Math 128, Spring 2014
Current problems in combinatorics:
Combinatorial representation theory of complex Lie algebras and related topics
LogisticsProfessor: Zajj Daugherty
Office hours: 314 Kemeny, by appointment or drop-in (see calendar below).
Class: MWF 1:45-2:50 in 028 Haldeman Center
NotesMain file (last updated: 5/19)
TeX files for the notes: m128s14Notes.tex, preamble.tex
(Notes handed out on the first day)
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
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
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 sl2. (.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.
- Due Monday 5/19: Some things about crystals.
- Due Monday 6/2: Some things about centralizer algebras.
- Due Monday 6/2: Some things about combinatorial representation theory.
Preamble for homework tex files.
CalendarAs people make appointments with me for general help, I will add them here so that others can also drop in.
ConferencesDiscrete 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).
[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.