Monotonicity of strata in the stratification of the cone of totally positive matrices

According to a theorem of Bjorner 5], there exists a stratified space whose strata are labeled by the elements of u, v] for every interval u, v] in the Bruhat order of a Coxeter group W, and each closed stratum (respectively the boundary of each stratum) has the homology of a ball (respectively of a sphere). In 6], Fomin and Shapiro suggest a natural geometric realization of these stratified spaces for a Weyl group W of a semi-simple Lie group G, and then prove its validity in the case of the symmetric group. The stratified spaces arise as links in the Bruhat decomposition of the totally non-negative part of the unipotent radical of G. In this article, we verify the topological regularity property of the strata formed as a result of Bruhat partial ordering on the elements of theWeyl group (of rank 4) of a semi-simple simply connected algebraic group G which is SL(4,?) in our case here. The Weyl group here is the Coxeter group S ₄.