Math 128, Spring 2014

Current problems in combinatorics:

Combinatorial representation theory of complex Lie algebras and related topics

### Logistics

**Professor:**Zajj Daugherty

**Office hours:**314 Kemeny, by appointment or drop-in (see calendar below).

**Class:**MWF 1:45-2:50 in 028 Haldeman Center

### Notes

**Main file**(last updated: 5/19)

TeX files for the notes: m128s14Notes.tex, preamble.tex

(Notes handed out on the first day)

**Slides:**

Monday 3/24:
Introduction, survey of the symmetric group Wednesday 3/26: Lie algebras and their enveloping algebras Friday 3/28: Representations and Hopf algebras Monday 3/31: Representations of sl_2 Wednesday 4/2: Finishing sl_2, semisimple Lie algebras, forms Friday 4/4: Jordan-Chevalley, Cartan subalgebras, weights, roots Monday 4/7: Some facts about roots Wednesday 4/9: More facts about roots Monday 4/14: More on roots Wednesday 4/16: A first look at root systems, reps of g Thursday 4/17: Representations of g continued Friday 4/18: Representations of g continued Monday 4/21: Representations of g continued |
Wednesday 4/30:
Bases and Weyl groups Friday 5/2: Weyl groups Monday 5/5: More on representation theory Wednesday 5/7: Weyl character formula Friday 5/9: Littelmann path model and crystals Monday 5/12: More on Littelmann path model and crystals Wednesday 5/14: Centralizer algebras Friday 5/16: Schur-Weyl duality Monday 5/19: Idempotents and the Temperley-Lieb algebra Wednesday 5/21: Diagram algebras Thursday 5/22: Quantum groups Friday 5/23: Braid groups and their quotients |

### Homework

Complete a minimum of five of the following.

- Due Friday 4/4: Some things about the classical Lie algebras. (.pdf) (.tex) (answers)
- Due Wednesday 4/9: Some things about sl
_{2}. (.pdf) (.tex) (answers) - Due Monday 4/14: Some things about NIBS forms. (.pdf) (.tex) (answers)
- Due Monday 4/21: Some things about roots. (.pdf) (.tex) (answers)
- Due Tuesday 4/29: Some things about mathematicians. [Credit given for CRT attendance] (.pdf) (.tex)
- Due Friday 5/2: Some things about weights and representations. (.pdf) (.tex) (answers)
- Due Friday 5/9: Some things about classification.
(.pdf)
(.tex)
(answers)

- Due Monday 5/19: Some things about crystals.
(.pdf)
(.tex)

- Due Monday 6/2: Some things about centralizer algebras.
(.pdf)
(.tex)

- Due Monday 6/2: Some things about combinatorial representation theory.
(.pdf)
(.tex)

Preamble for homework tex files.

### Calendar

As people make appointments with me for general help, I will add them here so that others can also drop in.

### Conferences

Discrete Math Day at Dartmouth, April 12.Conference on CRT at CRM, April 21-25.

### Papers and books

[CP94] A guide to quantum groups (Introduction), V. Chari and A. Pressley, Cambridge University Press (1994).[DRV1] Affine and degenerate affine BMW algebras: The center, Z. Daugherty, A. Ram, and R. Virk, Osaka J. Math., 51-1 (2014).

[DRV2] Affine and degenerate affine BMW algebras: actions on tensor space, Z. Daugherty, A. Ram, and R. Virk, Selecta Math., 19-2 (2013) 611--653.

[GP00] Characters of finite Coxeter groups and Iwahori-Hecke algebras, Sections 7.1--7.2, M. Geck, G. Pfeiffer, London Mathematical Society Monographs. New Series, 21. The Clarendon Press, Oxford University Press, New York, 2000.

[GW10] Symmetry, Representations, and Invariants, Sections 4.1--4.3, R. Goodman, N. Wallach, Graduate Texts in Mathematics, Springer, New York, 2010.

[HK02] Introduction to Quantum Groups and Crystal Bases, section 3, J. Hong and S-J. Kang, Graduate Studies in Mathematics, Vol 43, American Mathematical Society (2002).

[KR02] Representations of Graded Hecke algebras, C. Kriloff and A. Ram, Representation Theory (6) (2002), 31-69.

[Li95] Paths and root operators in representation theory, P. Littelmann, Ann. of Math. (2) 142 (1995), no. 3, 499--525.

[Na96] Young's orthogonal form for Brauer's centralizer algebra, M. Nazarov, Journal of Algebra 182 (3), 664-693.

[Ra06] Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux, A. Ram, Pure Appl. Math. Q. 2 (2006), no. 4, part 2, 963--1013.