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arxiv: 1907.10836 · v1 · pith:X6BD3EOLnew · submitted 2019-07-25 · 🧮 math.CO

Queer Supercrystal Structure for Increasing Factorizations of Fixed-Point-Free Involution Words

Toya Hiroshima This is my paper

Pith reviewed 2026-05-24 16:32 UTC · model grok-4.3

classification 🧮 math.CO
keywords fixed-point-free involution wordsincreasing factorizationsqueer supercrystalsprimed tableauxsymplectic shifted Hecke insertionCoxeter-Knuth relations
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The pith

Increasing factorizations of fixed-point-free involution words form queer supercrystals.

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

The paper proves that the set of increasing factorizations of fixed-point-free involution words stands in bijection with primed tableaux. The bijection arises from the symplectic shifted Hecke insertion algorithm by showing that Coxeter-Knuth related words produce identical insertion tableaux. Primed tableaux already carry queer supercrystal operators, so the factorizations inherit the same operators and structure. A reader would care because this supplies a direct combinatorial model for the supercrystals inside the language of involution words.

Core claim

The set of increasing factorizations of fixed-point-free involution words is in one-to-one correspondence with primed tableaux via symplectic shifted Hecke insertion, where the insertion tableau is an increasing shifted tableau and the recording tableau is primed, and therefore the factorizations carry the queer supercrystal structure.

What carries the argument

Symplectic shifted Hecke insertion, which sends Coxeter-Knuth equivalent FPF-involution words to the same increasing shifted tableau while recording a primed tableau.

If this is right

  • Queer supercrystal operators can be defined directly on the increasing factorizations.
  • Coxeter-Knuth equivalence classes become the connected components of the supercrystal.
  • The correspondence supplies a shifted-tableau model for the factorizations.

Where Pith is reading between the lines

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

  • The same insertion algorithm might be used to define supercrystal operators on other families of involution words.
  • Crystal operators could yield new recurrence relations or generating functions for counting the factorizations.

Load-bearing premise

The symplectic shifted Hecke insertion algorithm sends all words in the same Coxeter-Knuth class to the same insertion tableau.

What would settle it

Two Coxeter-Knuth related fixed-point-free involution words that produce different insertion tableaux under symplectic shifted Hecke insertion.

Figures

Figures reproduced from arXiv: 1907.10836 by Toya Hiroshima.

Figure 5.1
Figure 5.1. Figure 5.1: An example of q(3)-crystal structure of increasing factorizations of FPF-involution words, 2143, 2343, 2413, 2431, 2434, 4213, 4231, and 4234. The set of these words is Rˆ FPF(546213). This property is compatible with Eq (5.1), i.e, the gl(m)-crystal structure of RFm FPF(z) is in one-to-one correspondence with that of PTm(λ), and ˜e F i and ˜f F i (i = 1, . . . , m − 1, ¯1) satisfy conditions in Definiti… view at source ↗
read the original abstract

We show that the set of increasing factorizations of fixed-point-free (FPF) involution words has the structure of queer supercrystals. By exploiting the algorithm of symplectic shifted Hecke insertion recently introduced by Marberg, we establish the one-to-one correspondence between the set of increasing factorizations of fixed-point-free involution words and the set of primed tableau (semistandard marked shifted tableaux) and the latter admits the structure of queer supercrystals. In order to establish the correspondence, we prove that the Coxeter-Knuth related FPF-involution words have the same insertion tableau in the symplectic shifted Hecke insertion, where the insertion tableau is an increasing shifted tableau and the recording tableau is a primed tableau.

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 paper claims that the set of increasing factorizations of fixed-point-free (FPF) involution words carries the structure of queer supercrystals. It establishes this by constructing a bijection to primed tableaux (semistandard marked shifted tableaux) via Marberg's symplectic shifted Hecke insertion, after proving that Coxeter-Knuth equivalent FPF-involution words produce the same insertion tableau (an increasing shifted tableau) with the recording tableau being primed.

Significance. If the central bijection and invariance hold, the result supplies a combinatorial realization of queer supercrystals on a new family of objects arising from Coxeter groups, extending existing crystal structures on tableaux and words. The approach is a strength in that it reuses an independently introduced insertion algorithm rather than defining new operators, and the logical skeleton (invariance under equivalence classes followed by identification with an already-equipped set) avoids circularity or hidden parameters.

major comments (1)
  1. The invariance of the symplectic shifted Hecke insertion under Coxeter-Knuth moves for FPF-involution words is load-bearing for the bijection claim. The manuscript should isolate this as an explicit lemma (with verification steps) rather than subsuming it in the outline of the correspondence.
minor comments (2)
  1. The abstract states the proof strategy but does not name the section containing the invariance argument; adding a forward reference would improve readability.
  2. Notation for 'primed tableau' and 'increasing shifted tableau' should be fixed at first use to avoid ambiguity with standard shifted tableaux.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion. We agree that the requested change will improve clarity and will incorporate it in the revision.

read point-by-point responses

  1. Referee: The invariance of the symplectic shifted Hecke insertion under Coxeter-Knuth moves for FPF-involution words is load-bearing for the bijection claim. The manuscript should isolate this as an explicit lemma (with verification steps) rather than subsuming it in the outline of the correspondence.

    Authors: We agree that the invariance under Coxeter-Knuth moves is central and should be highlighted. In the revised manuscript we will extract the relevant argument as a standalone lemma (Lemma X.Y) that states: if two FPF-involution words are Coxeter-Knuth equivalent then their symplectic shifted Hecke insertion tableaux coincide. The proof will be written out in full detail, citing the relevant properties of Marberg’s insertion algorithm and the specific verifications needed for the FPF case that were previously only sketched in the correspondence outline. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by citing Marberg's external symplectic shifted Hecke insertion algorithm, proving invariance of the insertion tableau on Coxeter-Knuth classes of FPF-involution words (a direct combinatorial argument), and identifying the resulting primed tableaux with an independently equipped queer supercrystal structure. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and the central bijection does not rely on a self-citation chain or imported uniqueness theorem from the present author. The manuscript is self-contained against external benchmarks once the cited insertion algorithm is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the correctness of Marberg's symplectic shifted Hecke insertion algorithm and on the pre-existing queer supercrystal structure on primed tableaux; both are imported from prior literature rather than derived inside the paper.

axioms (1)
  • domain assumption Symplectic shifted Hecke insertion sends Coxeter-Knuth equivalent FPF-involution words to identical insertion tableaux.
    This preservation property is required to obtain a well-defined bijection with primed tableaux.

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Reference graph

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