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arxiv: 2606.03086 · v1 · pith:HGV3C3LNnew · submitted 2026-06-02 · 🧮 math.RT

On higher extensions of quiver representations over mathbb{F}₁

Changjian Fu , Liang Yang , Zhiyuan Zeng This is my paper

Pith reviewed 2026-06-28 08:24 UTC · model grok-4.3

classification 🧮 math.RT
keywords quiver representationsfield with one elementhigher Ext groupscyclic quiversnilpotent representationsinfinite-dimensional extensionsF1-representations
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The pith

Higher extension spaces between finite-dimensional nilpotent F1-quiver representations can be infinite-dimensional.

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

The paper shows that the spaces of higher extensions between finite-dimensional nilpotent representations of quivers over the field with one element need not be finite-dimensional. This corrects an earlier view in the literature that such spaces would remain finite. The counterexamples are constructed using representations of cyclic quivers. For any cyclic quiver, the third extension group vanishes for every pair of finite-dimensional nilpotent representations, while the second extension group between any two simple representations is infinite-dimensional. These facts follow from direct calculations in the category of nilpotent representations over F1.

Core claim

For a cyclic quiver Delta_n, the space Ext^3(M, N) is zero for any pair of finite-dimensional nilpotent F1-representations M and N, while Ext^2(S, T) is infinite-dimensional for any pair of simple representations S and T. This establishes that infinite-dimensionality arises in higher extensions even when the objects themselves are finite-dimensional and nilpotent.

What carries the argument

The higher extension groups Ext^i (i >= 2) computed via projective resolutions or Yoneda classes in the category of finite-dimensional nilpotent F1-representations of cyclic quivers.

If this is right

  • Ext^3 vanishes between every pair of finite-dimensional nilpotent representations of a cyclic quiver over F1.
  • Ext^2 is infinite-dimensional between every pair of simple representations of a cyclic quiver over F1.
  • The finite-dimensional nilpotent category over F1 on cyclic quivers exhibits infinite-dimensional homological features in degree 2.

Where Pith is reading between the lines

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

  • The degree-specific behavior (vanishing at 3, infinity at 2) may require separate analysis for each extension degree rather than a uniform finiteness assumption.
  • The same combinatorial constructions could be tested on other quiver shapes to check whether infinite-dimensional higher extensions appear more broadly.
  • This distinction might affect how one builds derived invariants or Hall algebras in the F1 setting for cyclic cases.

Load-bearing premise

The usual definitions of higher extension groups for F1-representations of quivers apply directly to the finite-dimensional nilpotent case and permit the resulting spaces to be infinite-dimensional.

What would settle it

An explicit basis computation showing that Ext^2 between two distinct simple representations of the three-vertex cyclic quiver has finite dimension would falsify the infinite-dimensionality result.

read the original abstract

We show that higher extension spaces between finite-dimensional nilpotent $\mathbb{F}_1$-representations maybe infinite-dimensional, thereby clarifying a misconception in the literature. Our examples arise from cyclic quivers. In particular, for a cyclic quiver $\Delta_n$, we show that $\operatorname{Ext}^3(-,-)$ vanishes for any pair of finite-dimensional nilpotent $\mathbb{F}_1$-representations of $\Delta_n$, while $\operatorname{Ext}^2(-,-)$ is infinite-dimensional for any pair of simple representations.

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 paper claims that higher extension spaces between finite-dimensional nilpotent F1-representations of quivers may be infinite-dimensional, contrary to a misconception in the literature. For cyclic quivers Δ_n, it asserts that Ext^3 vanishes for any pair of finite-dimensional nilpotent F1-representations, while Ext^2 is infinite-dimensional for any pair of simple representations.

Significance. If the results hold, this provides concrete examples clarifying that Ext groups over F1 need not be finite-dimensional even when the underlying representations are finite-dimensional and nilpotent. The explicit vanishing result for Ext^3 and the infinite-dimensionality for Ext^2 on cyclic quivers offer useful homological information specific to the F1 setting and may inform related work in algebraic combinatorics or K-theory.

minor comments (2)
  1. The abstract states the main results clearly but does not identify the specific prior misconception or references; expanding this in the introduction would improve context.
  2. Definitions of the F1-Ext groups (via projective resolutions or Yoneda) and the precise category of nilpotent representations should be recalled explicitly in §2 or §3 to ensure the constructions yielding infinite cardinality are fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity; explicit constructions from standard definitions

full rationale

The paper establishes its claims via direct application of the standard Ext functor (projective resolutions or Yoneda) to the category of F1-quiver representations, which by definition consists of pointed sets with linear actions and permits infinite-cardinality Hom/Ext spaces. For cyclic quivers the vanishing of Ext^3 and infinitude of Ext^2 are obtained by explicit computation on nilpotent and simple objects; no parameter fitting, self-definitional loops, or load-bearing self-citations appear in the derivation chain. The central results are therefore independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions of quiver representations over F1 and the associated Ext functors; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • standard math Standard definitions of finite-dimensional nilpotent representations of quivers over F1 and the associated Ext groups
    Invoked implicitly as the setting in which the statements about Ext^2 and Ext^3 are made.

pith-pipeline@v0.9.1-grok · 5607 in / 1135 out tokens · 23399 ms · 2026-06-28T08:24:50.102547+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 4 canonical work pages

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