Involution Schubert-Coxeter Combinatorics

Zachary Hamaker ; Eric Marberg ; Brendan Pawlowski

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

Suppose K is a closed subgroup of GL(n,C) which acts on the complete flag variety with finitely many orbits. When K is a Borel subgroup, these orbits are Schubert cells, whose study leads to Schubert polynomials and many connections to type A Coxeter combinatorics. When K is O(n,C) or Sp(n,C), the orbits are indexed by some involutions in the symmetric group. Wyser and Yong described polynomials representing the cohomology classes of the orbit closures, and we investigate parallels for these ``involution Schubert polynomials'' of classical combinatorics surrounding type A Schubert polynomials. We show that their stable versions are Schur-P-positive, and obtain as a byproduct a new Littlewood-Richardson rule for Schur P-functions.

A key tool is an analogue of weak Bruhat order on involutions introduced by Richardson and Springer. This order can be defined for any Coxeter group W, and its labelled maximal chains correspond to reduced words for distinguished elements of W which we call atoms. In type A we classify all atoms, generalizing work of Can, Joyce, and Wyser, and give a connection to the Chinese monoid of Cassaigne et al. We give a different description of some atoms in general finite W in terms of strong Bruhat order.