Brick Varieties And Toric Matrix Schubert Varieties

In the first part of this thesis we study brick varieties which are fibers of the Bott-Samelson varieties. Bott-Samelson varieties are a twisted product of CP1 's with a map into G/B. These varieties are mostly studied in the case in which the map into G/B is birational to the image; however in the first part of this thesis we study a fiber of this map when the map is not birational. We prove that in some cases the general fiber, which we christen a brick variety, is a toric variety. 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. In the second part of this thesis, which is joint work with Karola M´ sz´ ros, e a we show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of -Grothendieck polynomials. Finally, we explain this geometric realization by characterizing the matrix Schubert varieties which are the product of a toric variety and a vector space