RSK insertion and Knuth moves for Schubert polynomials

We introduce analogs of left and right RSK insertion for Schubert calculus of complete flag varieties. The objects being inserted are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. We adopt Lenart’s growth diagrams of permutations to give a combinatorial rule for Schubert structure constants in the separated descent case. We also study a class of biwords with an associativity property, called plactic biwords. We prove that any two plactic biwords with the same insertion bumpless pipe dream are connected by an analog of Knuth moves.