Séminaire Lotharingien de Combinatoire, 78B.51 (2017), 12 pp.
Zachary Hamaker, Eric Marberg and Brendan Pawlowski
Involution Schubert-Coxeter Combinatorics
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
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
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.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
The following versions are available: