We study the lifting of the Schubert stratification of the homogeneous space of complete real flags of $R^{ n+1}$ 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 $\sigma=a_{ i_1}... a_{ i_k}$ for permutations $\sigma\in S_{ n+1}$ 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.
Categories: math.GT