The Purity Conjecture in Type $C$

Rachel Karpman

arxiv: 1907.08275 · v1 · pith:TNJ3KPAWnew · submitted 2019-07-18 · 🧮 math.CO

The Purity Conjecture in Type C

Rachel Karpman This is my paper

Pith reviewed 2026-05-24 19:37 UTC · model grok-4.3

classification 🧮 math.CO

keywords weakly separated collectionsplabic graphsLagrangian GrassmannianPurity Conjecturesymmetric collectionstype Cpositroidscombinatorial coordinates

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The pith

A symmetric version of the Purity Conjecture holds for weakly separated collections corresponding to symmetric plabic graphs on the Lagrangian Grassmannian.

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

The paper constructs maximal weakly separated collections that are symmetric under an involution, linking them directly to symmetric plabic graphs. These graphs serve as coordinate charts specifically for the Lagrangian Grassmannian rather than the ordinary Grassmannian. It then proves that such a collection is maximal by inclusion exactly when it reaches the maximum possible size. This extends an earlier result of Oh, Postnikov and Speyer to the type C setting. Readers interested in combinatorial models for algebraic varieties would find the extension relevant because it supplies explicit charts and a size criterion for maximality in a new geometric context.

Core claim

Oh, Postnikov and Speyer constructed a correspondence between maximal weakly separated collections and reduced plabic graphs for the Grassmannian and proved Scott's Purity Conjecture as a corollary. In this note, maximal weakly separated collections corresponding to symmetric plabic graphs are described; these give coordinate charts on the Lagrangian Grassmannian, and a symmetric version of the Purity Conjecture is proved.

What carries the argument

Symmetric plabic graphs, which are plabic graphs invariant under the involution that defines the Lagrangian Grassmannian and correspond to maximal symmetric weakly separated collections.

If this is right

Maximal symmetric weakly separated collections give coordinate charts on the Lagrangian Grassmannian via symmetric plabic graphs.
A symmetric weakly separated collection is maximal by inclusion if and only if it is maximal by size.
The size of a maximal symmetric collection is determined by the same formula as in the non-symmetric case adjusted for symmetry.
Symmetric positroids or related objects admit combinatorial parametrizations through these collections.

Where Pith is reading between the lines

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

The result suggests that similar purity statements may hold for other classical groups or flag varieties with involutions.
Explicit constructions of the symmetric plabic graphs could lead to new algorithms for computing coordinates on the Lagrangian Grassmannian.
Testing the correspondence on small values of m and k would verify the bijection assumption in low dimensions.

Load-bearing premise

The natural extension of the Oh-Postnikov-Speyer bijection between maximal weakly separated collections and reduced plabic graphs holds when symmetry under the defining involution is imposed on both sides.

What would settle it

A counterexample would be a symmetric plabic graph whose corresponding collection is not maximal by inclusion, or a maximal-by-size symmetric weakly separated collection that cannot be realized by any symmetric plabic graph.

read the original abstract

A collection $\mathcal{C}$ of $k$-element subsets of $\{1,2,\ldots,m\}$ is weakly separated if for each $I, J \in \mathcal{C}$, when the integers $1,2,\ldots,m$ are arranged around in a circle, there is a chord separating $I \backslash J$ from $J \backslash I$. Oh, Postnikov and Speyer constructed a correspondence between weakly separated collections which are maximal by inclusion and reduced plabic graphs, a class of networks defined by Postnikov which give coordinate charts on the Grassmannian of $k$-planes in m-space. As a corollary, they proved Scott's Purity Conjecture, which states that a weakly separated collection is maximal by inclusion if and only if it is maximal by size. In this note, we describe maximal weakly separated collections corresponding to symmetric plabic graphs, which give coordinate charts on the Lagrangian Grassmannian, and prove a symmetric version of the Purity Conjecture.

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.

Desk Editor's Note private letter to a colleague

Karpman constructs symmetric plabic graphs and proves the corresponding purity statement for the Lagrangian Grassmannian.

read the letter

This note supplies the symmetric version of the Oh-Postnikov-Speyer result: it describes maximal weakly separated collections that arise from symmetric plabic graphs and proves that maximality by inclusion is equivalent to maximality by size in the type-C setting. The construction adapts the earlier bijection by enforcing the fixed-point-free involution that defines the Lagrangian Grassmannian, and the abstract indicates the author carries out the necessary verification rather than leaving the extension implicit. That is the main new content. The work is a short, direct extension of an established correspondence, and it does the job of stating the symmetric collections explicitly while discharging the purity claim through the same style of argument used in type A. No load-bearing step appears to be reduced to a fitted quantity or circular definition. The only soft spot worth noting is that the paper is concise, so the details of how the symmetry condition interacts with the reducedness of the graphs and the chord-separation property will require close reading to confirm there are no edge cases missed in the involution. That is a normal feature of a note rather than a flaw. The result is aimed at people already working with positroids, plabic graphs, and coordinate charts on Grassmannians in other classical types. It is a legitimate incremental resolution of a stated conjecture and deserves a serious referee.

Referee Report

0 major / 1 minor

Summary. The paper extends the Oh-Postnikov-Speyer correspondence to the symmetric setting by describing maximal weakly separated collections that correspond to symmetric plabic graphs (which parametrize coordinate charts on the Lagrangian Grassmannian) and proves the symmetric Purity Conjecture: such a collection is maximal by inclusion if and only if it is maximal by cardinality.

Significance. The result supplies an explicit combinatorial model for positroid strata in type C and confirms that the purity property survives the fixed-point-free involution defining the Lagrangian Grassmannian. This strengthens the link between weakly separated collections, plabic graphs, and cluster structures on Lagrangian Grassmannians, and the construction itself constitutes the main technical contribution.

minor comments (1)

[Introduction] The abstract states that the authors 'describe' the collections and 'prove' the conjecture, but the introduction would benefit from a short explicit example of a symmetric plabic graph and its corresponding weakly separated collection to orient the reader before the general construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; direct extension of external prior result

full rationale

The paper extends the Oh-Postnikov-Speyer bijection (external authors) between maximal weakly separated collections and reduced plabic graphs to the symmetric setting, then proves the corresponding purity statement. The base correspondence and conjecture are cited from independent prior work, not self-citation. The manuscript's own construction discharges the symmetric case without reducing any claim to a fitted input, self-definition, or ansatz smuggled via the authors' prior papers. No equations or steps in the provided text exhibit the enumerated circularity patterns; the result is self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the prior correspondence of Oh-Postnikov-Speyer and on the standard definition of plabic graphs; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The Oh-Postnikov-Speyer bijection between maximal weakly separated collections and reduced plabic graphs extends to the symmetric setting.
Invoked to transfer the purity statement from the ordinary to the symmetric case.

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