主 题: Hook length formulas: Combinatorics and Number Theory
报告人: Prof. Guo-Niu Han (CNRS 法国国家科研中心 / 南开大学)
时 间: 2008-11-13 上午 10:30 - 11:30
地 点: 理科一号楼 1572
(Combinatorics) We introduce the hook length expansion technique
and explain how to discover old and new hook length formulas for
partitions. In particular, we derive an expansion
formula for the powers of the Euler Product in terms of partition
hook lengths, discovered by Nekrasov and Okounkov. We also obtain
an extension by adding two more parameters, which appears to be a
discrete interpolation between the Macdonald identities and the
generating function for t-cores.
(Number Theory) Our formula unifies the classical Jacobi and
Gauss identities when t=2. We proved some arithmetical properties
for t=3 and made a general conjecture. In the case t=5, the
conjecture implies the long standing Lehmer conjecture which says
that the Ramanujan tau-function never takes the zero value.
