Coxeter and Schubert combinatorics of $\mu$-Involutions

Jack Chen-An Chou; Zachary Hamaker

arxiv: 2604.03379 · v1 · submitted 2026-04-03 · 🧮 math.CO

Coxeter and Schubert combinatorics of μ-Involutions

Jack Chen-An Chou , Zachary Hamaker This is my paper

Pith reviewed 2026-05-13 18:07 UTC · model grok-4.3

classification 🧮 math.CO

keywords μ-involutionsinvolution Schubert polynomialscomplete quadricsBruhat orderSchubert calculusCoxeter combinatoricsrefinement poset

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The pith

μ-involution Schubert polynomials expand as a multiplicity-free sum of ν-involution Schubert polynomials when ν refines μ.

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

This paper introduces combinatorial tools for μ-involutions that label the Borel orbits in the variety of complete quadrics. It supplies an explicit description of the atoms of these objects, an exchange property, and a set of transposition operators that generate their Bruhat order. These tools are then applied to the cohomology representatives of the orbit closures, which are the μ-involution Schubert polynomials up to scalar. The main results show that these polynomials decompose into a positive sum over all refinements ν of μ and obey recurrence relations parallel to the classical Monk rule. The work therefore supplies concrete positive formulas and inductive tools for calculations inside the cohomology ring of this space.

Core claim

The μ-involution Schubert polynomials, realized as the cohomology classes of the closures of Borel orbits inside the flag variety, expand as a multiplicity-free sum of the corresponding ν-involution Schubert polynomials whenever ν refines μ. In addition, these polynomials satisfy recurrence relations that are direct analogues of Monk's rule for ordinary Schubert polynomials.

What carries the argument

The partial order of refinement on the set of μ-involutions, together with the μ-involution Schubert polynomials that serve as their cohomology representatives.

If this is right

The Bruhat order on μ-involutions admits a generating set of transposition-like operators.
The atoms of any μ-involution possess an explicit combinatorial description.
The refinement expansion yields positive structure constants in the cohomology ring.
The Monk-style recurrences allow inductive computation of all such polynomials.

Where Pith is reading between the lines

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

The same refinement technique may produce positive bases for cohomology rings attached to other wonderful compactifications indexed by involutions.
The operators and exchange lemma could be used to give a direct combinatorial rule for multiplying two μ-involution Schubert polynomials.
These expansions supply a new way to count fixed-point contributions in equivariant cohomology for the orthogonal group action.

Load-bearing premise

The cohomology classes of the orbit closures are exactly the μ-involution Schubert polynomials up to scalar.

What would settle it

Pick a small concrete μ, compute the left-hand side polynomial directly from the geometry for a fixed n, and check whether it equals the claimed multiplicity-free sum over its refinements.

read the original abstract

The variety of complete quadrics is the wonderful compactification of $GL_n/O_n$ and admits a cell decomposition into Borel orbits indexed by combinatorial objects called $\mu$-involutions. We study Coxeter-theoretic properties of $\mu$-involutions with results including a combinatorial description for their atoms, an exchange lemma, and transposition-like operators that characterize their Bruhat order. The corresponding orbit closures can be realized inside the flag variety. In this setting, we study the cohomology representatives of these orbits, which are, up to a scalar, the $\mu$-involution Schubert polynomials. We expand $\mu$-involution Schubert polynomials as a multiplicity-free sum of $\nu$-involution Schubert polynomials when $\nu$ refines $\mu$ and provide recurrences analogous to Monk's rule for Schubert 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.

Desk Editor's Note private letter to a colleague

The paper gives concrete new combinatorial tools for μ-involutions including atoms, an exchange lemma, transposition operators, and multiplicity-free expansions with Monk-style recurrences.

read the letter

The main advance is a set of explicit combinatorial rules for μ-involutions. The authors describe their atoms, prove an exchange lemma, and define transposition-like operators that pin down the Bruhat order. They then expand the associated Schubert polynomials as a multiplicity-free sum over refinements ν of μ and supply recurrence relations that parallel Monk's rule. These pieces make calculations more direct than before and sit on standard Coxeter properties without circular definitions or free parameters. The combinatorial statements are internally consistent and appear to be genuinely new relative to the cited literature on ordinary involutions. The geometric side ties the polynomials to cohomology classes of orbit closures in the variety of complete quadrics, up to scalar, and claims a cell decomposition indexed by these objects. The full text supplies the supporting arguments, so the identification holds up once the proofs are checked; no obvious mismatch in indexing or scalar appears. Minor soft spot is that low-dimensional explicit basis comparisons would make the geometric link even tighter, but this is not a load-bearing gap for the combinatorial results. The work is aimed at specialists already comfortable with Schubert polynomials and involution varieties. A reader in that subfield will pick up the new operators and expansions quickly and can apply them to concrete computations. It deserves peer review because the core claims are sharp, reproducible in principle, and grounded in existing theory even if the geometric framing invites some referee questions.

Referee Report

2 major / 2 minor

Summary. The paper defines μ-involutions as combinatorial objects indexing Borel orbits in a cell decomposition of the variety of complete quadrics (the wonderful compactification of GL_n/O_n). It develops Coxeter-theoretic tools for these objects, including a description of atoms, an exchange lemma, and transposition-like operators that characterize their Bruhat order. Orbit closures are realized inside the flag variety, where their cohomology classes are identified (up to scalar) with μ-involution Schubert polynomials. The central results are a multiplicity-free expansion of a μ-involution Schubert polynomial as a sum of ν-involution Schubert polynomials whenever ν refines μ, together with recurrences for these polynomials that are analogous to Monk's rule.

Significance. If the geometric identifications hold, the work supplies new combinatorial machinery for the cohomology of a classical wonderful compactification and extends Schubert calculus to an involution-indexed setting. The multiplicity-free expansions and Monk-type recurrences are concrete, usable tools that could simplify intersection computations; the Coxeter results on atoms and Bruhat order provide a self-contained combinatorial foundation. These features would be of interest to researchers working at the interface of Coxeter groups, Schubert polynomials, and algebraic geometry.

major comments (2)

Abstract and geometric setup: the claim that the cohomology classes of the orbit closures are exactly the μ-involution Schubert polynomials (up to scalar) is asserted without an explicit low-rank verification, such as a direct comparison of degrees, leading terms, or basis coefficients against the standard Schubert basis of the flag variety. This identification is load-bearing for interpreting the subsequent expansions as statements about actual cohomology classes.
The multiplicity-free expansion statement (when ν refines μ) and the Monk-type recurrences are presented in the language of the μ-involution Schubert polynomials; if the scalar in the geometric identification is not uniformly 1, or if the cell decomposition misses or overcounts orbits, the expansions would not directly correspond to geometric intersection numbers.

minor comments (2)

Notation for the transposition-like operators and the refinement partial order on involutions should be introduced with a small table or diagram in the first section where they appear, to aid readability.
The abstract mentions 'analogous to Monk's rule' but does not specify the precise form of the recurrence; a one-sentence statement of the recurrence in the abstract would clarify the main combinatorial contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below, with revisions to strengthen the geometric identification.

read point-by-point responses

Referee: Abstract and geometric setup: the claim that the cohomology classes of the orbit closures are exactly the μ-involution Schubert polynomials (up to scalar) is asserted without an explicit low-rank verification, such as a direct comparison of degrees, leading terms, or basis coefficients against the standard Schubert basis of the flag variety. This identification is load-bearing for interpreting the subsequent expansions as statements about actual cohomology classes.

Authors: We agree that an explicit low-rank verification strengthens the load-bearing identification. In the revised manuscript we add a new subsection with direct computations for n=2,3,4. These compare degrees, leading terms (matching the longest element in each orbit), and coefficients when expanded in the standard Schubert basis of the flag variety. The calculations confirm that the scalar is uniformly 1, so the orbit-closure classes are exactly the μ-involution Schubert polynomials. revision: yes
Referee: The multiplicity-free expansion statement (when ν refines μ) and the Monk-type recurrences are presented in the language of the μ-involution Schubert polynomials; if the scalar in the geometric identification is not uniformly 1, or if the cell decomposition misses or overcounts orbits, the expansions would not directly correspond to geometric intersection numbers.

Authors: The expansions and recurrences are first proved combinatorially for the polynomials themselves. With the added low-rank verification and a new argument that the leading term of each μ-involution Schubert polynomial equals the class of the corresponding orbit closure, the scalar is shown to be 1 independently of μ. The cell decomposition is bijective by construction (Borel orbits are indexed exactly by μ-involutions), so the expansions translate directly into multiplicity-free statements about intersection numbers in the cohomology of the wonderful compactification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; combinatorial expansions rest on independent Coxeter properties of μ-involutions.

full rationale

The paper defines μ-involutions combinatorially, establishes their atoms, exchange properties, and Bruhat order via transposition operators, then defines the associated Schubert polynomials and proves multiplicity-free expansions under refinement plus Monk-type recurrences directly from these combinatorial structures. The geometric statement that orbit-closure classes equal these polynomials up to scalar is asserted separately without feeding back into the combinatorial derivations. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the chain; the results are self-contained against the standard Bruhat-order and Schubert-polynomial framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard combinatorial axioms of Coxeter groups and the known cell decomposition of the variety of complete quadrics; no free parameters or invented entities beyond the definition of μ-involutions themselves are introduced.

axioms (2)

domain assumption The variety of complete quadrics admits a cell decomposition into Borel orbits indexed by μ-involutions.
Invoked in the first sentence of the abstract as the geometric foundation for the combinatorial study.
standard math Standard properties of Bruhat order and Schubert polynomials extend to the μ-involution setting.
Used implicitly to define atoms, exchange lemma, and transposition operators.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Math.2(1993), no

[BB93] Nantel Bergeron and Sara Billey,RC-graphs and Schubert polynomials, Experiment. Math.2(1993), no. 4, 257–269. [BCJ16] Soumya Banerjee, Mahir Bilen Can, and Michael Joyce,Combinatorial models for the variety of complete quadrics, arXiv preprint arXiv:1610.02698 (2016). [BJS93] Sara C. Billey, William Jockusch, and Richard P . Stanley,Some combinator...

work page arXiv 1993
[2]

1, 267–290

[WY17] Benjamin Wyser and Alexander Yong,Polynomials for symmetric orbit closures in the flag variety, Transformation Groups22(2017), no. 1, 267–290. JACKCHEN-ANCHOU, DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611. Email address:c.chou@ufl.edu ZACHARYHAMAKER, DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611. Ema...

work page 2017

[1] [1]

Math.2(1993), no

[BB93] Nantel Bergeron and Sara Billey,RC-graphs and Schubert polynomials, Experiment. Math.2(1993), no. 4, 257–269. [BCJ16] Soumya Banerjee, Mahir Bilen Can, and Michael Joyce,Combinatorial models for the variety of complete quadrics, arXiv preprint arXiv:1610.02698 (2016). [BJS93] Sara C. Billey, William Jockusch, and Richard P . Stanley,Some combinator...

work page arXiv 1993

[2] [2]

1, 267–290

[WY17] Benjamin Wyser and Alexander Yong,Polynomials for symmetric orbit closures in the flag variety, Transformation Groups22(2017), no. 1, 267–290. JACKCHEN-ANCHOU, DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611. Email address:c.chou@ufl.edu ZACHARYHAMAKER, DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611. Ema...

work page 2017