Matrix majorizations and their endomorphisms

Vector and matrix majorizations are a wide class of important relations on linear spaces and algebras. This theory dates back to Muirhead 1903 and Lorenz 1905. It was later developed by Hardy, Littlewood and Polya. Modern theory of majorization has many applications in various branches of mathematics, economics and many other areas. At the same time, this theory contains many important algebraic questions. We investigate various majorization relations of matrices and matrix families and their geometric and combinatorial characterizations. In addition, we provide characterizations of linear operators preserving or converting majorizations.