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Coxeter and Schubert combinatorics of $μ$-Involutions

Jack Chen-An Chou, Zachary Hamaker

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 $μ$-involutions. We study Coxeter-theoretic properties of $μ$-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 $μ$-involution Schubert polynomials. We expand $μ$-involution Schubert polynomials as a multiplicity-free sum of $ν$-involution Schubert polynomials when $ν$ refines $μ$ and provide recurrences analogous to Monk's rule for Schubert polynomials.

Coxeter and Schubert combinatorics of $μ$-Involutions

Abstract

The variety of complete quadrics is the wonderful compactification of and admits a cell decomposition into Borel orbits indexed by combinatorial objects called -involutions. We study Coxeter-theoretic properties of -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 -involution Schubert polynomials. We expand -involution Schubert polynomials as a multiplicity-free sum of -involution Schubert polynomials when refines and provide recurrences analogous to Monk's rule for Schubert polynomials.

Paper Structure

This paper contains 16 sections, 27 theorems, 62 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Permutations
  4. Involutions
  5. $\mu$--involutions
  6. Schubert polynomials and transition equations
  7. Coxeter combinatorics for $\mu$--involutions
  8. Atoms of $\mu$--involutions
  9. From $\mu$ to $\nu$
  10. Transpositions for $\mu$--Bruhat order
  11. Transitions for $\mu$--involutions
  12. Future directions
  13. Shellability of $\mu$–Bruhat order
  14. $\mu$–involution Stanley symmetric functions
  15. Combinatorial formulas for $\mu$--involution Schubert polynomials
  16. ...and 1 more sections

Key Result

Theorem 1.1

For $\pi$ a $\mu$--involution with blocks $B_1, \dots,B_k$, where $Q_1Q_2 \dots Q_k$ is the concatenation of $Q_1,Q_2,\dots,Q_k$. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: The action of $t^\mathcal{I}_{ij}$ on $(b,a),(d,c) \in \mathrm{Cyc}(y)$ with $a < c$. When $a = b$ or $c=d$, we only use the earlier letter.
  • Figure 2: The black edges form the weak $\mu$-Bruhat order for $\mu = (3,1)$. Their edge labels correspond to the appropriate $s_i$ for the corresponding $0$--Hecke monoid action. The red edges are $\mu$-Bruhat order covers that are not weak $\mu$-Bruhat covers. They are labeled with transposition-like operators $t^\mu_{ij}$ witnessing these covers that are defined in Section \ref{['ss:transposition']}.
  • Figure 3: The cases for two cycles in $B$ with one split when restricting to $B_L$. For each possible inverse block atom, the third column depicts the restriction to $Q_L$.
  • Figure 4: The action when $t^\mu_{ij}(\pi) = \tau \neq \pi$ for nested cycles containing $i$ when $i,j$ are in different blocks in $\pi$. The third column depicts the possible subwords of the values in these cycles in $v\in \mathcal{A}_\mu^{-1}(\pi)$ where $t_{ij}v \in \mathcal{A}_\mu^{-1}(\tau)$, with crossed out subwords excluded by the fact that $v \lessdot t_{ij}v$. The fourth column depicts the corresponding subwords in $t_{ij}v$. We omit cases where there are no intermediate values between $i$ and $j$.
  • Figure 5: Non-trivial actions of $t_{ij}^\mu$ for $i,j$ in different blocks for pairs of cycles not nested. In this chart, we only depict cases where the pairwise structure of the cycle containing $i$ or its counterpart changes.

Theorems & Definitions (53)

  • Theorem 1.1: = Theorem \ref{['thm: mu involution atoms']}
  • Theorem 1.2: = Theorem \ref{['t:mu-transition']}
  • Lemma 2.1: Deletion Property, bjorner2005combinatorics*Prop. 1.4.7
  • Proposition 2.2: bjorner2005combinatorics*Prop. 2.4.4
  • Theorem 2.3: HMP2*Cor 5.13; see also CJW*Thm 2.5
  • Theorem 2.4: HMP2*Thm. 6.10
  • Example 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7: hamaker2018transition*Thm 3.23
  • ...and 43 more