On the Fourier coefficients of word maps on unitary groups

TL;DR

This paper investigates the expected Fourier coefficients of characteristic polynomials of word maps on Haar-random unitary matrices, revealing bounds and symmetries using Weingarten calculus and group-theoretic cancellations.

Contribution

It introduces bounds on the expected Fourier coefficients of word maps on unitary groups and employs Weingarten calculus with symmetry analysis to uncover cancellations.

Findings

Expected coefficients are bounded by a combinatorial expression with a positive exponent

Symmetries from Weingarten calculus lead to cancellations in the sums

The bounds depend only on the word and matrix dimension

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