Flag Varieties

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Preliminaries

   V. Lakshmibai, Justin Brown

   Abstract

   This chapter is a brief review of commutative algebra and algebraic geometry. We have included basic definitions and properties. For details in commutative algebra, we refer the reader to [17] and in algebraic geometry to [28, 67].

3. Chapter 2. Structure Theory of Semisimple Rings

   V. Lakshmibai, Justin Brown

   Abstract

   This chapter discusses the structure theory of semisimple rings. For further details, the reader may refer to [57].

4. Chapter 3. Representation Theory of Finite Groups

   V. Lakshmibai, Justin Brown

   Abstract

   This chapter is on the representation theory of finite groups. For further details, we refer the reader to [23]. Throughout, G is a finite group of cardinality n and K is algebraically closed of characteristic 0, or of characteristic p where p does not divide n.

5. Chapter 4. Representation Theory of the Symmetric Group

   V. Lakshmibai, Justin Brown

   Abstract

   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 [23].

6. Chapter 5. Symmetric Polynomials

   V. Lakshmibai, Justin Brown

   Abstract

   This chapter is a brief account on symmetric polynomials. For details, we refer the reader to [64].

7. Chapter 6. Schur-Weyl Duality and the Relationship Between Representations of Sd and GLn (ℂ)

   V. Lakshmibai, Justin Brown

   Abstract

   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 [23].

8. Chapter 7. Structure Theory of Complex Semisimple Lie Algebras

   V. Lakshmibai, Justin Brown

   Abstract

   This chapter is on the structure theory of complex, semisimple Lie algebras. We give complete details for sl_n(ℂ) and give a brief account for other semisimple Lie algebras. For details, see [35].

9. Chapter 8. Representation Theory of Complex Semisimple Lie Algebras

   V. Lakshmibai, Justin Brown

   Abstract

   In this chapter, we discuss the representation theory of complex semisimple Lie algebras. While we give full details for sl_n(ℂ), we only sketch the details for other semisimple Lie algebras. For more details, we refer the reader to [35].

10. Chapter 9. Generalities on Algebraic Groups

    V. Lakshmibai, Justin Brown

    Abstract

    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, 36].

11. Chapter 10. Structure Theory of Reductive Groups

    V. Lakshmibai, Justin Brown

    Abstract

    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, 36].

12. Chapter 11. Representation Theory of Semisimple Algebraic Groups

    V. Lakshmibai, Justin Brown

    Abstract

    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).

13. Chapter 12. Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory

    V. Lakshmibai, Justin Brown

    Abstract

    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 [30, 31]); 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 [44].

14. Chapter 13. Singular Locus of a Schubert Variety in the Flag Variety SLn/B

    V. Lakshmibai, Justin Brown

    Abstract

    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].

15. Chapter 14. Applications

    V. Lakshmibai, Justin Brown

    Abstract

    In this chapter, we discuss two important applications of standard monomial theory. The first application is to classical invariant theory. The categorical quotients occurring in classical invariant theory (cf. [82]) have a natural identification with certain open subsets of Schubert varieties in a suitable G/P.

16. Chapter 15. Free Resolutions of Some Schubert Singularities.

    V. Lakshmibai, Justin Brown

    Abstract

    A classical problem in commutative algebra and algebraic geometry is to describe the syzygies of the defining ideals of interesting varieties. Let k ≤ n ≤ m be positive integers. The space D_k of m × n matrices (over a field K) of rank at most k is a closed subvariety of the mn-dimensional affine space of all m×n matrices.

17. Chapter 16. Levi Subgroup Actions on Schubert Varieties, and Some Geometric Consequences

    V. Lakshmibai, Justin Brown

    Abstract

    In this chapter, we consider certain Levi subgroup actions on Schubert varieties in the Grassmannian, and derive some geometric consequences on the singularities of Schubert varieties as well as sphericity consequences. To be more precise, let L _ω be the Levi part of the stabilizer Q _ω in GL _N (ℂ) (for left multiplication) of a Schubert variety X(ω) in the Grassmannian G _d,N .

18. Backmatter