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arxiv: 2511.17723 · v4 · pith:WSCJD6IFnew · submitted 2025-11-21 · 🧮 math.CO · math.AG

Three formulas for CSM classes of open quiver loci

Moriah Elkin This is my paper

Pith reviewed 2026-05-25 07:07 UTC · model grok-4.3

classification 🧮 math.CO math.AG
keywords CSM classesopen quiver lociquiver polynomialspipe dreamslacing diagramsequivariant cohomologytype A quiversrank conditions
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The pith

Equivariant CSM classes of open quiver loci in type A representations are computed by one geometric formula and two combinatorial formulas that refine the Buch-Fulton quiver polynomials.

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

The paper works in the space of equioriented type A quiver representations and defines open quiver loci by imposing strict rank conditions on the maps between vector spaces. These open loci have closures that are the usual quiver loci, whose classes in equivariant cohomology are the known quiver polynomials. The central contribution is three explicit formulas for the equivariant Chern-Schwartz-MacPherson classes of the open loci themselves; these classes carry strictly more information than the polynomials of the closures. One formula is geometric while the other two are combinatorial, and the second combinatorial formula is expressed using chained generic pipe dreams obtained by adapting ordinary pipe dreams so that they match the structure of lacing diagrams.

Core claim

Open quiver loci are the subvarieties cut out by strict rank conditions inside the representation space of an equioriented type A quiver. Their closures are the quiver loci whose equivariant cohomology classes are the Buch-Fulton quiver polynomials. The paper supplies one geometric formula and two combinatorial formulas for the equivariant CSM classes of the open loci; the second combinatorial formula counts chained generic pipe dreams, which are a modification of Bergeron-Billey pipe dreams designed to resemble Abeasis-Del Fra lacing diagrams.

What carries the argument

Chained generic pipe dreams, a modification of Bergeron-Billey pipe dreams that more closely resemble Abeasis-Del Fra lacing diagrams, used to give a combinatorial count of the CSM classes.

If this is right

  • The CSM classes refine the quiver polynomials by recording data on the open loci rather than only on their closures.
  • The geometric formula expresses the CSM class directly in terms of the geometry of the open loci inside the representation space.
  • The combinatorial formulas, especially the one using chained generic pipe dreams, give an explicit counting rule for the coefficients.
  • These classes are defined for any equioriented type A quiver and any choice of strict rank conditions.

Where Pith is reading between the lines

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

  • The pipe-dream model may extend to give combinatorial rules for CSM classes of other degeneracy loci that admit lacing-diagram descriptions.
  • The refinement property suggests that the CSM classes could detect finer positivity or sign patterns not visible in the ordinary quiver polynomials.
  • Direct comparison of the three formulas on small examples could produce new identities between geometric and pipe-dream expressions.

Load-bearing premise

The closures of the open quiver loci defined by strict rank conditions are exactly the quiver loci whose classes are the Buch-Fulton quiver polynomials.

What would settle it

An explicit low-rank quiver example in which the CSM class produced by any of the three formulas differs from the class obtained by direct localization or resolution computation.

read the original abstract

In the space of equioriented type $A$ quiver representations, we define subvarieties called "open quiver loci" by placing strict rank conditions on the maps within representations. The closures of these subvarieties are the quiver loci, whose equivariant cohomology classes are the quiver polynomials of Buch and Fulton. We present one geometric formula and two combinatorial formulas that compute equivariant Chern-Schwartz-MacPherson (CSM) classes of open quiver loci; these classes refine the data of the quiver polynomials. The second combinatorial formula is in terms of "chained generic pipe dreams," which modify the pipe dreams of Bergeron and Billey to more strongly resemble the lacing diagrams of Abeasis and Del Fra.

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 / 3 minor

Summary. The paper defines open quiver loci in equioriented type A quiver representation spaces via strict rank conditions on the maps. Their closures recover the standard quiver loci whose equivariant cohomology classes are the Buch-Fulton quiver polynomials. The central claim is that one geometric formula and two combinatorial formulas (one via chained generic pipe dreams obtained by modifying Bergeron-Billey pipe dreams to align with Abeasis-Del Fra lacing diagrams) compute the equivariant CSM classes of these open loci, and that these classes refine the quiver polynomials.

Significance. If the formulas hold, the work supplies explicit, computable expressions for refined equivariant invariants of open quiver loci, extending the Buch-Fulton theory. The combinatorial formulas, especially the chained generic pipe dreams, furnish a new bridge between pipe-dream combinatorics and lacing diagrams, which may enable further explicit calculations and positivity results in this setting.

minor comments (3)
  1. [Geometric formula section] The geometric formula is stated in terms of the ambient representation space; a brief derivation sketch or reference to the precise localization or push-forward used would clarify how it directly yields the CSM class of the open locus.
  2. [Combinatorial formulas] The definition of chained generic pipe dreams modifies Bergeron-Billey pipes; an explicit small example (e.g., for a 2-step quiver) comparing the new objects to both standard pipe dreams and lacing diagrams would make the modification easier to verify.
  3. [Introduction / Definitions] Notation for the strict rank conditions and the resulting open loci should be introduced with a running example quiver to avoid ambiguity when the formulas are applied.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on equivariant CSM classes of open quiver loci and for recommending minor revision. The referee's summary and significance statement accurately reflect the paper's contributions. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines open quiver loci via strict rank conditions whose closures recover the standard Buch-Fulton quiver loci by the usual topology of rank varieties; this identification is definitional and external to the new formulas. The geometric formula is obtained directly from the definition of equivariant CSM classes in the ambient space, while the combinatorial formulas (including chained generic pipe dreams) are constructed as explicit representatives that refine the known quiver polynomials without the refinement being imposed by fitting or self-definition. No load-bearing step reduces the claimed CSM classes to the input polynomials by construction, and no self-citation chain or ansatz smuggling is required for the central results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; full manuscript required for a complete ledger.

pith-pipeline@v0.9.0 · 5637 in / 1092 out tokens · 47131 ms · 2026-05-25T07:07:05.009176+00:00 · methodology

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Reference graph

Works this paper leans on

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