Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures

Abstract

Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of "admissible sequences" by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on involutions. In particular, our definition of "involution Schubert polynomials" can be viewed as a Billey-Jockusch-Stanley type formula for cohomology class representatives of ${\mathrm{O}}_n$- and ${\mathrm{Sp}}_{2n}$-orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type $B$.

38 pages; v2: some revisions and corrections, with an expanded introduction; v3, v4: added remarks, attribution, and acknowledgements; v5: revised introduction, updated references; v6: various revisions and corrections, removed geometric appendix, added index of notation, final version

References in corpus (7)

• Hopf Algebras in Combinatorics
• Involution words II: braid relations and atomic structures
• Schur P-positivity and involution Stanley symmetric functions
• Fixed-point-free involutions and Schur P-positivity
• Transition formulas for involution Schubert polynomials
• Stanley symmetric functions and Peterson algebras
• Stanley symmetric functions for signed involutions

Cited by in corpus (22)

• Involution words II: braid relations and atomic structures
• Schur P-positivity and involution Stanley symmetric functions
• Fixed-point-free involutions and Schur P-positivity
• Transition formulas for involution Schubert polynomials
• K-theory formulas for orthogonal and symplectic orbit closures
• On some properties of symplectic Grothendieck polynomials
• A symplectic refinement of shifted Hecke insertion
• On involutions in Weyl groups
• Braid relations for involution words in affine Coxeter groups
• On some actions of the 0-Hecke monoids of affine symmetric groups
• Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
• Shifted insertion algorithms for primed words
• Involution pipe dreams
• Atoms for signed permutations
• Stanley symmetric functions for signed involutions
• On pattern avoidance in matchings and involutions
• Affine transitions for involution Stanley symmetric functions
• Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions
• Bialgebras for Stanley symmetric functions
• $DIII$ clan combinatorics for the orthogonal Grassmannian
• Crystals for shifted key polynomials
• Schubert polynomial analogues for degenerate involutions