Locally convex curves and the Bruhat stratification of the spin group

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 $Spi{n}_{n+1}$. We call the lifted strata the Bruhat cells of $Spi{n}_{n+1}$, 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 $\sigma =a_{i_1}\mathrm{.}\mathrm{.}\mathrm{.}{a}_{i_k}$ 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.