On the Fourier coefficients of word maps on unitary groups

Abstract

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 \[ \left|\mathbb{E}\left(c_{m}\left(w(X_{1},\ldots,X_{r})\right)\right)\right|\leq\left(\begin{array}{c} d\\ m \end{array}\right)^{1-ε(w)}, \] for every $d$ and every $1\le m\le d$. 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 $\mathbb{E}\left(c_m(w)\right)$. These cancellations, coming from averaging a Weingarten function over cosets, follow from Schur's orthogonality relations.

30 pages, second version following feedback from the referees. To appear in Compositio Mathematica

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