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Séminaire Lotharingien de Combinatoire, 78B.51 (2017), 12 pp.

# Zachary Hamaker, Eric Marberg and Brendan Pawlowski

# Involution Schubert-Coxeter Combinatorics

**Abstract.**
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.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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