A combinatorial proof of the skew K-saturation theorem

We give a combinatorial proof of the skew version of the K-saturation theorem. More precisely, for any positive integer kk, we give an explicit injection from the set of skew semistandard Young tableaux with skew shape kλ/kμkλ/kμ and type kνkν, to the set of skew semistandard Young tableaux of shape λ/μλ/μ and type νν.Based on this method, we pose some natural conjectural generalizations on related problems.