We study the lifting of the Schubert stratification of the homogeneous space of complete real flags of ℝⁿ⁺¹ to its universal covering group Spin_n+1. We call the lifted strata the Bruhat cells of Spin_n+1, in keeping with the homonymous classical decomposition of reductive algebraic groups. We present explicit parameterizations for these Bruhat cells in terms of minimal-length expressions \(\sigma = {a_{{i_1}}} \cdots {a_{{i_k}}}\) for permutations σ ∈ S_n+1 in terms of the n generators a_i = (i, i + 1). These parameterizations are compatible with the Bruhat orders in the Coxeter—Weyl group S_n+1. This stratification is an important tool in the study of locally convex curves; we present a few such applications.

我们研究了real ^{n +1}的完全实旗的齐次空间的Schubert分层对其通用覆盖群Spin _{n +1}的提升。我们将提升层称为自旋_{n +1}的Bruhat细胞，以与还原代数群的同构经典分解保持一致。我们在最小长度的表达式而言这些Bruhat细胞本明确的参数化\（\西格玛= {A _ {{I_1}}} \ cdots {一个_ {{I_K}}} \）用于置换σ＆Element;小号_{Ñ 1}中的术语所述的ñ发电机一个_我=（I，I+1）。这些参数化与Coxeter-Weyl组S _{n +1中}的Bruhat阶兼容。这种分层是研究局部凸曲线的重要工具。我们介绍了一些这样的应用程序。