arXiv subjects

Luc Vinet

Publications and source records attributed to Luc Vinet.

At least 127 records · Page 7Linked to original sources

arXiv2019

Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient of the Bannai-Ito algebra

We provide an explicit isomorphism between a quotient of the Bannai--Ito algebra and the Brauer algebra. We clarify also the connection with the action of the Lie superalgebra osp(1|2) on the threefold tensor product of its fundamental representation. Finally, a conjecture is proposed to describe the centralizer of osp(1|2) acting on three copies of an arbitrary finite irreducible representation in terms of a quotient of the Bannai-Ito algebra.

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↗

arXiv2019

Heun algebras of Lie type

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat. For $\mathfrak{su}(1,1)$, one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and Laguerre polynomials. These Heun algebras are shown to be isomorphic the the Hahn algebra. Focusing on the harmonic oscillator algebra $\mathfrak{ho}$ leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the $\mathfrak{su}(1,1)$ cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn and Confluent Heun operators respectively.

math.RA↗

arXiv2019

Perfect State Transfer on Weighted Graphs of the Johnson Scheme

We characterize perfect state transfer on real-weighted graphs of the Johnson scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$ and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show, using classical number theory results, that $X$ has perfect state transfer at time $τ$ if and only if $n=2k$, $m\ge 2^{\lfloor{\log_2(k)} \rfloor}$, and there are integers $c_1, c_2, \cdots, c_m$ such that (i) $c_j$ is odd if and only if $j$ is a power of $2$, and (ii) for $r=1,2,\cdots,m$, \[w_r = \fracπτ \sum_{j=r}^m \frac{c_j}{\binom{2j}{j}} \binom{k-r}{j-r}.\] We then characterize perfect state transfer on unweighted graphs of $\mathcal{J}(n,k)$. In particular, we obtain a simple construction that generates all graphs of $\mathcal{J}(n,k)$ with perfect state transfer at time $π/2$.

math.CO↗

arXiv2019

Perfect State Transfer in a Spin Chain without Mirror Symmetry

We introduce an analytical $XX$ spin chain with asymmetrical transport properties. It has an even number $N+1$ of sites labeled by $n=0,\cdots N$. It does not exhibit perfect state transfer (PST) from end-to-end but rather from the first site to the next to last one. In fact, PST of one-excitation states takes place between the even sites: $n\leftrightarrow N-n-1$, $n=0,2,\cdots, N-1$; while states localized at a single odd site undergo fractional revival (FR) over odd sites only. Perfect return is witnessed at double the PST/FR time. The couplings and local magnetic fields are related to the recurrence coefficients of the dual -1 Hahn polynomials.

quant-ph↗

arXiv2019

Truncation of the reflection algebra and the Hahn algebra

In the context of connections between algebras coming from quantum integrable systems and algebras associated to the orthogonal polynomials of the Askey scheme, we prove that the truncated reflection algebra attached to the Yangian of sl(2) is isomorphic to the Hahn algebra. As a by-product, we provide a general set-up based on Euler polynomials to study truncations of reflection algebras.

math-ph↗

arXiv2019

The $q$-Higgs and Askey-Wilson algebras

A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ in the $q$-oscillator representation of the quantized universal enveloping algebra $U_q(\mathfrak{u}(4))$. This $q$-Higgs algebra is also found as a specialization of the Askey--Wilson algebra embedded in the tensor product $U_q(\mathfrak{su}(1,1))\otimes U_q(\mathfrak{su}(1,1))$. The connection between these two approaches is established on the basis of the Howe duality of the pair $\big(\mathfrak{o}_{q^{1/2}}(4),U_q(\mathfrak{su}(1,1))\big)$.

math-ph↗

arXiv2019

Signal processing, orthogonal polynomials, and Heun equations

A survey of recents advances in the theory of Heun operators is offered. Some of the topics covered include: quadratic algebras and orthogonal polynomials, differential and difference Heun operators associated to Jacobi and Hahn polynomials, connections with time and band limiting problems in signal processing.

math-ph↗

arXiv2019

Spin Chains, Graphs and State Revival

Connections between the 1-excitation dynamics of spin lattices and quantum walks on graphs will be surveyed. Attention will be paid to perfect state transfer (PST) and fractional revival (FR) as well as to the role played by orthogonal polynomials in the study of these phenomena. Included is a discussion of the ordered Hamming scheme, its relation to multivariate Krawtchouk polynomials of the Tratnik type, the exploration of quantum walks on graphs of this association scheme and their projection to spin lattices with PST and FR.

math-ph↗

arXiv2018

Bargmann and Barut-Girardello models for the Racah algebra

The Racah algebra and its higher rank extension are the algebras underlying the univariate and multivariate Racah polynomials. In this paper we develop two new models in which the Racah algebra naturally arises as symmetry algebra, namely the Bargmann model and the Barut-Girardello model. We show how both models are connected with the superintegrable model of Miller et al. The Bargmann model moreover leads to a new realization of the Racah algebra of rank $n$ as $n$-variable differential operators. Our conceptual approach also allows us to rederive the basis functions of the superintegrable model without resorting to separation of variables.

math-ph↗

arXiv2018

$SU_q(3)$ corepresentations and bivariate $q$-Krawtchouk polynomials

The matrix elements of unitary $SU_q(3)$ corepresentations, which are analogues of the symmetric powers of the natural repesentation, are shown to be the bivariate $q$-Krawtchouk orthogonal polynomials, thus providing an algebraic interpretation of these polynomials in terms of quantum groups.

math-ph↗

arXiv2018

The dual pair $Pin(2n)\times\mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra

The Bannai-Ito algebra can be defined as the centralizer of the coproduct embedding of $\mathfrak{osp}(1|2)$ in $\mathfrak{osp}(1|2)^{\otimes n}$. It will be shown that it is also the commutant of a maximal Abelian subalgebra of $\mathfrak{o}(2n)$ in a spinorial representation and an embedding of the Racah algebra in this commutant will emerge. The connection between the two pictures for the Bannai-Ito algebra will be traced to the Howe duality which is embodied in the $Pin(2n)\times\mathfrak{osp}(1|2)$ symmetry of the massless Dirac equation in $\mathbb{R}^{2n}$. Dimensional reduction to $\mathbb{R}^{n}$ will provide an alternative to the Dirac-Dunkl equation as a model with Bannai-Ito symmetry.

math-ph↗

arXiv2018

Bivariate Bannai-Ito polynomials

A two-variable extension of the Bannai-Ito polynomials is presented. They are obtained via $q\to-1$ limits of the bivariate $q$-Racah and Askey-Wilson orthogonal polynomials introduced by Gasper and Rahman. Their orthogonality relation is obtained. These new polynomials are also shown to be multispectral. Two Dunkl shift operators are seen to be diagonalized by the bivariate Bannai-Ito polynomials and 3- and 9-term recurrence relations are provided.

math.CA↗

arXiv2018

A higher rank Racah algebra and the $\mathbb{Z}_2^{n}$ Laplace-Dunkl operator

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the $\mathbb{Z}_2^n$ root system. This algebra is also the invariance algebra of the generic superintegrable model on the $n$-sphere. Bases of Dunkl harmonics are constructed explicitly using a Cauchy-Kovalevskaia theorem. These bases consist of joint eigenfunctions of maximal Abelian subalgebras of the higher rank Racah algebra. A method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented.

math-ph↗

arXiv2018

The generalized Racah algebra as a commutant

The Racah algebra $R(n)$ of rank $(n-2)$ is obtained as the commutant of the \mbox{$\mathfrak{o}(2)^{\oplus n}$} subalgebra of $\mathfrak{o}(2n)$ in oscillator representations of the universal algebra of $\mathfrak{o}(2n)$. This result is shown to be related in a Howe duality context to the definition of $R(n)$ as the algebra of Casimir operators arising in recouplings of $n$ copies of $\mathfrak{su}(1,1)$. These observations provide a natural framework to carry out the derivation by dimensional reduction of the generic superintegrable model on the $(n-1)$ sphere which is invariant under $R(n)$.

math-ph↗

arXiv2018

The Racah algebra as a commutant and Howe duality

The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the $2$-sphere. It is further identified as the commutant of the $\mathfrak{o}(2) \oplus \mathfrak{o}(2) \oplus \mathfrak{o}(2)$ subalgebra of $\mathfrak{o}(6)$ in oscillator representations of the universal algebra of the latter. How this observation relates to the $\mathfrak{su}(1,1)$ Racah problem and the superintegrable model on the $2$-sphere is discussed on the basis of the Howe duality associated to the pair $\big(\mathfrak{o}(6)$, $\mathfrak{su}(1,1)\big)$.

math-ph↗

arXiv2018

The q-Heun operator of big q-Jacobi type and the q-Heun algebra

The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second order q-difference operator that maps polynomials of degree $n$ to polynomials of degree $n+1$. It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil made out of two q-Heun operators. The special case of the q-Heun operator associated to the little q-Jacobi polynomials is also treated.

math.CA↗

arXiv2018

The Heun operator of Hahn type

The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either Pochhammer or Hahn polynomials and to be bilinear in the operators of the Hahn algebra. The extension of this algebra that includes the Heun-Hahn operator as generator is described. Biorthogonal rational functions on uniform grids are shown to be related to this framework.

math.CA↗