Young Tableaux and Stratification of the Space of Square Complex Matrices

Abstract

A stratification of the manifold of all square matrices is considered. One equivalence class consists of the matrices with the same sets of values of rank(A − λ_iI)^j . The stratification is consistent with a fibration on submanifolds of matrices similar to each other, i.e., with the adjoint orbits fibration. Internal structures of matrices from one equivalence class are very similar; among other factors, their (co)adjoint orbits are birationally symplectomorphic. The Young tableaux technique developed in the paper describes this stratification and the fibration of the strata on (co)adjoint orbits.