Atoms for signed permutations

Abstract

There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general formula for the cohomology class of a $K$-orbit closure in an arbitrary flag variety, where $K$ is a symmetric subgroup of a complex algebraic group. In type A, the terms in this formula are indexed by atoms for permutations. We study the combinatorics of atoms for involutions in the group of signed permutations. In particular, we give a compact description of the atom set for any signed involution and endow it with the structure of a graded poset. Our main result, as an application, is to identify explicitly the terms in Brion's cohomology formula in types B and C. These descriptions apply to all $K$-orbits in these types and are the first of their kind outside of type A.