Intermediate symplectic characters and shifted plane partitions of shifted double staircase shape

TL;DR

This paper proves a conjecture on counting shifted plane partitions of a specific shape using intermediate symplectic characters, providing new character identities and generating functions.

Contribution

It introduces new character identities involving intermediate symplectic characters and derives generating functions for shifted plane partitions of shifted double staircase shape.

Findings

Proved Hopkins' conjecture on shifted plane partitions.

Derived a bialternant formula for intermediate symplectic characters.

Established generating functions for these partitions.

Abstract

We use intermediate symplectic characters to give a proof and variations of Hopkins' conjecture, now proved by Hopkins and Lai, on the number of shifted plane partitions of shifted double staircase shape with bounded entries. In fact, we prove some character identities involving intermediate symplectic characters, and find generating functions for such shifted plane partitions. The key ingredients of the proof are a bialternant formula for intermediate symplectic characters, which interpolates between those for Schur functions and symplectic characters, and the Ishikawa-Wakayama minor-summation formula.