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.
Eigenvalue problems on irreducible $\mathfrak{su}(2)$ modules and their adjoints are considered in the Bargmann, Barut-Girardello and finite difference models. The biorthogonality relations that arise between the corresponding generating functions of the Krawtchouk polynomials are sorted out. A link with Padé approximation is made.
We conjecture the existence of an embedding of the Racah algebra into the universal enveloping algebra of $\mathfrak{sl}_n$. Evidence of this conjecture is offered by realizing both algebras using differential operators and giving an embedding in this realization.
Recent results on the Racah algebra $\mathcal{R}_n$ of rank $n - 2$ are reviewed. $\mathcal{R}_n$ is defined in terms of generators and relations and sits in the centralizer of the diagonal action of $\mathfrak{su}(1,1)$ in $\mathcal{U}(\mathfrak{su}(1,1))^{\otimes n}$. Its connections with multivariate Racah polynomials are discussed. It is shown to be the symmetry algebra of the generic superintegrable model on the $ (n-1)$ - sphere and a number of interesting realizations are provided.
Université de Montréal（蒙特利尔大学）；Centre de Recherches Mathématiques, Université de Montréal（蒙特利尔大学数学研究中心）；CNRS（法国国家科学研究中心）；University of Miami（迈阿密大学）；Department of Physics, University of Miami（迈阿密大学物理系）；Laboratoire d’Annecy de Physique Théorique（阿讷西理论物理实验室）
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 represen…
Assume that ${\mathbb F}$ is a field with $\operatorname{char}{\mathbb F}\not=2$. The Racah algebra $\Re$ is a unital associative ${\mathbb F}$-algebra defined by generators and relations. The generators are $A$, $B$, $C$, $D$ and the relations assert that $[A,B]=[B,C]=[C,A]=2D$ and each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$ is central in $\Re$. The Bannai-Ito algebra $\mathfrak{BI}$ is a unital associative ${\mathbb F}$-algebra generated by $X$, $Y$, $Z$ and the relations assert that each of $\{X,Y\}-Z$, $\{Y,Z\}-X$, $\{Z,X\}-Y$ is central in $\mathfrak{BI}$. It was discovered that t…
Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C, D$ and the relations state that $$ [A,B]=[B,C]=[C,A]=2D $$ and each of \begin{gather*} [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC \end{gather*} is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak H$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $t_0,t_1,t_0^\vee,t_1^\vee$ and the relations …
BC Type Z-measures and Determinantal Point Processes
（BC型）z-测度是Jacobi边重数非负Gelfand-Tsetlin图的路空间上的一族四参数$z, z', a, b$概率测度.我们可以将$z$ -测度解释为穿孔正实数线$\mathfrak{X} = \mathbb{R}_+\setminus\{1\}$上的随机点过程$\mathcal{P}_{z, z', a, b}$.我们的主要结果是，这些随机过程是决定性的，而且我们计算的相关核显式的超几何函数。对于参数$z, z'$的特殊值，$\mathfrak{X}$上的过程$\mathcal{P}_{z, z', a, b}$本质上是Racah正交多项式系综的标度极限，其相关核可以简单地从Racah多项式的某些极限计算出来.因此，在随机矩阵的语言，我们研究某些解析延拓的过程是限制的拉卡合奏，使他们保留的行列式结构。我们的结果的另一种解释，本文的主要动机，是大群的表示理论。在表示论的术语，本文解决了几个无限维对称空间的调和分析的自然问题。
Algebraic interpretation of discrete families of matrix valued orthogonal polynomials
Centre de recherches mathématiques, Université de Montréal（蒙特利尔大学数学研究中心）；IVADO and Centre de recherches mathématiques, Université de Montréal（IVADO与蒙特利尔大学数学研究中心）
Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra generated by $A,B,C,D$ and the relations state that $[A,B]=[B,C]=[C,A]=2D$ and each of $$ [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC $$ is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak H$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i\}_{i=0}^3$ and the relations state that \begin{gather*} t_0+t_1+t_2+t_3 = -1, \\ \hbox{$t_i^2$ is central for al…
