Combinatorial proof of the skew K-saturation theorem

TL;DR

This paper presents a combinatorial proof of the skew K-saturation theorem, establishing an explicit injection between certain skew semistandard Young tableaux, and proposes conjectural refinements based on this method.

Contribution

It provides the first combinatorial proof of the skew K-saturation theorem and introduces explicit injections for skew semistandard Young tableaux.

Findings

Established an explicit injection between skew tableaux of scaled and original shapes.

Proposed conjectural refinements for related combinatorial problems.

Enhanced understanding of the structure of skew semistandard Young tableaux.

Abstract

We give a combinatorial proof of the skew Kostka analogue of the K-saturation theorem. More precisely, for any positive integer k, we give an explicit injection from the set of skew semistandard Young tableaux with skew shape k\lambda/k\mu{} and type k\nu{} to the set of skew semistandard Young tableaux of shape \lambda/\mu{} and type \nu. Based on this method, we pose some natural conjectural refinements on related problems.