University of Minnesota Combinatorics Seminar: 2022

Date: 02/25/2022

Speaker: Andy Hardt

(Super)symmetric Functions from Solvable Lattice Models and Discrete-Time Hamiltonian Operators
Lattice models are ice-like rectangular grids that originated in statistical physics. To a given lattice model, one associates a partition function, which in the physics context denotes the energy of the system. In our context, these functions often have special properties such as symmetry and Cauchy identities. Many important functions in representation theory and Schubert calculus can be expressed as a lattice model partition function, including Schur polynomials, Macdonald polynomials, metaplectic Whittaker functions, LLT polynomials, Schubert polynomials, and Grothendieck polynomials. Lattice models achieve their full mathematical power when they are solvable, a feature which allows us to compute the partition function and ensures it has nice properties. We will discuss two different methods of solvability: the Yang-Baxter equation, and discrete-time Hamiltonians coming from Heisenberg algebras. Both of these techniques involve representations of quantum groups in key but very different ways. We will explore these solvability methods in the context of some important lattice models and discuss two surprising (and unexplained!) facts: In certain cases the solvability conditions for these two methods turn out to be very similar. The resulting partition functions tend to be "supersymmetric". Hamiltonian operators are interesting combinatorial objects in their own right, and we will spend some time discussing Thomas Lam's formalism for them. For a deeper exploration of lattice model basics, come to the pre-talk in the Student Combinatorics and Algebra seminar, on Thursday.

(Super)symmetric Functions from Solvable Lattice Models and Discrete-Time Hamiltonian Operators

Lattice models are ice-like rectangular grids that originated in statistical physics. To a given lattice model, one associates a partition function, which in the physics context denotes the energy of the system. In our context, these functions often have special properties such as symmetry and Cauchy identities. Many important functions in representation theory and Schubert calculus can be expressed as a lattice model partition function, including Schur polynomials, Macdonald polynomials, metaplectic Whittaker functions, LLT polynomials, Schubert polynomials, and Grothendieck polynomials.

Lattice models achieve their full mathematical power when they are solvable, a feature which allows us to compute the partition function and ensures it has nice properties. We will discuss two different methods of solvability: the Yang-Baxter equation, and discrete-time Hamiltonians coming from Heisenberg algebras. Both of these techniques involve representations of quantum groups in key but very different ways. We will explore these solvability methods in the context of some important lattice models and discuss two surprising (and unexplained!) facts:

In certain cases the solvability conditions for these two methods turn out to be very similar.
The resulting partition functions tend to be "supersymmetric".

Hamiltonian operators are interesting combinatorial objects in their own right, and we will spend some time discussing Thomas Lam's formalism for them. For a deeper exploration of lattice model basics, come to the pre-talk in the Student Combinatorics and Algebra seminar, on Thursday.