Maki Nakasuji

Schur multiple zeta functions and their properties

Multiple zeta functions have been studied at least since Euler, who found many of their algebraic properties. In particular, they are greatly developed since the 1980s in several different contexts such as modular forms, mixed Tate motives, quantum groups, moduli spaces of vector bundles, scattering amplitudes, etc.

In this talk, we introduce a generalization of the Euler-Zagier type multiple zeta and zeta-star functions, that we call Schur multiple zeta functions. These functions are defined as sums over combinatorial objects called semi-standard Young tableaux. We will show the determinant formulas for Schur multiple zeta functions, which lead to quite non-trivial algebraic relations among multiple zeta and zeta-star functions. This is based on joint work with O. Phuksuwan and Y. Yamasaki. And we will also show relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras, which is a joint work with K. Matsumoto. Further, if time permits we will introduce Schur type poly-Bernoulli numbers and investigate their properties.