Math 263B: Pipedreams

We will discuss the combinatorial theory of various families of polynomials such as Schubert and Grothendieck polynomials, Schur polynomials and Stanley symmetric functions. We will focus on pipedreams, which are combinatorial objects representing Schubert polynomials, as instances of solvable lattice models, and related combinatorics.

This course will meet MWF at 11:30 AM in 380F. Lecture notes will be posted here.

Prerequisites are some familiarity with ideas from Lie theory (root systems, Weyl group). I will not assume that you know anything about crystals but will probably not devote too much lecture time to foundations of crystal theory.

Lectures

• Lecture 1: Schubert calculus.
• Lecture 2: Demazure operators; definition of Schubert polynomials.
• Lecture 3: Demazure characters.
• Lecture 4: Tokuyama Models I.
• Lecture 5: Tokuyama Models II.
• Lecture 6: States, Gelfand-Tsetlin Patterns, Tableaux and $q=0$ Tokuyama models.
• Lecture 7: Parametrized Yang-Baxter equations.
• Lecture 8: Demazure Characters and Crystals.
• Lecture 9: Schützenberger Involution, crystal structure on Tokuyama states.
• Lecture 10: Open models: how the Yang-Baxter equation leads to Demazure recursions.
• Lecture 11: Closed models.
• Lecture 12: Column parameters and parametrized Yang-Baxter equations.
• Lecture 13: Gamma and Delta ice, and hybrid models.
• Lecture 14: Classic Pipedreams: definitions and examples.
• Lecture 15: Proof using the Yang-Baxter equation that classic pipedreams represent Schubert polynomials.
• Lecture 16: Bumpless Pipedreams
• Lecture 17: Hybrid models

Links

• Solvable Lattice Models by notes in progress by Daniel Bump and Anne Schilling.
• Macdonald: Schubert Polynomials at LACIM, Montreal.

Graph Paper (for the lectures)

• Graph paper
• Crystal Graph paper
• Crystal Graph paper (with tableaux)
• Boltzmann weights and R-matrix for the open and closed models