The Knuth bijection, quantum matrices and free crystals
refreshments at 3:45pm
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.
Speaker's Contact Info: arkady(at-sign)math.cornell.edu
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