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arxiv: 2311.17733 · v4 · submitted 2023-11-29 · 🧮 math.GR · math.GT· math.PR· math.RT

Stable Invariants of Words from Random Matrices

Doron Puder , Yotam Shomroni , Danielle Ernst-West , Matan Seidel This is my paper

Pith reviewed 2026-05-24 05:27 UTC · model grok-4.3

classification 🧮 math.GR math.GTmath.PRmath.RT
keywords free group wordsstable commutator lengthrandom matricespermutation matricesorthogonal matricesstable primitivity rankFourier coefficientsword invariants
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The pith

Tweaking random matrix ensembles from unitaries to permutations and orthogonals produces new stable invariants of words that coincide with stable commutator length and stable primitivity rank in cases involving generalized symmetric groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the known link between stable commutator length and Fourier coefficients of random unitary matrices extends when the underlying ensemble is changed to permutations, orthogonal matrices or generalized symmetric groups. These changes define new invariants for any given word in a free group. In the cases built from generalized symmetric groups the new invariants equal the stable commutator length and the stable primitivity rank. A sympathetic reader cares because the construction supplies multiple algebraic routes to the same topological data and raises the possibility that many such invariants are actually independent of the chosen ensemble.

Core claim

Tweaking the random-matrix model used to define stable commutator length via unitaries to other ensembles such as the symmetric group, the orthogonal group and generalized symmetric groups produces new invariants of words. For generalized symmetric groups these invariants coincide with stable commutator length and with the stable primitivity rank. The paper defines these invariants via limits of Fourier coefficients and proves the coincidences while posing conjectures about their topological nature in other cases.

What carries the argument

The stable limit of Fourier-type coefficients extracted from the distribution of a word evaluated on random elements drawn from the tweaked matrix ensembles.

Load-bearing premise

The coefficients extracted from different random matrix ensembles are independent of the ensemble and possess intrinsic topological or combinatorial descriptions.

What would settle it

For a concrete word and a small generalized symmetric group, compute the stable limit of the coefficient from that ensemble and compare it to the known value of stable commutator length; systematic mismatch would show the claimed coincidence fails.

Figures

Figures reproduced from arXiv: 2311.17733 by Danielle Ernst-West, Doron Puder, Matan Seidel, Yotam Shomroni.

Figure 1.1
Figure 1.1. Figure 1.1: Known inequalities between some of the stable invariants discussed in this paper. In every [PITH_FULL_IMAGE:figures/full_fig_p010_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: On the left is a morphism from P, a union of two cycles, to ∆, a barbell-shaped graph. The edges of ∆ are oriented and labeled in order to depict the morphism: the image of every edge of P is marked using these labels and orientations. The two Whitehead graph of this morphism, one for each vertex of ∆, are drawn on the right. In each of the two Whitehead graphs, each vertex is labeled by the correspondin… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Folding and unfolding: consider an immersion of finite graphs [PITH_FULL_IMAGE:figures/full_fig_p016_2_2.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: A degree-2 ssql-extremal surface for w = a 2 b 2ab−1 , showing that ssql (w) = 1. |w| 2 · P v∈V (Γ)(deg(v)−2) P v∈V (Γ) deg(v) is a weighted average of the numbers deg(v)−2 deg(v) , each of which is at least 1 2 , a value obtained only when the degree is 4. As there exist 4-regular diagrams, where each of the nine possible 4-cycles appears the same number of times, we conclude the the infimum is |w| 2 · … view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Consider w = a 3 ba−1 b −1 . The upper left graph is Γw, and the bottom left one is (the decorated) Whηw (o) (the black oriented edge labeled by a or b at every vertex only marks the corre￾sponding half-edge of Ω, and is not a genuine part of the Whitehead graph). The other four graphs depict the four types of pieces which are valid for this word when m = 3. Here we restrict to pieces representing connec… view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: A homotopy-equivalent folding step cj : Σj−1 → Σj In this algebraic-free decomposition Σ is a core graph, bfree is an immersion (hence so is f ◦ bfree), and the efficiency of balg follows immediately from that of b. It remains to show that the condition about nb(e) is satisfied in (6.4) and that −χ(Σ) ≤ −χ(Γ). The latter follows from [HP23, Lem. 5.3]. By [HP23, §5], bfree : Σ → Γ can be obtained by a seq… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript defines stable invariants of words in free groups via limits of Fourier-type coefficients extracted from w-random matrices in tweaked ensembles (symmetric group, orthogonal group, generalized symmetric groups). It rigorously verifies that these invariants coincide with scl and Wilton's stable primitivity rank in specific cases for generalized symmetric groups, while posing detailed conjectures for intrinsic topological or combinatorial descriptions and model-independence in other cases.

Significance. If the conjectures hold, the work supplies a unified random-matrix framework recovering scl (as in the prior unitary case) and additional invariants including stable primitivity rank. The explicit case-by-case verifications and the decision to label remaining claims as conjectures rather than asserting intrinsic descriptions are strengths; the approach yields concrete, falsifiable predictions for further ensembles.

minor comments (2)
  1. Notation for the Fourier coefficients and the precise normalization in the limit definitions could be collected in a single preliminary section for easier reference across the case analyses.
  2. A short table summarizing which invariants are proven to match scl or stable primitivity rank, and in which ensembles, would improve readability of the results section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for highlighting its strengths in providing explicit verifications and in appropriately labeling remaining claims as conjectures, and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; definitions and verifications are independent

full rationale

The paper introduces new invariants as limits of Fourier-type coefficients extracted from tweaked random-matrix ensembles (permutations, orthogonals, generalized symmetric groups) and rigorously establishes coincidence with scl and Wilton's stable primitivity rank only in explicitly delimited cases for generalized symmetric groups. Outside those cases it states conjectures rather than asserting model-independence. No load-bearing step reduces by the paper's own equations to a fitted parameter, self-citation chain, or definitional tautology; the central objects are newly formalized and the verified equalities rest on direct computation within the chosen ensembles.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms or invented entities are identifiable from the given text.

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