Integrals of Irreducible Representations of Classical Groups

A new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group

We present a new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group. The identity is obtained from an integral involving Gegenbauer ultraspherical polynomials. A similar procedure can be applied for other types of integrals of such polynomials, and may therefore lead to the derivation of further new relations.

A method of integration for classical and quantum equations based on the connection between canonical transformations and irreducible representations of Lie groups

We propose a method for integrating the right-invariant geodesic flows on Lie groups based on the use of a special canonical transformation in the cotangent bundle of the group. We also describe an original method of constructing exact solutions for the Klein - Gordon equation on unimodular Lie groups. Finally, we formulate a theorem which establishes a connection between the special canonical transformation and irreducible representations of Lie group. This connection allows us to consider the proposed methods of integrating for classical and quantum equations in the framework of a unified ap…

酉群上的矩阵积分：Schur-Weyl对偶的一个应用

Matrix integrals over unitary groups: An application of Schur-Weyl duality

该研究运用Schur-Weyl对偶综述酉群矩阵积分公式，指出其计算与Weingarten函数相关，该函数求值依赖的置换群不可约特征标暂无通用闭式公式，相关信息可通过特征标表获取。

Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras

We consider finite W-algebras U(g,e) associated to even multiplicity nilpotent elements in classical Lie algebras. We give a classification of finite dimensional irreducible U(g,e)-modules with integral central character in terms of the highest weight theory for finite W-algebras. As a corollary, we obtain a parametrization of primitive ideals of U(g) with associated variety the closure of the adjoint orbit of e and integral central character.

经典半单代数不可约表示的Gel'fand-Tsetlin实现

To the Gel'fand-Tsetlin realization of irreducible representations of classical semisimple algebras

证明了$A_n$代数的不可约表示的Gel'fand-Tsetlin实现与n维空间中的线性精确可积系统直接相关.给出了该系统的一般解。

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

Irreducible representations (irreps) of $\mathbb{Z}_2^2$-graded supersymmetry algebra of ${\cal N}=2$ are obtained by the method of induced representation and they are used to derive $\mathbb{Z}_2^2$-graded supersymmetric classical actions. The irreps are four dimensional for $ λ= 0$ where $ λ$ is an eigenvalue of the Casimir element, and eight dimensional for $λ\neq 0.$ The eight dimensional irreps reduce to four dimensional ones only when $λ$ and an eigenvalue of Hamiltonian satisfy a particular relation. The reduced four dimensional irreps are used to define $\mathbb{Z}_2^2$-graded supersym…