Abstract
Fomin and Shapiro conjectured that the link of the identity in the Bruhat stratification of the totally nonnegative real part of the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over \({\mathbb{R}}\) is a regular CW complex homeomorphic to a ball. The main result of this paper is a proof of this conjecture. This completes the solution of the question of Bernstein of identifying regular CW complexes arising naturally from representation theory having the (lower) intervals of Bruhat order as their closure posets. A key ingredient is a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps; it most naturally applies to images of maps from regular CW complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology.
Highlights
In this paper, the following conjecture of Sergey Fomin and Michael Shapiro from [11] is proven.Conjecture 1.1 Let Y be the link of the identity in the totally nonnegative real part of the unipotent radical of a Borel subgroup B in a semisimple, connected algebraic group defined and split over R
Results of Björner [4] combine with results of Björner and Wachs [6] to imply that each interval of Bruhat order is the closure poset of a regular CW complex
The tools we develop in order to prove Conjecture 1.1 give a new approach to the general question of how to prove that the image of a map from a polytope which restricts to a homeomorphism on the interior but not necessarily on the boundary is a regular CW complex homeomorphic to a ball
Summary
The following conjecture of Sergey Fomin and Michael Shapiro from [11] is proven. We needed to develop a class of collapses that would only identify points having the same image under f(i1,...,id), while restricting ourselves to operations where we could control homeomorphism type and regularity To this end, we collapse cells across families of curves which seem typical enough of fibers of maps of interest arising e.g. in combinatorial representation theory to be likely to be useful for other examples of interest as well. Theorem 1.3 and Theorem 4.21 provide a fairly combinatorial general approach to proving that images of sufficiently nice maps from polytopes are regular CW complexes homeomorphic to balls Another crucial ingredient in the proof of Conjecture 1.1 is the 0-Hecke algebra associated to a Coxeter group W. Throughout the paper, we deliberately include a high level of detail, so as to help readers bridge between the combinatorics, topology, and representation theory
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