Alladi Ramakrishnan Hall

K-theory analogues of Schur’s P-and Q-functions

Takeshi Ikeda

Okayama University of Science, Japan

In 1911, I. Schur discovered a remarkable symmetric function known as Q-function in the study of projective representations of symmetric groups. Combinatorial aspects of Q-function has been intensively studied
too. Most notably, a connection to geometry was found by P. Pragacz who proved that the same function represents a Schubert class of the Lagrangian Grassmannian. Although this coincidence is still quite mysterious, H. Naruse and I managed to introduce an extension of Q- and P-functions in the context
of K-theory of Lagrangian Grassmannian and maximal orthogonal Grassmannian respectively. These functions called GQ- and GP-functions can be expressed as many different ways such as a sum of monomials related to “set-valued” shifted tableaux, and also as a single Pfaffian form that reminds us of Schur’s original definition of Q-functions. I am going to discuss the related results and open questions.