论文标题
在单一群体上的单词图的傅立叶系数上
On the Fourier coefficients of word maps on unitary groups
论文作者
论文摘要
给定一个单词$ w（x_ {1}，\ ldots，x_ {r}）$，即，$ r $元素的免费组中的一个元素，以及一个整数$ d \ geq1 $，我们研究了随机矩阵的特征性多项式， Haar-random独立$ d \ times d $统一矩阵。如果$ c_ {m}（x）$表示$ x $的特征多项式的$ m $ th系数，我们的主要定理意味着仅根据$ w $，有一个正常数$ε（w）$，因此\左| \ Mathbb { \]对于每$ d $，每$ 1 \ leq m \ leq d $。 我们的主要计算工具是Weingarten演算，它使我们能够在诸如上面的期望之类的统一组上表达积分，如对称组的某些总和。我们利用隐藏的对称性来在表示$ \ mathbb {e} \ left（c_ {m}（w）（w）\ right）$的总和中找到取消。这些取消是来自Schur的正交关系的平均素能函数的平均值。
Given a word $w(x_{1},\ldots,x_{r})$, i.e., an element in the free group on $r$ elements, and an integer $d\geq1$, we study the characteristic polynomial of the random matrix $w(X_{1},\ldots,X_{r})$, where $X_{i}$ are Haar-random independent $d\times d$ unitary matrices. If $c_{m}(X)$ denotes the $m$-th coefficient of the characteristic polynomial of $X$, our main theorem implies that there is a positive constant $ε(w)$, depending only on $w$, such that \[ \left|\mathbb{E}\left(c_{m}\left(w(X_{1},\ldots,X_{r})\right)\right)\right|\leq\left(\begin{array}{c} d\\ m \end{array}\right)^{1-ε(w)}, \] for every $d$ and every $1\leq m\leq d$. Our main computational tool is the Weingarten Calculus, which allows us to express integrals on unitary groups such as the expectation above, as certain sums on symmetric groups. We exploit a hidden symmetry to find cancellations in the sum expressing $\mathbb{E}\left(c_{m}(w)\right)$. These cancellations, coming from averaging a Weingarten function over cosets, follow from Schur's orthogonality relations.
