One variant of Fulton's K-saturation conjecture is as follows:

$K_{\lambda/\mu,w} > 0 \Leftrightarrow K_{n\lambda/n\mu,n w} > 0$ for any integer $n>0.$

Here $K_{\lambda/\mu,w}$ denotes the Kostka numbers (number of skew SSYT of shape $\lambda/\mu$ and weight $w.$

This has been proved in various ways, (Knutson, Tao), so it is no longe a conjecture, but to me the proofs are quite involved. The proof shows the similar statement for Littlewood-Richardson coefficients, using K-hives and the above follows as a corollary.

**Question:** Is there an elementary proof of the above statement?
Could one expect a short proof of this?

equivalentto Horn's conjecture (this is a simple, but great, observation of Belkale) and Horn's conjecture has a very natural proof due to Purbhoo and Sottile. – Allen Knutson May 6 '13 at 13:43