In representation theory of Lie algebras combinatorial tools play a very important role.
The combinatorics allows to describe some complicated algebraic structures in simple explicit terms. In many cases it also gives an algebraic meaning to combinatorial constructions and put them in a larger mathematical framework.
In this course we will focus on combinatorial aspects of representation theory, i.e. on explicit realizations of representations and related structures. An important part of it will be bijective correspondences. For example we will see how to parametrize a basis in an irreducible representation of by lattice paths.
We will start with the representation theory of Lie algebra , will study the Gelfand-Zetlin basis, and will see how the semistandard Young tableaux parametrize this basis. After this we will learn how to construct irreducible representations of the symmetric group using standard Young tableaux. Then we will focus on the Schur-Weyl duality between and the symmetric group . Here
the combinatorial tools will be standard and semistandard Young tableaux.
Other possible topics are RSK correspondence, Littelmann path model for and the combinatorics of lattice paths. If time permits, we will discuss quantized universal enveloping algebras and crystal bases of Kashiwara and Lustig.