# Locally convex curves and the Bruhat stratification of the spin group

## TL;DR

This paper explores the lifting of the Schubert stratification to the spin group, providing explicit parameterizations of Bruhat cells and demonstrating their relevance to locally convex curves.

## Contribution

It introduces explicit parameterizations of Bruhat cells in the spin group and applies these to the study of locally convex curves, extending classical stratifications.

## Key findings

- Explicit parameterizations of Bruhat cells in Spin_{n+1}
- Compatibility of parameterizations with Bruhat order
- Applications to the study of locally convex curves

## Abstract

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.