Atoms for signed permutations

TL;DR

This paper explores the combinatorics of atoms for involutions in signed permutations, providing a compact description and poset structure, and applies these findings to explicitly identify terms in Brion's cohomology formula for types B and C.

Contribution

It introduces a new combinatorial framework for atoms in signed involutions and applies it to extend Brion's cohomology formula beyond type A.

Findings

Compact description of atom sets for signed involutions

Poset structure on atom sets

Explicit identification of cohomology terms in types B and C

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.