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

TL;DR

This paper introduces involution words and their enumerative properties, connecting combinatorics with geometry and Schubert calculus, and provides formulas for orbit closure cohomology classes.

Contribution

It defines involution analogues of key permutation objects and links them to geometric interpretations and representation theory, extending Schubert calculus to involutions.

Findings

Involution Stanley symmetric function for the longest element factors into staircase-shaped Schur functions.

Number of involution words equals the dimension of a Weyl group representation.

Provides formulas for cohomology classes of orbit closures in flag varieties.

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 $B$.