Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Word Measures on $GL_N(q)$ and Free Group Algebras   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.

1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys