Lattice models as Poincaré pairings
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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 Maulik, Okounkov, and others, creating a bridge between geometry and integrable lattice models. One nice aspect of this bridge is that the "hard" basis of one theory corresponds to the "easy" basis of the other.
We will analyze some of the lattice models we have already encountered in this seminar from this perspective. 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. This analysis is a straightforward generalization of results in a paper of 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 and colored Iwahori Whittaker model. 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 Gindikin-Karpelevich formula and Bump-Nakasuji-Naruse conjecture. These ideas will be detailed in my forthcoming doctoral thesis.
Relevant papers: Maulik and Okounkov, a more accessible summary of Maulik and Okounkov, Rimanyi (h-deformed Schubert calculus), Gorbunov, Korff, and Stroppel, Zinn-Justin's notes, Tokuyama model, Iwahori Whittaker model, Frozen Pipes, Motivic Chern classes and Iwahori invariants
Zoom link: https://stanford.zoom.us/j/91274693500?pwd=bmtHZnhTMG1HQ3pTOHYxUWJ2Z1ZjQT09
Zoom ID: 912 7469 3500
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