Locally convex curves and the Bruhat stratification of the spin group

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 σ = a i 1 • • • 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.

Grassmann convexity and multiplicative Sturm theory, revisited

“…In §2 we will introduce our main technical tool which is the rank function for a certain type of cyclic words using which we prove Theorem 1. In §3 we recall several results from [6], prove some additional statements and settle Theorem 2. (Notice that besides the references we already mentioned other relevant results can be found in e.g.…”

“…In the following Lemma we provide an alternative characterization of the subsets Pos, Neg ⊂ Lo 1 n ; here id ∈ Lo 1 n is the identity matrix. Lemma 2.2 (see [10], [6]).…”

“…In this section we follow the notation of [5,6] and use matrix realizations of flag curves obtained as osculating curves of convex projective curves. (Such realizations were frequently used in the earlier papers by the authors).…”

“…This group can be interpreted as the tangent space to Fl n at any fixed chosen flag f . Alternatively, the usual LU and QR decompositions define diffeomorphisms Q : Lo 1 n → U 1 and L : U 1 → Lo 1 n , where U 1 ⊂ Fl n is a top dimensional cell (see [6]). The following statement can be found in e.g.…”

“…To settle Theorem 3 we need more notation. As in [6] (see especially Sections 2 and 7), let S n be the symmetric group with generators a i = (i i + 1). The symmetric group is endowed with the usual Bruhat order.…”