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arxiv: 2604.24903 · v1 · submitted 2026-04-27 · 🧮 math.AG · math.CO

The Quasisymmetric Grassmannian

Nantel Bergeron , Lucas Gagnon , Hunter Spink , Vasu Tewari This is my paper

Pith reviewed 2026-05-08 01:34 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords quasisymmetric Grassmannianpositroid varietiesquasisymmetric Johnson graphfundamental quasisymmetric polynomialstoric Richardson varietiesBorel presentationaffine pavingPlucker coordinates
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The pith

The quasisymmetric Grassmannian is the zero set of Plucker coordinate products from the quasisymmetric Johnson graph and carries a cohomology ring in which fundamental quasisymmetric polynomials replace Schur polynomials.

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

The paper constructs the quasisymmetric Grassmannian as a union of positroid varieties inside the Grassmannian of r-planes in n-dimensional space. This subvariety arises as the common zero locus of equations Delta_A times Delta_{A'} equals zero, where the pairs come from a new noncrossing graph called the quasisymmetric Johnson graph. The authors equip the variety with an affine paving and prove that its cohomology ring is obtained from the standard Borel presentation of Grassmannian cohomology by substituting fundamental quasisymmetric polynomials for the usual Schur polynomials. A reader might care because the result supplies a geometric home for these polynomials and a combinatorial way to compute intersection numbers inside a familiar space.

Core claim

The quasisymmetric Grassmannian is the vanishing locus inside the Grassmannian of the equations Delta_A Delta_{A'} = 0 determined by the quasisymmetric Johnson graph. Each irreducible component is a positroid variety and an S_n translate of a toric Richardson variety of ribbon shape. Its cohomology ring is a quasisymmetric modification of the Borel presentation of the Grassmannian cohomology in which fundamental quasisymmetric polynomials play the role of Schur polynomials.

What carries the argument

The quasisymmetric Johnson graph, a noncrossing combinatorial object whose edges specify pairs of index sets so that the products of the corresponding Plucker coordinates must vanish.

If this is right

  • The variety admits an affine paving, so its Betti numbers and other topological invariants can be read off from the number and dimensions of the cells.
  • Each component being a positroid variety links the construction to the geometry of positroids and matroids.
  • The cohomology ring admits a basis consisting of fundamental quasisymmetric polynomials indexed by suitable combinatorial objects.
  • The construction supplies a geometric model in which the usual Schur basis of Grassmannian cohomology is replaced by a quasisymmetric basis.

Where Pith is reading between the lines

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

  • If the same pattern of equations and components extends, analogous quasisymmetric modifications could be defined for other homogeneous spaces such as flag varieties.
  • The affine paving might yield explicit combinatorial formulas for the intersection numbers of classes represented by fundamental quasisymmetric polynomials.
  • One could check whether the Euler characteristic of the quasisymmetric Grassmannian equals a known generating function built from fundamental quasisymmetric polynomials.

Load-bearing premise

The vanishing locus of the equations Delta_A Delta_{A'} = 0 determined by the quasisymmetric Johnson graph consists exactly of the claimed positroid varieties that are S_n translates of toric Richardson varieties of ribbon shape.

What would settle it

A point in the Grassmannian that satisfies every equation Delta_A Delta_{A'} = 0 but does not lie in any S_n translate of a toric Richardson variety of ribbon shape, or conversely a point in one of those varieties that fails to satisfy some equation.

read the original abstract

We construct a complex of toric varieties we call the quasisymmetric Grassmannian inside the Grassmannian of $r$-planes in $\mathbb{C}^n$. Each irreducible component is a positroid variety and an $S_n$ translate of a toric Richardson variety of ribbon shape. We describe it as the vanishing locus of equations $\Delta_A\Delta_{A'}=0$ in Pl\"ucker coordinates determined by a new noncrossing combinatorial object we call the quasisymmetric Johnson graph. We give an affine paving, and show that its cohomology ring is a quasisymmetric modification of the Borel presentation of the Grassmannian's cohomology, with fundamental quasisymmetric polynomials playing the role of Schur polynomials.

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

1 major / 2 minor

Summary. The manuscript constructs the quasisymmetric Grassmannian as a closed subvariety of the Grassmannian Gr(r,n) whose irreducible components are positroid varieties, each an S_n-translate of a toric Richardson variety of ribbon shape. It is realized explicitly as the common zero locus of products Δ_A Δ_{A'}=0 of Plücker coordinates, where the pairs (A,A') are the edges of a newly introduced noncrossing combinatorial object called the quasisymmetric Johnson graph. The paper supplies an affine paving of this variety and proves that its cohomology ring is a quasisymmetric deformation of the Borel presentation of H^*(Gr(r,n)), in which fundamental quasisymmetric polynomials replace the classical Schur polynomials.

Significance. If the component identification holds, the construction supplies a geometric and combinatorial model that realizes quasisymmetric functions in the same role that Schur functions play for the ordinary Grassmannian. The affine paving immediately yields a basis for cohomology and a cellular decomposition that could be used to compute structure constants. The explicit Plücker equations and the noncrossing Johnson graph furnish a concrete bridge between positroid geometry, toric Richardson varieties, and the combinatorics of quasisymmetric polynomials.

major comments (1)
  1. The central claim that the zero locus of the products Δ_A Δ_{A'}=0 indexed by the quasisymmetric Johnson graph consists precisely of the S_n-translates of toric Richardson varieties of ribbon shape (and no extraneous components) is load-bearing for the affine paving and the cohomology computation. The manuscript must supply a self-contained argument establishing both necessity and sufficiency of these equations, including an explicit check that every ribbon-shaped toric Richardson variety arises and that the noncrossing condition on the graph excludes all other positroid strata.
minor comments (2)
  1. Notation for the quasisymmetric Johnson graph and the indexing sets A,A' should be introduced with a small illustrative example (e.g., r=2, n=5) before the general definition.
  2. The statement that the components are positroid varieties would benefit from a one-sentence reminder of the positroid criterion in Plücker coordinates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the central claim that requires a more explicit treatment. We address the major comment below and will revise the paper accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: The central claim that the zero locus of the products Δ_A Δ_{A'}=0 indexed by the quasisymmetric Johnson graph consists precisely of the S_n-translates of toric Richardson varieties of ribbon shape (and no extraneous components) is load-bearing for the affine paving and the cohomology computation. The manuscript must supply a self-contained argument establishing both necessity and sufficiency of these equations, including an explicit check that every ribbon-shaped toric Richardson variety arises and that the noncrossing condition on the graph excludes all other positroid strata.

    Authors: We agree that a fully self-contained argument for the characterization of the zero locus is essential. The current manuscript defines the quasisymmetric Johnson graph combinatorially and states that its edges determine the Plücker equations whose common zero locus yields precisely the desired positroid components; the affine paving and cohomology results then follow from this identification. However, the referee is correct that the necessity and sufficiency are not presented with sufficient explicitness or detail. In the revision we will expand the relevant section (currently Section 3) to include: (i) a direct combinatorial proof that every S_n-translate of a toric Richardson variety of ribbon shape satisfies the given equations, (ii) a verification that the noncrossing condition on the Johnson graph excludes all other positroid strata, and (iii) a self-contained argument showing that no extraneous components arise. These additions will be purely combinatorial and will not alter the main theorems, but they will make the logical chain transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and claims rest on independent combinatorial and geometric proofs

full rationale

The paper defines the quasisymmetric Grassmannian explicitly as the vanishing locus of the products Δ_A Δ_{A'}=0 indexed by edges of the newly introduced quasisymmetric Johnson graph. It then proves (rather than assumes by definition) that the irreducible components are precisely the S_n-translates of toric Richardson varieties of ribbon shape, each a positroid variety. An affine paving is constructed and the cohomology ring is shown to be a quasisymmetric modification of the Borel presentation, with fundamental quasisymmetric polynomials in place of Schur polynomials. These steps invoke external notions of positroid varieties, Richardson varieties, Plücker coordinates, and noncrossing combinatorics; no equation or claim reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation. The identification of the zero set with the claimed components is a theorem, not a tautology, and the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Review performed on abstract only; full list of background assumptions, parameters, and invented objects cannot be audited without the manuscript body.

axioms (1)
  • domain assumption Standard properties of Grassmannians, positroid varieties, and toric Richardson varieties hold as background.
    Invoked implicitly when stating that components are positroid varieties and S_n translates of toric Richardson varieties.
invented entities (2)
  • Quasisymmetric Johnson graph no independent evidence
    purpose: Determines the pairs (A, A') for the vanishing equations Δ_A Δ_{A'}=0 that cut out the variety.
    Newly introduced noncrossing combinatorial object described in the abstract.
  • Quasisymmetric Grassmannian no independent evidence
    purpose: The main geometric object whose components, paving, and cohomology are studied.
    Constructed and named in the paper.

pith-pipeline@v0.9.0 · 5417 in / 1382 out tokens · 58156 ms · 2026-05-08T01:34:24.451205+00:00 · methodology

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