The entries in the LR-tableau

Let $\mathrm{\Gamma }$ be the Littlewood-Richardson tableau corresponding to an embedding $M$ of a subgroup in a finite abelian $p$-group. Each individual entry in $\mathrm{\Gamma }$ 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.