The socle tableau as a dual version of the Littlewood-Richardson tableau
Justyna Kosakowska, Markus Schmidmeier
TL;DR
The paper introduces the socle tableau as a dual counterpart to the Littlewood-Richardson tableau for embeddings of finite-length Λ-modules, proving that every socle tableau is realized by some embedding and that the socle tableau of an embedding uniquely determines, and is determined by, the dual LR-tableau through the Hom-matrix. It establishes a socle- Green–Klein type correspondence, connects tableau data to the Hom-matrix, and provides explicit conversion formulas between the socle tableau, the dual LR-tableau, and the Hom-matrix. The work also links these invariants to the position of objects in the Auslander-Reiten quiver and the representation-space geometry, and highlights a tableau-switching phenomenon that conjecturally converts the socle-tableau into the dual LR-tableau. Overall, this duality yields a unified framework for understanding embeddings via tableau combinatorics and representation-theoretic geometry. The results offer a new combinatorial handle on LR-coefficients through socle-tableaux, and reveal structural parallels between LR and socle data in module embeddings.$
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.