Beijing Institute of Mathematical Sciences and Applications

A symmetric function is, roughly, a function of variables (x_1,x_2, ....) that is unchanged when the variables are permuted.
For example,
x_1 + x_2 + x_3,
x_1 x_2 + x_1 x_3 + x_2x_3,
x_1 x_2 x_3
are symmetric.

The subject began with polynomial equations: the coefficients are the elementary symmetric functions of the roots are the elementary symmetric functions of the roots.
But eventually it became a common language connecting
algebra(representation of symmetric group, characters of representations of many algebras),
combinatorics(Young tableaux, plane partitions, permutations, lattice paths and tilings),
geometry and topology(cohomology of Grassmannians, Schubert calculus, cohomology and \(K\)-theory of flag varieties),
probability(random matrices, random tilings and plane partitions, interacting particle systems such as TASEP and \(q\)-TASEP;
asymptotic representation theory, KPZ universality, Schur measures and Schur processes),
and mathematical physics(tau functions of the integrable hierarchies, eigenfunctions of quantum many-body systems, partition functions of vertex models).


During the nineteenth century, symmetric functions became objects of study in their own right.
In the 20th century the subject was absorbed into algebraic combinatorics and representation theory, and then by algebraic geometry.
Over the last half century, symmetric functions have become an important part of mathematical physics and probability theory.


Nowadays, an active area of research in symmetric functions involves studying them through certain vertex models—statistical physics models that possess integrability properties.
This connection allows to prove properties and identities for symmetric functions in an elegant way,
finding new properties of symmetric functions, defining new classes of symmetric functions, and also investigating the properties of the latter.


In this course, I want to show how the theory of vertex models naturally becomes a generating theory for the theory of symmetric functions. In the sense that simple and natural identities for the former generate properties and identities for the latter.
The main examples of symmetric functions we will consider are Schur polynomials, Grothendieck polynomials, and Hall-Littlewood polynomials, which relate to and 5- and 6-vertex models on the vertex models side.