We study matrix integration over the classical Lie groups U ( N ), Sp (2 N ), SO (2 N ) and SO (2 N + 1), using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz ± Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large N the partition functions, Wilson loops and Hopf links of Chern–Simons theory on S 3 with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern–Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom.

中文翻译：

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古典群体和托普利兹±汉高未成年人的矩阵模型及其在切恩·西蒙斯理论和费米离子模型中的应用

我们使用对称函数理论和等价公式在行列式和次要矩阵方面研究对称李群U（N），Sp（2 N），SO（2 N）和SO（2 N +1）上的矩阵积分±Hankel矩阵。我们为此类积分建立了许多分解和扩展，同时还插入了不可约字符。作为一个具体的例子，我们在有限的和大的N值下计算S 3上的Chern–Simons理论的分区函数，Wilson循环和Hopf链以及上述对称群。在这种情况下，为通用模型找到的身份转化为该理论的可观察者之间的关系。最后，我们使用字符展开来评估Chern-Simons类型的随机矩阵合奏中的平均值，描述具有矩阵自由度的可解费米离子模型的光谱。