A Proof of the Box Conjecture for Commuting Pairs of Matrices

We prove the Box Conjecture for pairs of commuting nilpotent matrices, as formulated by Iarrobino et al {IKvSZ}. This describes the Jordan type of the dense orbit in the nilpotent commutator of a given nilpotent matrix. Our main tool is the Burge correspondence between the set of all partitions and a set of binary words {Bur-1, Bur-2}. For connection with the algebraic and geometric setup of matrices and orbits we employ some of Shayman's results on invariant subspaces of a nilpotent matrix {Sha-1,Sha-2}. Our proof is valid over an arbitrary field.