Word Measures on $GL_N(q)$ and Free Group Algebras
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| Format: | Preprint |
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2021
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| author | Ernst-West, Danielle Puder, Doron Seidel, Matan |
| author_facet | Ernst-West, Danielle Puder, Doron Seidel, Matan |
| contents | Fix a finite field $K$ of order $q$ and a word $w$ in a free group $F$ on $r$ generators. A $w$-random element in $GL_N(K)$ is obtained by sampling $r$ independent uniformly random elements $g_1,\ldots,g_r\in GL_N(K)$ and evaluating $w\left(g_1,\ldots,g_r\right)$. Consider $\mathbb{E}_w\left[\mathrm{fix}\right]$, the average number of vectors in $K^{N}$ fixed by a $w$-random element. We show that $\mathbb{E}_{w}\left[\mathrm{fix}\right]$ is a rational function in $q^{N}$. Moreover, if $w=u^{d}$ with $u$ a non-power, then the limit $\lim_{N\to\infty}\mathbb{E}_{w}\left[\mathrm{fix}\right]$ depends only on $d$ and not on $u$. These two phenomena generalize to all stable characters of the groups $\left\{ GL_N(K)\right\}_{N}$.
A main feature of this work is the connection we establish between word measures on $GL_N(K)$ and the free group algebra $K\left[F\right]$. A classical result of Cohn [1964] and Lewin [1969] is that every one-sided ideal of $K\left[F\right]$ is a free $K\left[F\right]$-module with a well-defined rank. We show that for $w$ a non-power, $\mathbb{E}_{w}\left[\mathrm{fix}\right]=2+\frac{C}{q^{N}}+O\left(\frac{1}{q^{2N}}\right)$, where $C$ is the number of rank-2 right ideals $I\le K\left[F\right]$ which contain $w-1$ but not as a basis element. We describe a full conjectural picture generalizing this result, featuring a new invariant we call the $q$-primitivity rank of $w$.
In the process, we prove several new results about free group algebras. For example, we show that if $T$ is any finite subtree of the Cayley graph of $F$, and $I\le K\left[F\right]$ is a right ideal with a generating set supported on $T$, then $I$ admits a basis supported on $T$. We also prove an analogue of Kaplansky's unit conjecture for certain $K\left[F\right]$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_11099 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Word Measures on $GL_N(q)$ and Free Group Algebras Ernst-West, Danielle Puder, Doron Seidel, Matan Group Theory Combinatorics Rings and Algebras 20E05, 20c07 (Primary) 16S34, 20g40, 20H30, 68r15, 20c33 (Secondary) Fix a finite field $K$ of order $q$ and a word $w$ in a free group $F$ on $r$ generators. A $w$-random element in $GL_N(K)$ is obtained by sampling $r$ independent uniformly random elements $g_1,\ldots,g_r\in GL_N(K)$ and evaluating $w\left(g_1,\ldots,g_r\right)$. Consider $\mathbb{E}_w\left[\mathrm{fix}\right]$, the average number of vectors in $K^{N}$ fixed by a $w$-random element. We show that $\mathbb{E}_{w}\left[\mathrm{fix}\right]$ is a rational function in $q^{N}$. Moreover, if $w=u^{d}$ with $u$ a non-power, then the limit $\lim_{N\to\infty}\mathbb{E}_{w}\left[\mathrm{fix}\right]$ depends only on $d$ and not on $u$. These two phenomena generalize to all stable characters of the groups $\left\{ GL_N(K)\right\}_{N}$. A main feature of this work is the connection we establish between word measures on $GL_N(K)$ and the free group algebra $K\left[F\right]$. A classical result of Cohn [1964] and Lewin [1969] is that every one-sided ideal of $K\left[F\right]$ is a free $K\left[F\right]$-module with a well-defined rank. We show that for $w$ a non-power, $\mathbb{E}_{w}\left[\mathrm{fix}\right]=2+\frac{C}{q^{N}}+O\left(\frac{1}{q^{2N}}\right)$, where $C$ is the number of rank-2 right ideals $I\le K\left[F\right]$ which contain $w-1$ but not as a basis element. We describe a full conjectural picture generalizing this result, featuring a new invariant we call the $q$-primitivity rank of $w$. In the process, we prove several new results about free group algebras. For example, we show that if $T$ is any finite subtree of the Cayley graph of $F$, and $I\le K\left[F\right]$ is a right ideal with a generating set supported on $T$, then $I$ admits a basis supported on $T$. We also prove an analogue of Kaplansky's unit conjecture for certain $K\left[F\right]$-modules. |
| title | Word Measures on $GL_N(q)$ and Free Group Algebras |
| topic | Group Theory Combinatorics Rings and Algebras 20E05, 20c07 (Primary) 16S34, 20g40, 20H30, 68r15, 20c33 (Secondary) |
| url | https://arxiv.org/abs/2110.11099 |