Magnetic interpretation of the Robinson–Schensted–Knuth algorithm

Abstract

Combinatorial aspects of the Robinson–Schensted–Knuth (RSK) algorithm have been discused in the context of a Heisenberg magnetic ring with N nodes, each with the spin s. Each magnetic configuration acquires a natural interpretation as a word of the length N in the alphabet of spins, consisting of n = 2s+1 letters. We demonstrate that the construction of n-tuple cover of the ring, with a separate copy for each letter of the alphabet of spins, allows for a transparent determination of maximal length of non-decreasing subwords. Moreover, it yields completeness of the RSK correspondence in classification of the irreducible basis of the Weyl duality between actions of unitary and symmetric groups in the space spanned on all magnetic configurations.