# Summer Graduate School

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Location: |
MSRI: Simons Auditorium, Atrium |

This summer school will give an introduction to **representation stability**, the study of algebraic structural properties and stability phenomena exhibited by sequences of representations of finite or classical groups -- including sequences arising in connection to hyperplane arrangements, configuration spaces, mapping class groups, arithmetic groups, classical representation theory, Deligne categories, and twisted commutative algebras. Representation stability incorporates tools from commutative algebra, category theory, representation theory, algebraic combinatorics, algebraic geometry, and algebraic topology. This workshop will assume minimal prerequisites, and students in varied disciplines are encouraged to apply.

**Suggested prerequisites:**

Representations of finite groups over C and the classification of Sn-irreps

Representation Theory of Finite Groups:

- Fulton--Harris, "Representation Theory, A first course", Chapters 1-3

- Serre, "Linear representations of finite groups", Parts I and II

Representations of S_n:

- Fulton--Harris, "Representation Theory, A first course", Chapter 4

- James, "The representation theory of symmetric groups"

Commutative algebra (Noetherian rings, tensor product, free resolutions)

Tensor products:

- Dummit--Foote, "Abstract Algebra", Chapter 10.4

- Atiyah--MacDonald, "Introduction to Commutative Algebra", Chapter 2

Noetherian rings:

- Dummit--Foote, "Abstract Algebra", Chapter 15.1

- Atiyah--MacDonald, "Introduction to Commutative Algebra", Chapter 6-7

Gröbner bases

- Dummit--Foote, "Abstract Algebra", Chapter 9.5-9.6

- Cox--Little--O'Shea, "Ideals, Varieties, and Algorithms", Chapters 2.1-2.6

- Eisenbud, "Commutative algebra" Chapter 15

Representation theory of GLnC

- Henderson, "Representations of Lie Algebras", whole book

- Fulton--Harris, "Representation Theory, A first course", Chapter 15

Homological algebra (Tor, Ext, derived functors)

- Dummit--Foote, "Abstract Algebra", Chapter 10.5, 17.1

- Rotman, "An Introduction to Homological Algebra", Chapter 6.1-6.2 & 7.1-7.2

- Weibel, "An introduction to homological algebra", Chapters 2, 3

Topology (homology and cohomology of spaces and/or groups)

Spaces:

- Hatcher, "Algebraic Topology", Chapters 2-3

Groups:

- Brown, "Cohomology of Groups", Chapters I-III

- Dummit--Foote, "Abstract Algebra", Chapter 17.2

- Weibel, "An introduction to homological algebra", Chapters 6, 7

Symmetric functions

- MacDonald, "Symmetric Functions and Hall Polynomials", Chapter I

- Stanley, "Enumerative Combinatorics, Vol 2", Chapter 7

Category theory

- Mac Lane, "Categories for the Working Mathematician", Chapter

I.1-I.5, II.1-II.4, & IV.1-IV.4.

Backgroud articles:

- Church--Ellenberg--Farb, "FI-modules and stability for

representations of symmetric groups"

- Sam--Snowden, "Introduction to twisted commutative algebras"

- Draisma, "Noetherian up to symmetry"

For **eligibility** and **how to apply**, see the **Summer Graduate Schools homepage**

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

Representation stability

Representation theory

homological stability

functor categories

Gröbner methods

Noetherian

group cohomology

Schur-Weyl duality

twisted commutative algebra

FI-module

VI-module

VIC-module

pure braid groups

hyperplane arrangements

mapping class groups

Moduli space

Torelli groups

configuration spaces

congruence subgroups

Deligne categories

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**

13P10 - Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

14M15 - Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]

14M17 - Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]

05E10 - Combinatorial aspects of representation theory [See also 20C30]