arXiv subjects

Yotam Shomroni

Publications and source records attributed to Yotam Shomroni.

6 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

Probabilistic Hanna Neumann Conjectures

We develop a theory of polymatroids on Stallings core graphs, which provides a new technique for proving lower bounds on stable invariants of words and subgroups in free groups $F$, and for upper bounds on their probability for mapping, under a random homomorphism from $F$ to a finite group $G$, into some subgroup of $G$. As a result, we prove the gap conjecture on the stable $K$-primitivity rank by Ernst-West, Puder and Seidel, prove a conjecture of Reiter about the number of solutions to a system of equations in a finite group action, and give a unified proof of the "rank-1 Hanna Neumann conjecture" by Wise and its higher rank analogue. We further show that the stable compressed rank and its $q$-analogue coincide with the decay rate of many-words measure on stable actions of finite simple groups of large rank. Finally, we conjecture an analogue of the Hanna Neumann conjecture over fields, and suggest that every finite group action is associated to some version of the HNC.

math.GR↗

arXiv2025

Word Measures on Wreath Products II

Every word $w$ in $F_r$, the free group of rank $r$, induces a probability measure (the $w$-measure) on every finite group $G$, by substitution of random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $E_w[χ]$ of the irreducible characters of $G$. For every finite group $G$, every stable character $χ$ of $G\wr S_n$ (trace of a finitely generated $FI_G$-module), and every word $w\in F_r$, we approximate $E_w[χ]$ up to an error term of $O(n^{-π(w)})$, where $π(w)$ is the primitivity rank of $w$. This generalizes previous works by Puder, Hanany, Magee and the author. As an application we show that random Schreier graphs of representation-stable actions of $G\wr S_n$ are close-to-optimal expanders. The paper reveals a surprising relation between stable representation theory of wreath products and not-necessarily connected Stallings core graphs.

math.GR↗

arXiv2025

Stable Invariants of Words from Random Matrices II: Formulas and Extensions

Let $w$ be a word in a free group. As was revealed by Magee and Puder in [arXiv:1802.04862], the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain stable Fourier coefficients of $w$-random unitary matrices. In the first part of the current work [arXiv:2311.17733], we demonstrated how this phenomenon is much broader: we proved more instances of such results and conjectured others. These new results and conjectures involved other topological invariants (relatives of scl) and different families of groups. In the current paper we further extend and support this theory. We provide another instance of the theory and prove that the stable primitivity rank, too, can be expressed in terms of stable Fourier coefficients of $w$-random elements of groups. We introduce concrete formulas for stable Fourier coefficients of $w$-random elements in the symmetric group $S_N$ and its generalizations in the form of the wreath products $G\wr S_N$ where $G$ is an arbitrary compact group. We also define new stable invariants related to these groups, and prove they give bounds to many of the stable Fourier coefficients. As an aside, we generalize to tuples of words a result of Puder and Parzanchevski [arXiv:1202.3269] about the expected number of fixed points of $w$-random permutations.

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↗

arXiv2023

Word Measures On Wreath Products I

Every word $w$ in the free group $F_r$ of rank $r$ induces a probability measure (the $w$-measure) on every compact group $G$, by substitution of Haar-random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $\mathbb{E}_w[χ]$ of the irreducible characters of $G$. For every compact group $G$, the wreath product with the symmetric group $G\wr S_n$ has some natural irreducible characters $χ$, and we approximate $\mathbb{E}_w[χ]$ for every word $w\in F_r$, revealing new automorphism-invariant quantities of words that generalize the primitivity rank $π(w)$. This generalizes previous works by Parzanchevsky-Puder and Magee-Puder. We demonstrate applications to automorphism groups of trees, investigate properties of the new invariants, and show polynomial decay of $\mathbb{E}_w[χ]$ also for wreath products with more general actions.

math.GR↗