Schedule:

• Tuesday, Jan. 14. José spoke about basics on the representation theory of finite-dimensional algebras: the radical, semisimple algebras, Schur's lemma. Also, the basics of the representations of groups over algebraically closed fields of characteristic zero. Exercises.

• Tuesday, Jan. 21. Alex started speaking about the representation theory of the symmetric group in characteristic 0. In particular, he produced a combinatorial basis in terms of paths in the branching graph, introduced the notion of weights (i.e. simultaneous eigenvalues for Jucys-Murphy elements) and finished by defining the rank 2 degenerate affine Hecke algebra.

• Tuesday, Jan. 28. Alex finished with the representation theory of the symmetric group. In particular, he showed how the representation theory of the rank 2 degenerate affine Hecke algebra gives severe restrictions on the weights that can appear in a representation, defined the combinatorial equivalence on weights and give a bijection between equivalence classes of weights and partitions, as well as between weights and standard Young tableaux. Exercises.

• Tuesday, Feb. 4. Milo started on Schur-Weyl duality. He proved that, given a vector space V, the actions of S_n and GL(V) on the n-fold tensor product of copies of V are centralizes of each other. This implies, in particular, that to each irreducible representation of S_n we can attach an irreducible (or zero) representation of GL(V) -- its multiplicity space in the n-fold tensor product. Next week, we will study these representations of GL(V) in detail.

• Tuesday, Feb. 11 Milo continued with the study of irreducible representations of GL(V). In particular, he showed that any polynomial representation can be embedded in the n-fold product of copies of V, so Schur-Weyl duality indeed gives every polynomial representation. Then he prepared the way to classify these irreducile representations via their highest-weight. More precisely, the representation decomposes as a direct sum of irreducible representations of the torus T of diagonal matrices, and isotypic components of these decomposition are known as weight spaces -- they are precisely simultaneous eigenvectors for all diagonal matrices.

• Tuesday, Feb. 18. José reaped the benefits of Milo's work from last week. In particular, we showed that polynomial irreducible representations of GL(V) are indexed by partitions with at most dim(V) parts. We also introduced fundamental weights (the highest weights of the wedge powers of V) and saw that the set of weights of a representation is closed under the action of the symmetric group of rank dim(V). Exercises.

• Tuesday, Feb. 25. Joseph computed the characters of irreducible representations of the symmetric group S_n. He did this by first computing the character of an induced representation, which is much easier. The character is given by certain coefficients in a symmetric polynomial. He also introduced Kostka numbers.

• Tuesday, March 3. Joseph computed the characters of the irreducible representations of the general linear group GL_n via Schur-Weyl duality. He also provided a formula for their q-dimension.

• Tuesday, March 10. José introduced Lie algebras, proved Schur-Weyl duality for the Lie algebra gl_n and introduced universal enveloping algebras, as well as a sketch of a proof of the PBW theorem.