On the Fourier coefficients of word maps on unitary groups

Given a word $w(x_1,\dots ,x_r)$, i.e., an element in the free group on $r$ elements, and an integer $d\ge 1$, we study the characteristic polynomial of the random matrix $w(X_1,\dots ,X_r)$, 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 $\epsilon (w)$, depending only on $w$, such that $$\mid \mathbb{E}\left(c_m(w(X_1,\dots ,X_r))\right)\mid \le {\left(\begin{array}{c}d\\ m\end{array}\right)}^{1-\epsilon (w)},$$ for every $d$ and every $1\le m\le d$. 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 $\mathbb{E}(c_m(w))$. These cancellations, coming from averaging a Weingarten function over cosets, follow from Schur’s orthogonality relations.