We compute the auto-correlations functions of order \(m\ge 1\) for the characteristic polynomials of random matrices from certain subgroups of the unitary groups \({\text {U}}(2)\) and \({\text {U}}(3)\) by establishing new branching rules. These subgroups can be understood as certain analogues of Sato–Tate groups of \({\text {USp}}(4)\) in our previous paper. Our computation yields symmetric polynomial identities with m-variables involving irreducible characters of \({\text {U}}(m)\) for all \(m \ge 1\) in an explicit, uniform way.

中文翻译：

---

酉群的自相关函数

我们计算酉群的某些子群\({\text {U}}(2)\)和\({\ ) 的随机矩阵特征多项式的阶\(m\ge 1\)自相关函数通过建立新的分支规则来文本{U}}(3)\) 。这些子群可以理解为我们之前论文中\({\text {USp}}(4)\)的 Sato–Tate 群的某些类似物。我们的计算以显式、统一的方式产生具有m变量的对称多项式恒等式，其中涉及所有\(m \ge 1\)的不可约字符\({\text {U}}(m)\) 。