Hook-Lengths, Symplectic/Orthogonal Contents and Amdeberhan's Conjectures
arXiv:2404.01973 · doi:10.1112/plms.70026
Abstract
The symplectic/orthogonal contents of partitions are related to the dimensions of irreducible representations of symplectic/orthogonal groups. In 2012, motivated by Nekrasov--Okounkov's hook-length formula and Stanley's hook-content formula, Amdeberhan proposed several conjectures about infinite product formulas for certain generating functions of hook-lengths and symplectic/orthogonal contents. Some special cases of his conjectures were recently proved by Amdeberhan, Andrews and Ballantine. In this paper, we prove the general cases of Amdeberhan's conjectures.
18 pages, accepted for publication in Proceedings of the London Mathematical Society