Rsk via Local Transformations

RSK is normally defined in terms of a somewhat complicated procedure called row-insertion. This definition unfortunately masks the remarkable symmetry property that if A RSK 7−−−→ (P,Q) then At RSK 7−−−→ (Q,P ). Here we present an alternate description of RSK via local transformations that makes the symmetry obvious. Our approach is thus similar in spirit to the growth diagrams of Fomin [6]. In order to motivate this description, we first examine the Gelfand-Tsetlin patterns associated to the Young tableaux that result from RSK, and show how RSK can be thought of as a map between matrices.