The entries in the LR-tableau

TL;DR

This paper explores the relationship between Littlewood-Richardson tableaux and subgroup embeddings in finite abelian p-groups, revealing how tableau entries encode homomorphism information and categorify subgroup embedding structures.

Contribution

It establishes a categorification of LR-tableaux via subgroup embeddings, linking tableau entries to homological properties in the category of subgroup embeddings.

Findings

Entries in LR-tableaux encode homomorphism information.

Subgroup embeddings sharing an LR-tableau have shared homological properties.

The work provides a categorification linking combinatorics and algebraic structures.

Abstract

Let $\Gamma$ be the Littlewood-Richardson tableau corresponding to an embedding $M$ of a subgroup in a finite abelian $p$-group. Each individual entry in $\Gamma$ yields information about the homomorphisms from $M$ into a particular subgroup embedding, and hence determines the position of $M$ within the category of subgroup embeddings. Conversely, this category provides a categorification for LR-tableaux in the sense that all subgroup embeddings corresponding to a given LR-tableau share certain homological properties.