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

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 $p$ -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 — a correspondence which appears to be related to the tableau switching studied by Benkart, Sottile and Stroomer. We illustrate how the entries in the socle tableau position the object within the Auslander–Reiten quiver. Like the LR-tableau, the socle tableau determines the irreducible component of the object in representation space.