arXiv subjects

Danielle Ernst-West

Publications and source records attributed to Danielle Ernst-West.

4 recordsLinked to original sources

arXiv2026

Stable Invariants of Words from Random Matrices

Let $w$ be a word in a free group. A few years ago, Magee and the first named author discovered that the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain Fourier coefficients of $w$-random unitary matrices [arXiv:1802.04862]. But the random-matrix side of this equality can be naturally tweaked by considering $w$-random permutations, $w$-random orthogonal matrices and so on, to produce new invariants for any given word. Are these invariants new? interesting? Do they admit an intrinsic topological description as in the case of $w$-random unitaries and scl? The current paper formalizes the definition of these invariants coming from $w$-random matrices, answers the above questions in certain cases involving generalized symmetric groups, and poses detailed conjectures in many others. In particular, we present a plethora of topological, combinatorial and algebraic invariants of words which play, or are at least conjectured to play, a similar role to the one played by scl in the above-mentioned result. Among others, these invariants include two invariants recently defined by Wilton [arXiv:2210.09853]: the stable primitivity rank and a non-oriented analog of scl.

math.GR↗

arXiv2025

Primitivity Testing in Free Group Algebras via Duality

Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\leq K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$. As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module. A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces.

math.GR↗

arXiv2024

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.

math.GR↗

arXiv2024

The ring of stable characters over $\text{GL}_\bullet(q)$

For a fixed prime power $q$, let $\text{GL}_\bullet(q)$ denote the family of groups $\text{GL}_N(q)$ for $N \in \mathbb{Z}_{\geq 0}$. In this paper we study the $\mathbb{C}$-algebra of "stable" class functions of $\text{GL}_\bullet(q)$, and show it admits four different linear bases, each arising naturally in different settings. One such basis is that of stable irreducible characters, namely, the class functions spanned by the characters corresponding to finitely generated simple $\mathrm{VI}$-modules in the sense of [arXiv:1408.3694,arXiv:1602.00654]. A second one comes from characters of parabolic representations. The final two, one originally defined in [arXiv:1803.04155] and the other in [arXiv:2110.11099], are more combinatorial in nature. As corollaries, we clarify many properties of these four bases and prove a conjecture from [arXiv:2106.11587].

math.CO↗