Beijing Institute of Mathematical Sciences and Applications

The course is devoted to the properties of symmetric functions. These functions are used in many areas of mathematics, see, for example, courses by Anatoli Kirillov, Anton Nazarov, Yevhen Makedonskyi, myself and others. But usually one has only time to state the corresponding facts and use them. The plan for this course is to address specifically the symmetric functions themselves.
We will talk about the main bases: elementary and complete symmetric functions, Schur functions, etc. And then discuss their basic properties: orthogonality, Jacobi-Trudy formulas, Pieri formulas, Kostka numbers, RSK-correspondence, Littlewood-Richardson rule. In the second part of the course we discuss the connections with the representation theory, with main focus on symmetric group and general linear group, including Schur-Weyl duality and it's generalizations.