Brick Manifolds and Toric Varieties of Brick Polytopes

In type A, Bott-Samelson varieties are posets in which ascending chains are flags of vector spaces. They come equipped with a map into the flag variety G/B. These varieties are mostly studied in the case in which the map into G/B is birational to the image. In this paper we study Bott-Samelsons for general types, more precisely, we study the combinatorics a fiber of the map into G/B when it is not birational. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the “subword complexes” of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion’s resolutions of Richardon varieties.