On the Fourier coefficients of word maps on unitary groups

Abstract 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 $\epsilon(w)$ , depending only on w , such that \[|\mathbb{E}(c_{m}(w(X_{1},\ldots,X_{r})))|\leq\binom{d}{m}^{\!\!1-\epsilon(w)},\] 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 $\mathbb{E}(c_{m}(w))$ . These cancellations, coming from averaging a Weingarten function over cosets, follow from Schur’s orthogonality relations.