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arxiv: 2510.14971 · v3 · submitted 2025-10-16 · 🧮 math.GR · math.RT· quant-ph

On the invariants of finite groups arising in a topological quantum field theory

Christopher A. Schroeder , Hung P. Tong-Viet This is my paper

Pith reviewed 2026-05-18 06:15 UTC · model grok-4.3

classification 🧮 math.GR math.RTquant-ph
keywords finite groupsDijkgraaf-Witten invariantscommuting probabilitycharacter degreesnilpotencysolvabilitysupersolvabilitytopological quantum field theory
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The pith

Dijkgraaf-Witten invariants in 1+1 dimensions detect commutativity, nilpotency, supersolvability and solvability of finite groups.

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

This paper studies numerical invariants that finite groups acquire when a topological quantum field theory is applied to closed orientable surfaces in one space and one time dimension. The invariants are built directly from the degrees of the group's complex irreducible characters and generalize the classical commuting probability, the chance that two randomly chosen elements commute. By examining the versions of these invariants that correspond to surfaces of higher genus, the authors obtain explicit numerical thresholds and bounds that distinguish groups with the listed structural properties. A sympathetic reader cares because the work supplies computable, character-table-based tests for deep algebraic features that previously required direct examination of the group's multiplication table or subgroup lattice. If the claims hold, group theorists gain a new family of quantitative tools that link representation-theoretic data to solvability-type questions.

Core claim

By analyzing these higher-genus analogues, we establish new quantitative criteria relating the values of these invariants to key structural features of finite groups, such as commutativity, nilpotency, supersolvability and solvability. These results generalize several classical theorems concerning the commuting probability, thereby linking ideas from finite group theory and topological quantum field theory.

What carries the argument

The family of numerical invariants associated with closed orientable surfaces of various genera, each expressed in terms of the degrees of the complex irreducible characters of the finite group G; the invariants extend the commuting probability and serve as detectors for the listed structural properties.

Load-bearing premise

The Dijkgraaf-Witten invariants in 1+1 dimensions are completely determined by the degrees of the complex irreducible characters of G and that these numerical values directly encode the listed structural properties without additional assumptions on the group or the TQFT construction.

What would settle it

Compute the character degrees of a concrete finite group known to be non-nilpotent (or non-solvable) and check whether its higher-genus invariants violate the numerical bounds or thresholds that the paper claims hold only for nilpotent (or solvable) groups.

Figures

Figures reproduced from arXiv: 2510.14971 by Christopher A. Schroeder, Hung P. Tong-Viet.

Figure 1
Figure 1. Figure 1: Cobordisms for calculating (a) the “cup”, (b) the “cap” and (c) the comultiplication. Decomposing the genus h-surface in (d) into cobordisms calculated in [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
read the original abstract

In this paper, we investigate structural properties of finite groups that are detected by certain group invariants arising from Dijkgraaf--Witten theory, a topological quantum field theory, in one space and one time dimension. In this setting, each finite group $G$ determines a family of numerical invariants associated with closed orientable surfaces, expressed in terms of the degrees of the complex irreducible characters of $G$. These invariants can be viewed as natural extensions of the commuting probability $d(G)$, which measures the probability that two randomly chosen elements of $G$ commute and has been extensively studied in the literature. By analyzing these higher-genus analogues, we establish new quantitative criteria relating the values of these invariants to key structural features of finite groups, such as commutativity, nilpotency, supersolvability and solvability. Our results generalize several classical theorems concerning the commuting probability, thereby linking ideas from finite group theory and topological quantum field theory.

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 higher-genus invariants of finite groups G arising from the untwisted Dijkgraaf-Witten TQFT in 1+1 dimensions. These are expressed as normalized power sums over the degrees of the irreducible complex characters, generalizing the commuting probability d(G) via Z(Σ_g) proportional to sum (deg χ)^{2-2g}. The authors derive quantitative criteria that relate the values of these invariants to the group being commutative, nilpotent, supersolvable, or solvable, using classical bounds on character degrees and composition factors.

Significance. If the derivations hold, this establishes a direct link between TQFT partition functions and structural properties of finite groups through standard representation theory. The explicit construction from character degrees and the generalization of known results on d(G) are strengths, as is the absence of hidden parameters or twisting assumptions. The work may interest researchers at the interface of group theory and topological invariants.

minor comments (2)
  1. [§2] §2: the precise normalization factor relating the raw sum ∑ (deg χ)^{2-2g} to the invariant Z(Σ_g) should be stated explicitly, including any division by |G| or by the number of irreps, to facilitate direct comparison with d(G).
  2. The statement of the criterion for supersolvability would benefit from a short reminder of the relevant bound on character degrees or composition factors used in the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report, including the supportive summary, significance assessment, and recommendation for minor revision. We are pleased that the connection between the higher-genus Dijkgraaf-Witten invariants and classical group-theoretic properties is viewed as a strength. No specific major comments were listed in the report, so we have no point-by-point rebuttals to provide at this stage. We will incorporate any minor suggestions that may arise during the revision process.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the higher-genus invariants directly as normalized power sums over the multiset of irreducible character degrees via the standard untwisted 1+1D Dijkgraaf-Witten partition functions, then derives quantitative criteria for commutativity, nilpotency, supersolvability and solvability by applying classical bounds on character degrees and composition factors. These steps rest on independent results from finite group representation theory rather than any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations; the central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard representation theory of finite groups and the definition of Dijkgraaf-Witten TQFT; no free parameters, ad-hoc axioms, or new entities are indicated in the abstract.

axioms (1)
  • standard math Every finite group possesses a finite set of irreducible complex characters whose degrees determine the surface invariants.
    Standard fact from representation theory of finite groups invoked to express the invariants.

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Forward citations

Cited by 1 Pith paper

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  1. Higher Commutativity in Finite Groups, Rigidity, Extremal bounds, and Heisenberg-Type Families

    math.GR 2026-05 unverdicted novelty 7.0

    Establishes closed all-rank formulas for P_r(G) in cyclic-index groups and F_q-Heisenberg families, plus rigidity and isoclinism determination from low-rank values.

Reference graph

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