Thursday 29 September

I'll give an update on my latest work with Allen Knutson concerning the application of quantum integrable systems to Schubert calculus. I'll present two new models for the structure constants of Schubert classes and their generalizations (motivic Segre classes).

We introduce rational symmetric functions generated by the Izergin-Korepin vertex model and discuss some properties of these functions. It is quite well-known [Borodin, 14] that rational symmetric functions derived from the six vertex model have several nice properties such as a Cauchy identity, symmetrization formula and orthogonality. The proof of these results stems from the underlying Yang-Baxter integrability. One can extend many of these results to the Izergin-Korepin model, and we discuss how different techniques which were used in the six vertex model can be generalized. This work is joint with Sasha Garbali and Michael Wheeler.

Symplectic Q-functions are a family of Laurent polynomials obtained by specializing t=-1 in the Hall-Littlewood functions associated to the root system of type C, and they can be considered as a type C analogue of Schur's Q-functions. In this talk, we explain they share many of the properties of Schur's Q-functions including a tableau formula. Also we discuss several positivity conjectures on expansion coefficients, such as the structure constants, involving symplectic Q-functions.

In the study of the projective representations of the symmetric groups, Schur introduced a family of symmetric functions, called the Q-functions. The aim of this talk is to present *geometric* applications of the Q-function and its variants. The first such connection to geometry was discovered by Pragacz in the cohomology ring of the maximal isotropic Grassmannians (both symplectic and orthogonal). When we consider the Grothendieck ring rather than the cohomology ring, a natural question is to find a (symmetric) function representing the structure sheaf of the Schubert variety in the corresponding Grassmannian. Such a function was given by Naruse and myself in 2013. Naruse and Nakagawa also generalized the theory of such functions to more general cohomology theory in particular for the K-homology. In the first part of the talk I will review the results on the classical Q-functions, and K-theoretic analogues too. The second part is about more recent topics on some variants of Q-functions related to the affine Grassmannian of the symplectic group for both (co)homology and K-theory.