Solvable Lattice Models and the Geometry of Flag Varieties

Abstract

Geometric constructions of quantum groups and their associated R-matrices arose in the early 90's and have been generalized further in recent works of Aganagic, Maulik, and Okounkov, creating a bridge between geometry and solvable lattice models. One nice aspect of this bridge is that the "hard" basis of one theory corresponds to the "easy" basis of the other. In this thesis, we explore various lattice models using this perspective as guidance. We first describe how both the torus fixed point basis and the basis of Schubert classes in the equivariant cohomology of the flag variety are manifest in the "Frozen Pipes" lattice model of Brubaker, Frechette, Hardt, Tibor, and Weber. This analysis is a straightforward generalization of results due to Gorbunov, Korff, and Stroppel (see also the notes of Zinn-Justin) for the Grassmannian. Then we describe how the fixed point basis and the basis of motivic Chern classes in the equivariant K-theory of the cotangent bundle of the flag variety appear (in a more novel way) in the Tokuyama model of Brubaker, Bump, and Friedberg and colored Iwahori Whittaker model of Brubaker, Buciumas, Bump, and Gustafsson. Recent work of Aluffi, Mihalcea, Schürmann, and Su identifies these geometric bases with the Casselman and standard bases, respectively, of the Iwahori fixed vectors in the principal series representation, so this perspective allows us to make contact with formulas from p-adic representation theory, such as the Langlands-Gindikin-Karpelevich formula.