与非平凡表示相关的紧群上独立均匀变量的强渐近自由性
Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations
AI总结：
本文研究紧群表示论衍生的新型矩阵酉模型，证明非平凡标记对应不可约表示下的酉商群满足强渐近自由性，同时得到正交群版本，并开发四种具独立理论价值的新工具。
摘要：
独立Haar分布酉矩阵的渐近自由性由Voiculescu发现。目前已有诸多改进成果，包括随机酉矩阵的强渐近自由性，以及作用于Perron-Frobenius特征向量正交补空间的随机置换的强渐近自由性。本文研究了一类从紧群表示论中自然产生的新型矩阵酉模型。我们固定一个非平凡signature（标记）$\rho$，即两组非递增自然数的有限序列，当$n$足够大时，考虑酉群$\mathbb{U}_n$对应于标记$\rho$的不可约表示$V_{n,\rho}$。我们将$\mathbb{U}_n$的商群$\mathbb{U}_{n,\rho}$视为$\mathbb{U}(V_{n,\rho})$的矩阵子群，并证明在该广义框架下，抽取Haar测度的独立副本时强渐近自由性成立。我们还得到了该结果的正交群版本。借助表示论中的经典结论，该结果与张量的强渐近自由性密切相关，我们将其作为预备结论予以证明。为得到这一结果，我们需要开发四种新工具，每种都具有独立的理论价值：（i）中心化Weingarten（温加滕）演算及其一致估计；（ii）矩阵的高斯矩与酉矩的系统性一致比较；（iii）一般$C^*$-代数中广义且简化的算子值非回溯理论；最后（iv）张量矩矩阵的组合学。
英文摘要：
Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature $ρ$, i.e. two finite sequences of non-increasing natural numbers, and for $n$ large enough, consider the irreducible representation $V_{n,ρ}$ of $\mathbb{U}_n$ associated to the signature $ρ$. We consider the quotient $\mathbb{U}_{n,ρ}$ of $\mathbb{U}_n$ viewed as a matrix subgroup of $\mathbb{U}(V_{n,ρ})$, and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. In order to achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator valued non-backtracking theory in a general $C^*$-algebra, and finally, (iv) combinatorics of tensor moment matrices.
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