圈积表示的高斯涨落
Gaussian fluctuations of representations of wreath products
摘要：
本文研究了圈积GPDMATHABMM S_q=G^q \rtimes S_q的可约表示的渐近性，其中G是固定有限群，S_q是q元上的对称群，特别是当G=Z/2 Z时，我们恢复了超八面体群.我们将G \wr S_q的这种可约表示分解为不可约分量的和（或者等价地，分解为Young图的元组的集合），并且我们询问随机选择的分量的特征是什么（或者，在随机选择的元组中Young图的形状是什么）。我们的主要结果是，对于一大类表示字符的波动（和波动的形状的杨图）是渐近高斯。所考虑的类由表示的字符渐近几乎因式分解，它包括，除其他外，左正则表示，因此，我们证明了类似的Kerov的中心极限定理的圈积。
英文摘要：
We study the asymptotics of the reducible representations of the wreath products G\wr S_q=G^q \rtimes S_q for large q, where G is a fixed finite group and S_q is the symmetric group in q elements; in particular for G=Z/2Z we recover the hyperoctahedral groups. We decompose such a reducible representation of G\wr S_q as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products.
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