On the Fourier coefficients of word maps on unitary groups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913962741727232 |
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| author | Avni, Nir Glazer, Itay |
| author_facet | Avni, Nir Glazer, Itay |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04164 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Fourier coefficients of word maps on unitary groups Avni, Nir Glazer, Itay Probability Group Theory 60B15, 60B20 (Primary) 43A75, 20P05, 20B30 (Secondary) 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. |
| title | On the Fourier coefficients of word maps on unitary groups |
| topic | Probability Group Theory 60B15, 60B20 (Primary) 43A75, 20P05, 20B30 (Secondary) |
| url | https://arxiv.org/abs/2210.04164 |