Multiplication theorems for self-conjugate partitions

TL;DR

This paper extends addition-multiplication theorems to self-conjugate partitions, enabling modular identities involving hook-lengths and BG-rank, using Littlewood decomposition techniques.

Contribution

It introduces addition-multiplication theorems for self-conjugate partitions and incorporates BG-rank, advancing classical hook-length identities in this subset.

Findings

Established addition-multiplication theorems for self-conjugate partitions

Derived modular hook-length identities for self-conjugate partitions

Utilized Littlewood decomposition to handle parity issues

Abstract

In 2011, Han and Ji proved addition-multiplication theorems for integer partitions, from which they derived modular analogues of many classical identities involving hook-length. In the present paper, we prove addition-multiplication theorems for the subset of self-conjugate partitions. Although difficulties arise due to parity questions, we are almost always able to include the BG-rank introduced by Berkovich and Garvan. This gives us as consequences many self-conjugate modular versions of classical hook-lengths identities for partitions. Our tools are mainly based on fine properties of the Littlewood decomposition restricted to self-conjugate partitions.