Beijing Institute of Mathematical Sciences and Applications

Representation theory of symmetric groups is a fundamental example of finite group representations, with important connections to combinatorics, Lie theory (via Schur-Weyl correspondence), physics (quantum mechanics in particular), asymptotic analysis and various other areas. Classical methods treat each symmetric group separately, this course takes a more powerful, modern approach: we will study them as an inductive chain.

This approach allows us to proceed inductively, and we recover the parametrization of irreducible representations and their bases in a natural way. The key to this inductive procedure is the use of Jucys-Murphy elements, which generate a maximal commutative subalgebra — think of it as our version of a Cartan subalgebra, borrowing a powerful idea from Lie theory.

We will then construct seminormal and orthonormal forms of these representations. A careful look at the representation of the degenerate affine Hecke algebra H(2) will show the hidden structure behind the theory. This analysis opens the door to a similar treatment of Hecke algebras, which will be the final topic of the course.