Justin Brown

Structure Theory of Complex Semisimple Lie Algebras

This chapter is on the structure theory of complex, semisimple Lie algebras. We give complete details for ( mathfrak{s}{{mathfrak{l}}_{n}} )(ℂ) and give a brief account for other semisimple Lie algebras. For details, see [26].

Representation Theory of the Symmetric Group

This chapter is on the representation theory of the symmetric group. We describe two constructions of irreducible S n -modules: Frobenius-Young construction and Specht module construction. For further details, we refer the reader to [17].

Singular Locus of a Schubert Variety in the Flag Variety SL n / B

In this chapter, we present the results on the singular locus of a Schubert variety in the flag variety. For an in-depth discussion on the singular locus of Schubert varieties for other semisimple algebraic groups, we refer the reader to [2].

Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory

In this chapter, we first introduce the Grassmannian variety and its Schubert varieties. We then present the details on the standard monomial theory (cf. Hodge [23, 24]); we also present a proof of “vanishing theorems.” We then sketch the details of standard monomial theory for the flag variety and its Schubert varieties. For further details, refer to [31].

Generalities on Algebraic Groups

In this chapter, we first discuss the generalities on algebraic groups — the Lie algebra of an algebraic group, Jordan decomposition in an algebraic group, etc. We then discuss the structure theory of connected solvable groups. We also introduce the variety of Borel subgroups. For details, refer to [5, 25].

Structure Theory of Reductive Groups

In this chapter, we discuss the structure theory of reductive algebraic groups, root systems and Bruhat decomposition in reductive algebraic groups. For details, refer to [5, 25].

Representation Theory of Semisimple Algebraic Groups

In this chapter, we discuss the representation theory of semisimple algebraic groups. We also sketch the construction of finite dimensional irreducible representations of semisimple algebraic groups. We further discuss the geometric realization of finite dimensional irreducible representations of a semisimple algebraic group (over ℂ).

Representation Theory of Complex Semisimple Lie Algebras

In this chapter, we discuss the representation theory of complex semisimple Lie algebras. While we give full details for ( mathfrak{s}{{mathfrak{l}}_{n}} )(ℂ), we only sketch the details for other semisimple Lie algebras. For more details, we refer the reader to [26].

Schur-Weyl Duality and the Relationship Between Representations of S d and GL n (ℂ)

In this chapter, we discuss the representation theory of GL n (ℂ). Schur modules are introduced and are shown to be irreducible GL n (ℂ)-modules, using Schur-Weyl duality. We then discuss the representation theory of SL n (ℂ), and deduce the representation theory of GL n (ℂ). For further details, refer to [17].