From Littlewood-Richardson sequences to subgroup embeddings and back

TL;DR

This paper explores the relationship between Littlewood-Richardson sequences and subgroup embeddings in finite abelian p-groups, providing new insights and simplified proofs of existing theorems.

Contribution

It establishes a novel connection between LR-sequences of length two and subgroup embeddings, leading to streamlined proofs of Green and Klein's theorems.

Findings

LR-sequences of length two correspond to subgroup embeddings in finite abelian p-groups.

A subsequence of length 2 determines the entire LR-sequence.

New, simplified proofs of Green and Klein's theorems are derived.

Abstract

Let $\alpha$, $\beta$, and $\gamma$ be partitions describing the isomorphism types of the finite abelian $p$-groups $A$, $B$, and $C$. From theorems by Green and Klein it is well-known that there is a short exact sequence $0\to A\to B\to C\to 0$ of abelian groups if and only if there is a Littlewood-Richardson sequence of type $(\alpha,\beta,\gamma)$. Starting from the observation that a sequence of partitions has the LR property if and only if every subsequence of length 2 does, we demonstrate how LR-sequences of length two correspond to embeddings of a $p^2$-bounded subgroup in a finite abelian $p$-group. Using the known classification of all such embeddings we derive short proofs of the theorems by Green and Klein.