The socle tableau as a dual version of the Littlewood-Richardson tableau

TL;DR

This paper introduces socle tableaux as a dual concept to Littlewood-Richardson tableaux, establishing their properties, relation to embeddings of subgroups in finite abelian p-groups, and their mutual determination.

Contribution

It demonstrates that every socle tableau can be realized by an embedding and that socle tableaux and LR-tableaux of dual embeddings uniquely determine each other.

Findings

Socle tableaux are characterized by weakly increasing rows and strictly increasing columns.

Each socle tableau corresponds to some embedding of a subgroup in a finite abelian p-group.

Socle tableaux and LR-tableaux of dual embeddings are mutually determinative.

Abstract

Like the LR-tableau, a socle tableau is given as a skew diagram with certain entries. Unlike in the LR-tableau, the entries in the socle tableau are weakly increasing in each row, strictly increasing in each column and satisfy a modified lattice permutation property. In the study of embeddings of a subgroup in a finite abelian $p$-group, socle tableaux occur as isomorphism invariants, they are given by the socle series of the subgroup. We show that each socle tableau can be realized by some embedding. Moreover, the socle tableau of an embedding and the LR-tableau of the dual embedding determine each other.