发表机构
Université de Montréal; Centre de Recherches Mathématiques, Université de Montréal; CNRS; University of Miami; Department of Physics, University of Miami; Laboratoire d’Annecy de Physique Théorique（蒙特利尔大学; 蒙特利尔大学数学研究中心; 法国国家科学研究中心; 迈阿密大学; 迈阿密大学物理系; 阿讷西理论物理实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究了para族正交多项式（源于自旋链设计），证明其对应代数表示可约，分解为两个不可约模，并用经典多项式表达，统一解释了其性质与构造。
AI 中文摘要
para-Krawtchouk多项式源于对具有完美态转移的自旋链的探索；随后，通过Askey方案各族的奇异截断，得到了para-Racah、$q$-para-Racah和para-Bannai--Ito多项式，并依次用于设计自旋链。所有这些多项式都在双格点上正交；该前缀可追溯至para-Krawtchouk情形，其谱为parabose振子的谱。这些多项式为相应代数——Hahn、Racah、$q$-Racah、Bannai--Ito及互补Bannai--Ito——所提供的表示被证明是可约的。每个表示分解为同一代数的两个不可约模的直和，一个模对应于两个交错子格点之一，并给出了子模的参数。由此，para多项式可用子模所承载的经典多项式来表达，且其特征值的配对$\lambda_n=\lambda_{N-n}$得以解释；同时识别出该约化所依赖的双格点结构的三个特征。随后，在此分解的视角下，以统一的方式呈现了para族的性质、这些族产生的构造以及它们已被应用的场景。
英文摘要
The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, $q$-para-Racah and para-Bannai--Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras --- Hahn, Racah, $q$-Racah, Bannai--Ito and complementary Bannai--Ito --- are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing $λ_n=λ_{N-n}$ of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.
Comments44 pages
