The Knuth bijection, quantum matrices and free crystals

We explicitly compute the celebrated Robinson-Schensted-Knuth correspondence (RSK) between the set of the matrices with non-negative integer entries, and the set of the plane partitions. More precisely, in suitable linear coordinates on both sets, the RSK is expressed via minima of linear forms, i.e, in the piecewise-linear terms. In particular, we answer the following question by Curtis Grene: ``What shape corresponds to a given permutation under the Robinson correspondence? Our main tool in establishing these formulas is the quantum matrices and crystal bases. Based on the ideas of M. Kashiwara, we introduce the notion of free crystals, and build two such crystals on the set of all matrices with integer entries. The RSK is completely described as a canonical isomorphism between these two crystals.