Author(s): Steven V. Sam
Edition: version 2017-12-11
Year: 2017
Language: English
Commentary: Downloaded from https://www.math.ucsd.edu/~ssam/old/19W-201A/notes-old.pdf
1. Introduction
1.1. Prerequisites
1.2. Motivation
2. Representation theory
2.1. Schur–Weyl duality
2.2. Symmetric functions
2.3. Polynomial representations of general linear groups
2.4. Partitions
2.5. Bases for
2.6. Schur functors
2.7. Pieri's rule
2.8. Tensor categories
2.9. Categorical version of Schur–Weyl duality
2.10. Infinite number of variables
2.11. Littlewood–Richardson rule
2.12. A few more formulas
3. Representations of combinatorial categories
3.1. Twisted commutative algebras
3.2. Alternative models for tca's generated in degree 1
3.3. Bounded tca's
3.4. Noetherianity in general
3.5. Noetherian posets
3.6. Monomial representations and Gröbner bases
3.7. Gröbner categories
3.8. Example: FId-modules
4. Homological stability for symmetric groups
4.1. The complex of injective words
4.2. Nakaoka's theorem
4.3. Twisted homological stability
5. Representation stability for configuration spaces
5.1. Definitions
5.2. A spectral sequence
6. Algebraic geometry from tensors
6.1. Review of Zariski topology
6.2. Border rank
6.3. Hillar–Sullivant theorem
6.4. Proof of Theorem 6.2.2
6.5. Variants
7. More on FI-modules
7.1. Asymptotic combinatorial properties
7.2. Serre quotient categories
7.3. Generic FI-modules
7.4. Semi-induced FI-modules
7.5. Cohomology of FI-modules
8. -modules
8.1. Segre embeddings
8.2. -modules
Appendix A. Review of homological algebra
A.1. Exact sequences and exact functors
A.2. Derived functors
A.3. (Co)homology of groups
A.4. Spectral sequences
References
