A note on $\mathfrak{su}(2)$ models and the biorthogonality of generating functions of Krawtchouk polynomials

Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. In the present paper we give an elementary but comprehensive account of how the following are related: (i) Krawtchouk polynomials; (ii) finite-dimensional irreducible modules for the Lie algebra ${\mathfrak{sl}_2}$; (iii) a class of Leonard pairs said to have Krawtchouk type. Along the way we obtain elementary proofs of some well-known facts about Krawtchouk polynomials, such as the three-term recurr…

Racah problems for the oscillator algebra, the Lie algebra $\mathfrak{sl}_n$, and multivariate Krawtchouk polynomials

The oscillator Racah algebra $\mathcal{R}_n(\mathfrak{h})$ is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra $\mathfrak{h}$. An embedding of the Lie algebra $\mathfrak{sl}_{n-1}$ into $\mathcal{R}_n(\mathfrak{h})$ is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for $\mathfrak{h}$ and matrix elements of $\mathfrak{sl}_n$-representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.

$U(\mathfrak{sl}_2)$的Clebsch-Gordan规则、Krawtchouk代数与Hamming图

The Clebsch-Gordan Rule for $U(\mathfrak{sl}_2)$, the Krawtchouk Algebras and the Hamming Graphs

本文研究Hamming图的Terwilliger代数，利用对应Krawtchouk多项式的Askey-Wilson代数与$U(\mathfrak{sl}_2)$的同构关系，应用Clebsch-Gordan规则将标准模分解为不可约模的直和。

Branching rules for finite-dimensional $\mathcal{U}_q(\mathfrak{su}(3))$-representations with respect to a right coideal subalgebra

We consider the quantum symmetric pair $(\mathcal{U}_q(\mathfrak{su}(3)), \mathcal{B})$ where $\mathcal{B}$ is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of $\mathcal{B}$ are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of $\mathcal{U}_q(\mathfrak{su}(3))$ to $\mathcal{B}$ decomposes multiplicity free into irreducible representations of $\mathcal{B}$. Furthermore we give explicit expressions for the highest weight vectors in th…

广义约翰逊结合方案的多元$P$-多项式与$Q$-多项式结构

Multivariate $P$- and $Q$-polynomial structures of generalized Johnson association schemes

本文针对广义约翰逊结合方案$\boldsymbol{\frak J}_{n,h}(\boldsymbol{\frak X})$，证明其具有$(d+1)$元$P$-、$Q$-多项式结构，确定了本原幂等元与本征矩阵，回应了有限Gelfand对相关问题。

矩阵值正交多项式代数解释

Algebraic interpretation of discrete families of matrix valued orthogonal polynomials

本文通过李代数表示构造矩阵值正交多项式，探讨su(2)、su(1,1)及q-变形so_q(3)代数的矩阵类比，生成Krawtchouk、Meixner和离散Chebyshev多项式。