Tomaž Košir: Dokaz domneve o škatli za pare komutirajočih matrik / A proof of the Box Conjecture for commuting pairs of matrices

Povzetek: Iarrobino, Khatami, Van Steirtenghem in Zhao so leta 2014 postavili domnevo o škatli za pare komutirajočih matrik. Ta opisuje Jordanovo formo goste orbite v nilpotentnem komutatorju dane nilpotentne matrike. Dokaz domneve sloni na Burgeovi bijekciji med razčlembami in besedami na dveh generatorjih. Kot povezava med algebraičnim in geometričnim delom dokaza je uporabljen Shaymanov opis raznoterosti invariantnih podprostorov nilpotentne matrike. Dokaz je skupno delo z Johnom Irvingom in Mitjem Mastnakom.

Abstract: In 2014, Iarrobino, Khatami. Van Steirtenghem, and Zhao formulated the Box Conjecture for commuting pairs of matrices.
This describes the Jordan type of the dense orbit in the nilpotent commutator of a given nilpotent matrix. Our main tool in the proof is the Burge correspondence between the set of all partitions and a set of binary words. 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. The proof is a joint work with John Irving and Mitja Mastnak.