An extended generalization of RSK via the combinatorics of type $A$ quiver representations

The classical Robinson--Schensted--Knuth correspondence is a bijection from nonnegative integer matrices to pairs of semi-standard Young tableaux. Based on the work of, among others, Burge, Hillman, Grassl, Knuth and Gansner, it is known that a version of this correspondence gives, for any nonzero integer partition $\lambda$, a bijection from arbitrary fillings of $\lambda$ to reverse plane partitions of shape $\lambda$, via Greene--Kleitman invariants. By bringing out the combinatorial aspects of our recent results on quiver representations, we construct a family of bijections from fillings of $\lambda$ to reverse plane partitions of shape $\lambda$ parametrized by a choice of Coxeter element in a suitable symmetric group. We recover the above version of the Robinson--Schensted--Knuth correspondence for a particular choice of Coxeter element depending on $\lambda$.