiv2026

Twisted symmetric exclusion processes and set-theoretical $R$-matrices

We investigate periodic integrable Markov models, constructed from set-theoretical solutions of the Yang-Baxter equation. We first focus on the simplest class of solutions, called Lyubashenko solutions. We show that the resulting models are equivalent to some twisted Symmetric Simple Exclusion Process (SSEP), which are usual periodic SSEP models where a twist is added on a bond of the ring. We also provide various possible interpretations for these Markov models. Then, we study the long time dynamics of the twisted SSEP, characterising its different stationary states and counting them. Allowing the twist to vary, we examine the possible transitions between the different stationary states. Finally, we extend our construction of Markov models to set-theoretical solutions that are more general than Lyubashenko solutions and show that such models are not equivalent to a twisted SSEP in general.

math-ph↗

arXiv2026

Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$

This paper presents explicit formulas for intertwining operators of the quantum group $U_q(sl_2)$ acting on tensor products of Verma modules. We express a first set of intertwining operators (the holographic operators) in terms of the little $q$-Jacobi polynomials, and we obtain for the dual set (the symmetry breaking operators) a $q$-deformation of the Rankin--Cohen operators. The Verma modules are realised on polynomial spaces and, interestingly, we find along the way the need to work with non-commuting variables. Explicit connections are given with the Clebsch--Gordan coefficients of $U_q(sl_2)$ expressed with the $q$-Hahn polynomials.

math.RT↗

arXiv2023

Framization of Schur--Weyl duality and Yokonuma--Hecke type algebras

We study framizations of algebras through the idea of Schur--Weyl duality. We provide a general setting in which framizations of algebras such as the Yokonuma--Hecke algebra naturally appear and we obtain this way a Schur--Weyl duality for many examples of these algebras which were introduced in the study of knots and links. We thereby provide an interpretation of these algebras from the point of view of representations of quantum groups. In this approach the usual braid groups is replaced by the framed braid groups. This gives a natural procedure to construct framizations of algebras and we discuss in particular a new framized version of the Birman--Murakami--Wenzl algebra. The general setting is also extended to encompass the situation where the usual braid group is replaced by the so-called tied braids algebra, and this allows to collect in our approach even more examples of algebras introduced in the knots and links setting.

math.RT↗

arXiv2023

Matrix elements of $SO(3)$ in $sl_3$ representations as bispectral multivariate functions

We compute the matrix elements of $SO(3)$ in any finite-dimensional irreducible representation of $sl_3$. They are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials which generalize the Griffiths-Krawtchouk polynomials. Their recurrence and difference relations are obtained as byproducts of our construction. The proof is based on the decomposition of a general three-dimensional rotation in terms of elementary planar rotations and a transition between two embeddings of $sl_2$ in $sl_3$. The former is related to monovariate Krawtchouk polynomials and the latter, to monovariate Racah polynomials. The appearance of Racah polynomials in this context is algebraically explained by showing that the two $sl_2$ Casimir elements related to the two embeddings of $sl_2$ in $sl_3$ obey the Racah algebra relations. We also show that these two elements generate the centralizer in $U(sl_3)$ of the Cartan subalgebra and its complete algebraic description is given.

math.RT↗

arXiv2023

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

math.QA↗

arXiv2023

Entanglement of inhomogeneous free fermions on hyperplane lattices

We introduce an inhomogeneous model of free fermions on a $(D-1)$-dimensional lattice with $D(D-1)/2$ continuous parameters that control the hopping strength between adjacent sites. We solve this model exactly, and find that the eigenfunctions are given by multidimensional generalizations of Krawtchouk polynomials. We construct a Heun operator that commutes with the chopped correlation matrix, and compute the entanglement entropy numerically for $D=2,3,4$, for a wide range of parameters. For $D=2$, we observe oscillations in the sub-leading contribution to the entanglement entropy, for which we conjecture an exact expression. For $D>2$, we find logarithmic violations of the area law for the entanglement entropy with nontrivial dependence on the parameters.

cond-mat.stat-mech↗

arXiv2022

The Askey-Wilson algebra and its avatars

The original Askey-Wilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name 'Askey-Wilson algebra' is currently used to refer to a variety of related structures that appear in a large number of contexts. We review these versions, sort them out and establish the relations between them. We focus on two specific avatars. The first is a quotient of the original Zhedanov algebra; it is shown to be invariant under the Weyl group of type $D_4$ and to have a reflection algebra presentation. The second is a universal analogue of the first one; it is isomorphic to the Kauffman bracket skein algebra (KBSA) of the four-punctured sphere and to a subalgebra of the universal double affine Hecke algebra $(C_1^{\vee},C_1)$. This second algebra emerges from the Racah problem of $U_q(\mathfrak{sl}_2)$ and is related via an injective homomorphism to the centralizer of $U_q(\mathfrak{sl}_2)$ in its threefold tensor product. How the Artin braid group acts on the incarnations of this second avatar through conjugation by $R$-matrices (in the Racah problem) or half Dehn twists (in the diagrammatic KBSA picture) is also highlighted. Attempts at defining higher rank Askey-Wilson algebras are briefly discussed and summarized in a diagrammatic fashion.

math.QA↗

arXiv2021

Racah algebras, the centralizer $Z_n(\mathfrak{sl}_2)$ and its Hilbert-Poincaré series

The higher rank Racah algebra $R(n)$ introduced recently is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra $sR(n)$, is then introduced. Using results from classical invariant theory, this $sR(n)$ algebra is shown to be isomorphic to the centralizer $Z_{n}(\mathfrak{sl}_2)$ of the diagonal embedding of $U(\mathfrak{sl}_2)$ in $U(\mathfrak{sl}_2)^{\otimes n}$. This leads to a first and novel presentation of the centralizer $Z_{n}(\mathfrak{sl}_2)$ in terms of generators and defining relations. An explicit formula of its Hilbert-Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.

math.RT↗

arXiv2020

Morita equivalences for cyclotomic Hecke algebras of type B and D

We give a Morita equivalence theorem for so-called cyclotomic quotients of affine Hecke algebras of type B and D, in the spirit of a classical result of Dipper-Mathas in type A for Ariki-Koike algebras. As a consequence, the representation theory of affine Hecke algebras of type B and D reduces to the study of their cyclotomic quotients with eigenvalues in a single orbit under multiplication by $q^2$ and inversion. The main step in the proof consists in a decomposition theorem for generalisations of quiver Hecke algebras that appeared recently in the study of affine Hecke algebras of type B and D. This theorem reduces the general situation of a disconnected quiver with involution to a simpler setting. To be able to treat types B and D at the same time we unify the different definitions of generalisations of quiver Hecke algebra for type B that exist in the literature.

math.RT↗

arXiv2019

Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)

In the spirit of the Schur-Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of su(2). As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley-Lieb algebra, the Brauer algebra and the one-boundary Temperley-Lieb algebra as quotients of the Racah algebra.

math.RT↗

arXiv2017

Inductive basis on Birman-Murakami-Wenzl algebras and (transverse) Markov traces

We construct a new inductive basis of the Birman-Murakami-Wenzl algebra. Using it, we provide a new proof of the existence of the Markov trace on the BMW algebras affording the two-variable Kauffman polynomial. We prove also that all the transverse Markov traces on the BMW algebras are determined by the self-linking number, the HOMFLY-PT polynomial and the two-variable Kauffman polynomial.

math.GT↗

arXiv2015

Markov traces on affine and cyclotomic Yokonuma-Hecke algebras

In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can define Markov traces on them, which we in turn use to construct invariants for framed and classical knots in the solid torus. Finally, we study the Markov trace with zero parameters on the cyclotomic Yokonuma-Hecke algebras and determine the Schur elements with respect to that trace.

math.RT↗

arXiv2014

Representation theory of the Yokonuma-Hecke algebra

We develop an inductive approach to the representation theory of the Yokonuma-Hecke algebra ${\rm Y}_{d,n}(q)$, based on the study of the spectrum of its Jucys-Murphy elements which are defined here. We give explicit formulas for the irreducible representations of ${\rm Y}_{d,n}(q)$ in terms of standard $d$-tableaux; we then use them to obtain a semisimplicity criterion. Finally, we prove the existence of a canonical symmetrising form on ${\rm Y}_{d,n}(q)$ and calculate the Schur elements with respect to that form.

math.RT↗

arXiv2014

Fusion Procedure for Cyclotomic Hecke Algebras

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.

math.RT↗