#### Winter 2022/2023 V4A4 Representation Theory II

(Introduction to geometric representation theory)

**Announcements:**

*There will be no lecture on Friday, November 25. Due to this, Exercise Sheet #6 will be due the following week on Friday, December 2, and it will be longer than usual.*

**Instructor:** Prof. Dr. Catharina Stroppel

**Assistant:** Dr. Jacob Matherne

**Course content:** This course will give an introduction to some aspects of geometric representation theory. We will start by studying flag varieties and Grassmannians. Important in particular are the Schubert cells, which are the cells in some affine cellular decomposition of these varieties. Moreover, we will spend some time describing these varieties as varieties with a torus action. Then the connection to cohomology theories will be made. We will study them from a combinatorial, representation theoretic, and geometric point of view.
The geometric constructions will be connected with the combinatorics of Weyl groups and Coxeter groups.
In the second part, we will introduce the so-called Bott–Samelson varieties, which are certain resolutions of singularities arising in this context. If times allows, we will introduce equivariant cohomology and explain the important localization theorem and its relevance in representation theory.
The motto throughout the course will be that we want to use geometry to do representation theory.

**Lectures:**

Wednesdays 10–12 and Fridays 10–12 in the Zeichensaal at Wegeler Str. 10.

The lectures are planned to be held in person at the location above.

*The first lecture is on October 12.*

**Tutorials:**

The tutorials are planned to be held in person in the Neubau of the MI in Room N 0.007.

Tutorial times:

- Tuesday 8–10
- Tuesday 12–14

*The first tutorial will take place in the second week of the semester (October 17–21).*

*The distribution of the tutorial groups can now be found in the Sciebo folder, inside the folder "Material from tutorial".*

**Exam:**

*The details of the exam will be announced in the lecture.*

**Exercise sheets:**

Sheet | Due date | Comments |
---|---|---|

Exercise sheet 1 | Friday, October 21 at 10:15am | |

Exercise sheet 2 | Friday, October 28 at 10:15am | |

Exercise sheet 3 | Friday, November 4 at 10:15am | |

Exercise sheet 4 | Friday, November 11 at 10:15am | |

Exercise sheet 5 | Friday, November 18 at 10:15am | |

Exercise sheet 6 | Friday, December 2 at 10:15am | Note that you have two weeks for this sheet, and that it is longer than usual. |

Exercise sheet 7 | Wednesday, December 14 at 10:15am | Update: There have been substantial changes to Problem 3. Also, note that you have one and a half weeks for this sheet, but you will receive a new sheet on Friday, December 9 as usual. |

Exercise sheet 8 | Friday, December 16 at 10:15am | Problem 4(b) was deleted. (It was not the intended question and was unrelated to Problem 4(a).) |

Exercise sheet 9 | Friday, December 23 at 10:15am | |

Exercise sheet 10 | Friday, January 13 at 10:15am | |

Exercise sheet 11 | Friday, January 20 at 10:15am | |

Exercise sheet 12 | Friday, January 27 at 10:15am |

**Prerequisites for the course:** Basic knowledge on algebraic varieties and projective spaces and familiarity with representation theory. The material from courses like Algebra I and Algebra II are assumed to be known. Knowledge in Lie theory is required.

**Some references:**

- N. Chriss and V. Ginzburg - Representation theory and complex geometry (zbMATH Open link)
- W. Fulton - Young tableaux: With applications to representation theory and geometry (zbMATH Open link)
- V. Lakshmibai and J. Brown - Flag varieties: An interplay of geometry, combinatorics, and representation theory (zbMATH Open link)
- A. Björner and F. Brenti - Combinatorics of Coxeter groups (zbMATH Open link)