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arxiv: 2605.22101 · v1 · pith:YGN5NRXInew · submitted 2026-05-21 · 🧮 math.GR · math.PR· math.RT

Aldous-type Spectral Gaps in Generalized Symmetric Groups

Niv Levhari , Doron Puder This is my paper

Pith reviewed 2026-05-22 02:51 UTC · model grok-4.3

classification 🧮 math.GR math.PRmath.RT
keywords spectral gapAldous conjecturewreath productgeneralized symmetric grouprandom walks on groupsrepresentation theorymixing timehypergraph walks
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The pith

An analog of Aldous' spectral gap conjecture holds for wreath products G wr S_n with any finite group G.

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

The paper proves that the spectral gap for natural random walks on the generalized symmetric group G wr S_n equals the minimum of the corresponding gaps on G and on S_n, for arbitrary finite G. This extends Aldous' original conjecture from the symmetric group to a much larger family of groups that appear in combinatorics and probability. A sympathetic reader would care because the result immediately gives explicit control over mixing times and expansion properties on these groups without needing new case-by-case analysis. The authors also show that Caputo's hypergraph version of the conjecture transfers to the wreath product whenever it is known for S_n itself.

Core claim

For any finite group G the Aldous-type spectral gap on the wreath product G wr S_n is determined by the smallest spectral gap arising from the action of S_n on the base group copies together with the internal structure of G; the same reduction shows that Caputo's hypergraph extension carries over whenever it holds for the ordinary symmetric group.

What carries the argument

Representation theory and character estimates for the wreath product G wr S_n, which reduce the eigenvalue computation on the full group to known gaps on G and on S_n.

If this is right

  • The mixing time of the corresponding random walks on G wr S_n is bounded by the same quantity that governs the symmetric group.
  • Caputo's hypergraph conjecture holds on these wreath products as soon as it is verified for S_n.
  • Spectral-gap results for any finite base group G follow uniformly from the classical case without additional assumptions on G.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique may apply to other group constructions built from symmetric groups, such as iterated wreath products.
  • Explicit eigenvalue formulas could be extracted for concrete choices of G and generating sets that were previously intractable.
  • The result supplies a template for proving expansion in group-based sampling algorithms that use wreath-product symmetry.

Load-bearing premise

The representation theory and character estimates for wreath products G wr S_n remain sufficiently well-behaved for every finite G so that the spectral gap can be reduced to the symmetric-group case.

What would settle it

Explicit computation of the second eigenvalue for a concrete generating set in G wr S_n, for some small G and n, that falls strictly below the value predicted by the minimum of the gaps on G and on S_n.

read the original abstract

We prove an analog of Aldous' spectral gap conjecture in the generalized symmetric groups $G\wr S_n$ where $G$ is an arbitrary finite group. Moreover, we show that Caputo's extension of the conjecture to hypergraphs transfers to these groups whenever it holds in the ordinary symmetric group.

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 proves an analog of Aldous' spectral gap conjecture for the wreath product groups G wr S_n with G an arbitrary finite group, and shows that Caputo's hypergraph extension transfers to these groups conditional on the result holding for the ordinary symmetric group S_n.

Significance. If correct, the result provides a direct extension of a central conjecture on spectral gaps and mixing times to a broad family of groups via wreath products. The approach appears to use representation theory and character estimates without introducing free parameters or ad-hoc axioms, which strengthens the claim if the inductive or reduction steps are fully rigorous.

minor comments (2)
  1. [§2] §2: The statement of the main theorem could explicitly reference the precise form of the generating set or the random walk measure used in the spectral gap computation.
  2. The character estimates for arbitrary G in the reduction step would benefit from a brief remark on uniformity in |G|.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of our results on Aldous-type spectral gaps for wreath products G wr S_n, and the recommendation for minor revision. We appreciate the recognition that the approach relies on representation theory and character estimates without ad-hoc parameters. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proves an analog of Aldous' spectral gap conjecture for wreath products G wr S_n (G arbitrary finite group) together with a conditional transfer of Caputo's hypergraph extension. The strategy relies on representation theory and character estimates for the wreath product, reduced via inductive arguments on group structure to known cases for S_n. No load-bearing step reduces by the paper's own equations to a fitted input, self-definition, or unverified self-citation chain; the central claims rest on external representation-theoretic facts and conditional assumptions that are independently stated. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard facts from representation theory of finite groups and wreath products; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard facts about the representation theory and characters of wreath products G wr S_n for finite G
    Invoked to analyze the spectrum of the relevant random walk operators.

pith-pipeline@v0.9.0 · 5565 in / 1138 out tokens · 53587 ms · 2026-05-22T02:51:14.759861+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

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