Representation theory: 797RT

The class meets on MoWe 8:40AM - 9:55AM

LGT 1334

My office: 1238 LGRT

Office Hours: Mo, We 2:30PM-3:30PM or by appointment

Course webpage: http://people.math.umass.edu/~oblomkov/RepTheory.html

** Overview **

We are planning to cover basics of representation theory as well as key methods.
The key examples that are covered in class are finite groups, $GL_n(\mathbb{C})$, $S_n$, $GL_2$ over finite field and quiver algebras.
If time permits, we cover theory of Soergel bimodules and applications to knot homology.

**Main Textbook **

Introduction to Representation Theory,
by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner

It is available online: https://math.mit.edu/~etingof/repb.pdf

**Other books and further reading**

Representations of $SL_2(F_q)$, by Cedric Bonnafe

Quiver Representations and Quiver Varieties, by Alexander Krillov Jr.

$SL_2(R)$, by Serge Lang.

Introduction to Soergel bimodules, by Ben Elias, Shotaro Makisumi, Ulrich Thiel, Geordie Williamson.

Syllabus

**Homework**

Homework will be assigned bi-weekly. We will have group solving sessions in class for the problem sets.
A convenient time for homework meeting will be discussed in class.

**Grades **

Grade is assigned based on homework: 60%, Final presentation: 30% and class discussion participation 10%.

** Presentation topics **

1.Representation theory of SL(2,F_q), Drinfeld curve. Source: Bonnafe, Representation theory of SL_2(F_q)

2. BMW algebras. Source: Survey by Geordie Williamson available at http://people.mpim-bonn.mpg.de/geordie/BMW.pdf

3. Quantum groups: U_q(sl_2) and R-matrices. Source: Cassel, Quantum groups

4. Compact Lie groups and Lie algebras exponential map. Source: Hall, Lie groups, Lie algebras and representations: an elementary intrduction

5. Lie algebra of type G_2, roots and representation theory. Source: Humphreys, Introduction to Lie Algebras and Representation theory

6. Lattices, E_8, classification of indefinite lattices. Source: Serre, Course of arithmetics.

7. Inductive approach to representatio theory of S_n. Source: Okounkov, Vershik https://arxiv.org/abs/math/0503040

8. Homology of Lie algebras, central extensions. Source: Weibel, An introduction to homological algebra.

9. Temperley-Lieb algebras, graphical calculus and Jones-Wentzel projectors. Source: Khovanov's thesis, https://www.math.columbia.edu/~khovanov/research/thesis.pdf