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.

在任何Coxeter族群的对合中，都有弱Bruhat阶的自然类似物。间隔的饱和链按此顺序对应于一组称为原子的组元素的简化词。Brion给出了a的同调类的一般公式$ķ$任意标志变体中的-轨道闭合，其中 $ķ$是复数代数群的对称子群。在类型A中，此公式中的术语由原子索引以进行排列。我们研究了有符号排列组中对合的原子组合。尤其是，我们对任何有符号对合的原子集进行了简要描述，并赋予其渐变的波峰结构。作为应用，我们的主要结果是明确识别出B类型和B类型的Brion同调公式中的术语。这些描述适用于所有$ķ$-这些类型的轨道，是A类以外的同类轨道中的第一个。