KAIST Discrete Math Seminar

It is well-known that the ring of symmetric polynomials has a special basis, consisting of Schur polynomials and being indexed by Young diagrams. There underlies the representation theory of general linear groups and the combinatorics of Young diagrams. The geometry behind it is the geometry of Schubert varieties of Grassmannians of subspaces in a complex vector space. If we change the geometric setup to the case of Grassmannians of isotropic subspaces of a symplectic vector space, we have analogous functions called Schur Q-functions and their slight generalization. I will explain an explicit closed formula to express those functions as sums of Pfaffians and also the geometry and combinatorics behind it. If time allows, I will mention the technique of how we obtained the formula, that involves the so-called divided difference operators. This is a joint work with Takeshi Ikeda.