Atoms for signed permutations

Zachary Hamaker, Eric Marberg

Published 2018-02-27Version 1

Consider the Bruhat order of a Coxeter group restricted to its involutions. The maximal chains in this order going from the identity to a given involution are in bijection with the reduced words for a certain set of group elements, which we call atoms. We study the combinatorics of atoms for involutions in the group of signed permutations. This builds on prior work concerning atoms for involutions in the symmetric group, which was motivated by connections to the geometry of certain spherical varieties. Most notably, we prove that the set of atoms for any signed involution naturally has the structure of a graded poset whose maximal elements are counted by Catalan numbers. We also characterize the signed involutions with exactly one atom and prove some enumerative results about reduced words for signed permutations.