Announcement
The main aim of this seminar is to read the last part of the book
- Kirrilov Jr. Quiver representation and Quiver varieties.
As the organizor, I hope to cover more topics besides the book including.
Participants
Liu. Lu. Ma. Wang. Wei. Xiong.
About the main text
> The 9th chapter involves Symplectic Geomtry and Geometric Invariant Theory (GIT)
> The 10th chapter defines Quiver varieties by GIT.
> The 11th is about Hilbert schemes, for which algebraic geomtry is required.
> The 12th is relative to the classification of singularities, so it also needs some algebraic geomtry.
> The 13th chapter will be easy if knowing some background of geometric representation theory.
Log
| 2020 Oct 25th | Liu | Geometric Invariant Theory (first half of Chapter 9) | |
| 2020 Nov 1st | Xiong | Symplectic Calculus (middle half of Chapter 9) | |
| 2020 Nov 8th | Xiong | Symplectic Calculus (last half of Chapter 9) | |
| 2020 Nov 15th | Xiong | Introduction to Springer Theory | slide |
The references for the Topics
> Hilbert schemes of points over surface
- Nakajima. Lectures on Hilbert schemes of points on surfaces
- Nakajima. More lectures on Hilbert schemes of points on surfaces [arXiv]
> Quiver representations
- Kirrilov Jr. Quiver representation and Quiver varieties.
- Derksen, Weyman. An Introduction to Quiver Representations.
- Reineke. Moduli of representations of quivers [arXiv]
Categorification
> General
- Mazorchuk. Lectures on Algebraic Categorification.
> Geometric realization of Hecke algebras
- Chiss, Ginzburg. Representation theory and Complex Geometry.
- Lusztig, Bases in equivariant K-theory
- Lusztig, Bases in equivariant K-theory II
- Lusztig, Equivariant K-theory and representations of Hecke algebras
- Lusztig, Kazhdan. Equivariant K-theory and representations of Hecke algebras II
- Kazhdan and Lusztig, Proof of Delign–Langlands Conjecture.
- Kazhdan and Lusztig. Representations of Coxeter groups and Hecke algebras.
- Kazhdan and Lusztig. Schubert varieties and Poincar´e duality.
> Geometric realization of Quantum groups
- Lusztig. Quantum groups.
- Kashiwara, Saito. Geometric Construction of Crystal Bases [arXiv]
- Varagnolo, Vasserot. Canonical bases and KLR-algebras.
> Geometric realization of Kac--Moody algebras
- Kirrilov Jr. Quiver representation and Quiver varieties.
- Nakajima. Quiver varieties and Kac-Moody algebras.
- Nakajima. Quiver varieties and finite-dimensional representations of quantum affine algebras.
- Ginzburg. Lectures on Nakajima varieties. [arXiv]
Geometric Foundations
> Intersection homology and perverse sheaves
- Goresky, MacPherson. Intersection homology I. [pdf]
- Goresky, MacPherson. Intersection homology II. [pdf]
- MacPherson. Intersection cohomology and Perverse Sheaves. [pdf]
- Lusztig. Intersection cohomology methods in representation theory. [pdf]
- Hotta, Takeuchi, Tanisaki. D-Modules, Perverse Sheaves, and Representation Theory.
- Etingof. Introduction to Algebra $mathcal{D}$-module. [pdf]
- Goresky. Introduction to Perverse Sheaves, lecture notes [pdf]
- Kirwan, Woolf, An introduction to intersection homology.
- Tsai. Introduction to Perverse Sheaves [pdf]
- Beilinson, Bernstein, Deligne and Gabber. Faisceaux pervers.
- Borel. Intersection Cohomology.
- Maxim. Intersection Homology & Perverse Sheaves. GTM 281.
- Ginzburg. Geometric methods in the representation theory of Hecke algebras and quantum groups. [arXiv]
- Jantzen. Moment graphs and representations. [pdf]
> Symplectic Geometry
- da Silva. Lectures on Symplectic Geometry. [pdf]
- McDuff, Salamon. Introduction to symplectic topology.
> Geometric Invariant Theory.
- Milne. Algebraic Groups.
- Milne. Reductive Groups. [pdf]
- Mumford. Geometric invariant theory.
- Mukai. An introduction to invariants and Moduli. [pdf]
TBA