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Matchings in Corona graph and classical symmetric varieties

Yau Wing Li

TL;DR

The paper develops a combinatorial parametrization of finite $B(k)$-orbits in classical symmetric varieties via matchings on corona graphs and their double variants, establishing exact correspondences for types $AIII$ and $CII$ through a quiver-variety framework and Gabriel’s theorem. This yields ultra log-concavity and unimodality results for orbit counts and proves Can–Ugurlu’s conjecture about non-integrality in the BI case, with precise recurrences and asymptotics derived for the associated counting polynomials. The method extends to other classical types (symplectic and orthogonal) using minus-involution actions on matchings and double coset descriptions, providing a unified combinatorial approach to orbit enumeration in symmetric varieties. Together, these results offer a concrete, graph-theoretic lens on Borel orbit structure and illuminate the arithmetic nature of orbit-count polynomials across types.”

Abstract

We introduce an alternative combinatorial parametrization of Borel orbits in classical symmetric varieties using matchings of the Corona graph. As an application, we obtain ultra log-concavity and unimodality for the number of Borel orbits in Types AIII and CII. Moreover, we prove a conjecture of Can and Ugurlu concerning the non-integrality of the coefficients of the polynomial that interpolates the number of orbits in Type BI.

Matchings in Corona graph and classical symmetric varieties

TL;DR

The paper develops a combinatorial parametrization of finite -orbits in classical symmetric varieties via matchings on corona graphs and their double variants, establishing exact correspondences for types and through a quiver-variety framework and Gabriel’s theorem. This yields ultra log-concavity and unimodality results for orbit counts and proves Can–Ugurlu’s conjecture about non-integrality in the BI case, with precise recurrences and asymptotics derived for the associated counting polynomials. The method extends to other classical types (symplectic and orthogonal) using minus-involution actions on matchings and double coset descriptions, providing a unified combinatorial approach to orbit enumeration in symmetric varieties. Together, these results offer a concrete, graph-theoretic lens on Borel orbit structure and illuminate the arithmetic nature of orbit-count polynomials across types.”

Abstract

We introduce an alternative combinatorial parametrization of Borel orbits in classical symmetric varieties using matchings of the Corona graph. As an application, we obtain ultra log-concavity and unimodality for the number of Borel orbits in Types AIII and CII. Moreover, we prove a conjecture of Can and Ugurlu concerning the non-integrality of the coefficients of the polynomial that interpolates the number of orbits in Type BI.
Paper Structure (14 sections, 26 theorems, 30 equations, 6 figures)

This paper contains 14 sections, 26 theorems, 30 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notations
  4. Algebraic group
  5. Lie algebra
  6. Quiver
  7. Others
  8. Review on quiver representations
  9. Proofs of the result
  10. Other classical types
  11. General theory
  12. Symplectic group
  13. Orthogonal group
  14. Applications

Key Result

Theorem 1.2

Let $k$ be a field and $\overline k$ be an algebraic closure of $k$. Let $X$ be the homogeneous space $G/K$.

Figures (6)

  • Figure 2: Dimension vector $d_{m,n}$
  • Figure 3: A $2$-matching $\mathcal{S}$ and its dual $\mathbb{D} \mathcal{S}$ in $C_6$
  • Figure 4: A $2$-matching $\mathcal{S}$ and its minus $-\mathcal{S}$ in $C_6$
  • Figure : (a) $C_6$
  • Figure : (a) $C_6$
  • ...and 1 more figures

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Lemma 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 43 more