arXiv subjects

Meri Zaimi

Publications and source records attributed to Meri Zaimi.

18 recordsLinked to original sources

arXiv2026

Knot contact homology as a planar limit of Chern-Simons theory

We prove a conjecture relating augmentation varieties to the large $N$ limit of Chern-Simons theory. Although this does not directly establish that the augmentation polynomial of a knot is the classical limit of a deformed $\hat{A}$-polynomial - as suggested by Aganagi\'c and Vafa - it reduces the problem to characterizing certain algebraic properties of a module over the quantum torus, introduced in work of Gaiotto, Kannagi, and Sanjurjo. We term this the \emph{HOMFLYPT difference module}, which captures relations between the colored HOMFLYPT polynomials of different antisymmetric colorings. We demonstrate that the classical limit of this difference module for a knot is precisely the degree 0 abelianized knot contact homology of the knot, and we provide a natural extension of this result to links.

math.GT↗

arXiv2025

Contiguity relations for finite families of orthogonal polynomials in the Askey scheme

This paper classifies the contiguity relations for finite families of polynomials within the ($q$-)Askey scheme. The necessary and sufficient conditions for the existence of these contiguity relations are presented first. These conditions are then solved, yielding a comprehensive list of contiguity relations for these various families of polynomials. Furthermore, we demonstrate that all contiguity relations correspond to spectral transforms.

math.CA↗

arXiv2024

Meta algebras and biorthogonal rational functions: the $q$-Hahn case

A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases.

math.RT↗

arXiv2024

Bivariate $P$- and $Q$-polynomial structures of the association schemes based on attenuated spaces

The bivariate $P$- and $Q$-polynomial structures of association schemes based on attenuated spaces are examined using recurrence and difference relations of the bivariate polynomials which form the eigenvalues of the scheme. These bispectral properties are obtained from contiguity relations of univariate dual $q$-Hahn and affine $q$-Krawtchouk polynomials. The bispectral algebra associated to the bivariate polynomials is investigated, as well as the subconstituent algebra of the schemes. The properties of the schemes are compared to those of the non-binary Johnson schemes through a limit.

math.CO↗

arXiv2024

Griffiths polynomials of Racah type

Bivariate Griffiths polynomials of Racah type are constructed from univariate Racah polynomials. The bispectral properties of the former are deduced from simple properties of the latter. A duality relation and the orthogonality of these polynomials are provided. The domain of validity for the indices and variables of these polynomials is also determined. Particular limits on the parameters entering the polynomials allow to define several Griffiths polynomials of other types. One special limit connects them to the original Griffiths polynomials (of Krawtchouk type). Finally, a connection with the $9j$ symbols is made.

math-ph↗

arXiv2023

Factorized $A_2$-Leonard pair

The notion of factorized $A_2$-Leonard pair is introduced. It is defined as a rank 2 Leonard pair, with actions in certain bases corresponding to the root system of the Weyl group $A_2$, and with some additional properties. The functions arising as entries of transition matrices are bivariate orthogonal polynomials (of Tratnik type) with bispectral properties. Examples of factorized $A_2$-Leonard pairs are constructed using classical Leonard pairs associated to families of orthogonal polynomials of the ($q$-)Askey scheme. The most general examples are associated to an intricate product of univariate ($q$-)Hahn and dual ($q$-)Hahn polynomials.

math.RA↗

arXiv2023

m-distance-regular graphs and their relation to multivariate P-polynomial association schemes

An association scheme is $P$-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate $P$-polynomial association schemes of type $(\alpha,\beta)$ were introduced by Bernard et al., and multivariate $P$-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of $m$-distance-regular graph is defined and shown to give a graph interpretation of the multivariate $P$-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate $P$-polynomial association schemes and $m$-distance-regular graphs are also considered. In particular, bivariate $P$-polynomial schemes of type $(\alpha, \beta)$ are discussed, and their connection to 2-distance-regular graphs is established.

math.CO↗

arXiv2023

A bivariate $Q$-polynomial structure for the non-binary Johnson scheme

The notion of multivariate $P$- and $Q$-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate $P$-polynomial association scheme. We show here that it is also a bivariate $Q$-polynomial association scheme for some parameters. This provides, with the $P$-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.

math.CO↗

arXiv2022

An algebraic treatment of the Pastro polynomials on the real line

The properties of the Pastro polynomials on the real line are studied with the help of a triplet of $q$-difference operators. The $q$-difference equation and recurrence relation these polynomials obey are shown to arise as generalized eigenvalue problems involving the triplet of operators, with the Pastro polynomials as solutions. Moreover, a discrete biorthogonality relation on the real line for the Pastro polynomials is obtained and is then understood using adjoint operators. The algebra realized by the triplet of $q$-difference operators is investigated.

math.CA↗

arXiv2022

$R_I$ biorthogonal polynomials of Hahn type

A finite family of $R_I$ polynomials is introduced and studied. It consists in a set of polynomials of $_{3}F_{2}$ form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of $R_I$ type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions.

math.CA↗

arXiv2022

Askey-Wilson braid algebra and centralizer of $U_q(\mathfrak{sl}_2)$

A presentation of the centralizer of the three-fold tensor product of the spin $s$ representation of the quantum group $U_q(\mathfrak{sl}_2)$ is provided. It is expressed as a quotient of the Askey-Wilson braid algebra. This newly defined algebra combines the Askey-Wilson relations with the braid group relations, on three strands, together with a characteristic equation of degree $2s+1$ for the braid generators. Explicit bases are given for the centralizer.

math.RT↗

arXiv2022

Chern-Simons theory, link invariants and the Askey-Wilson algebra

The occurrence of the Askey-Wilson (AW) algebra in the $SU(2)$ Chern-Simons (CS) theory and in the Reshetikhin-Turaev (RT) link invariant construction with quantum algebra $U_q(\mathfrak{su}_2)$ is explored. Tangle diagrams with three strands with some of them enclosed in a spin-$1/2$ closed loop are associated to the generators of the AW algebra. It is shown in both the CS theory and RT construction that the link invariant of these tangles obey the relations of the AW generators. It follows that the expectation values of certain Wilson loops in the CS theory satisfy relations dictated by the AW algebra and that the link invariants do not distinguish the corresponding linear combinations of links.

math-ph↗

arXiv2021

Braid group and $q$-Racah polynomials

The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(\mathfrak{sl}_2)$ are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the $q$-Racah polynomials which appear as $6j$-symbols in the Racah problem for $U_q(\mathfrak{sl}_2)$. Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.

math.RT↗

arXiv2020

Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q(\mathfrak{sl}_2)$

The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey-Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley-Lieb, Birman-Murakami-Wenzl and one-boundary Temperley-Lieb algebras as quotients of the Askey-Wilson algebra.

math.QA↗

arXiv2019

Qubits as edge state detectors: illustration using the SSH model

As is well known, qubits are the fundamental building blocks of quantum computers, and more generally, of quantum information. A major challenge in the development of quantum devices arises because the information content in any quantum state is rather fragile, as no system is completely isolated from its environment. Generally, such interactions degrade the quantum state, resulting in a loss of information. Topological edge states are promising in this regard because they are in ways more robust against noise and decoherence. But creating and detecting edge states can be challenging. We describe a composite system consisting of a two-level system (the qubit) interacting with a finite Su-Schrieffer-Heeger chain (a hopping model with alternating hopping parameters) attached to an infinite chain. In this model, the dynamics of the qubit changes dramatically depending on whether or not an edge state exists. Thus, the qubit can be used to determine whether or not an edge state exists in this model.

cond-mat.mes-hall↗

arXiv2019

Detecting topological edge states with the dynamics of a qubit

We consider the Su-Schrieffer-Heeger (SSH) chain, which has 0, 1, or 2 topological edge states depending on the ratio of the hopping parameters and the parity of the chain length. We couple a qubit to one edge of the SSH chain and a semi-infinite undimerized chain to the other, and evaluate the dynamics of the qubit. By evaluating the decoherence rate of the qubit we can probe the edge states of the SSH chain. The rate shows strong even-odd oscillations with the number of sites reflecting the presence or absence of edge states. Hence, the qubit acts as an efficient detector of the topological edge states of the SSH model. This can be generalized to other topological systems.

cond-mat.mes-hall↗

arXiv2019

Bannai-Ito algebras and the universal R-matrix of osp(1|2)

The Bannai-Ito algebra $BI(n)$ is viewed as the centralizer of the action of $\mathfrak{osp}(1|2)$ in the $n$-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal $R$-matrix of $\mathfrak{osp}(1|2)$. The specific structure of the $\mathfrak{osp}(1|2)$ embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of $BI(n)$ are derived from those of $BI(3)$ by repeated action of the coproduct and using properties of the $R$-matrix and of the generators of the symmetric group $\mathfrak S_n$.

math.RT↗

arXiv2019

Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$

A description of the embedding of the universal Askey--Wilson algebra, AW(3), in $U_q(sl_2)^{\otimes 3}$ is given in terms of the universal R-matrix of $U_q(sl_2)$. The generators of the centralizer of $U_q(sl_2)$ in its three-fold product are naturally expressed through conjugations of Casimir elements with R. They are identified as the images of the generators of AW(3) under the embedding map by showing that they obey the AW(3) relations. This is achieved by introducing a natural coaction also constructed with the help of the R-matrix.

math.QA↗