The skew diagram poset and components of skew characters

TL;DR

This paper explores the structure of skew diagrams and their relation to skew characters, establishing bounds on components and constituents based on diagram convexity properties.

Contribution

It introduces a new poset structure on skew diagrams and derives bounds on skew character components using convex corner properties.

Findings

Skew diagrams with many convex corners are larger than disconnected single boxes.

Lower bounds are established for the number of components and constituents of skew characters.

The poset structure helps understand the combinatorial complexity of skew characters.

Abstract

We investigate the poset of skew diagrams ordered by adding or forming the union of skew diagrams. We will show that a skew diagram which has at least n convex corners to the upper left and also to the lower right is larger than the skew diagram consisting of n disconnected single boxes. Using this property, we obtain lower bounds for the number of components, constituents and pairs of components which differ by one box in a given skew character.