Fall 2023 MATH 155r Algebraic Combinatorics

This course will discuss several important classes of combinatorial objects such as permutations, set partitions, noncrossing partitions, posets, lattices, tableaux, and Dyck paths. Many of these objects can be associated in some way to symmetric groups. Symmetric groups are prototypical examples of Coxeter groups, which will be our primary focus in this class. Coxeter groups are central in modern algebraic combinatorics because they beautifully relate combinatorics, algebra, and geometry. We will discuss how several classical combinatorial objects associated to symmetric groups can be generalized to other Coxeter groups.

Meeting times: T & TH 1:30-2:45 PM 

Contact Information: 

Office: SC 235 

Email: colindefant@gmail.com

Office Hours: TH 3:00-4:00 PM or by appointment (this might change)

Textbook: Combinatorics of Coxeter Groups by Anders Björner and Francesco Brenti 

Syllabus

Course Notes (written up by Eliot Hodges)

Problem Set 1      (LaTex file)   (Solutions by Katherine Tung)

Problem Set 2    (LaTex file)

Problem Set 3    (LaTex file) 

Problem Set 4    (LaTex file)

Problem Set 5    (LaTex file) 

Recorded Lectures

Potential Sources for Final Projects:

1. The intermediate orders of a Coxeter group https://arxiv.org/abs/2203.00405

2. Lattices of acyclic pipe dreams https://arxiv.org/abs/2303.11025

3. Rowmotion in slow motion https://arxiv.org/abs/1712.10123

4. Acyclic reorientation lattices and their lattice quotients https://arxiv.org/abs/2111.12387

5. The canonical complex of the weak order https://arxiv.org/abs/2111.11553

6. Hopf dreams and diagonal harmonics https://arxiv.org/abs/1807.03044

7. Permutrees https://arxiv.org/abs/1606.09643

8. Lorentzian polynomials https://arxiv.org/abs/1902.03719 

9. Faces of generalized permutohedra https://arxiv.org/abs/math/0609184

10. Positroids and noncrossing partitions https://arxiv.org/abs/1308.2698

11. Dyck paths and positroids from unit interval orders https://arxiv.org/abs/1611.09279

12. Balance constants for Coxeter groups https://arxiv.org/abs/2005.09719

13. Castelnuovo–Mumford regularity of matrix Schubert varieties https://arxiv.org/abs/2111.10681

14. Permutohedra, associahedra, and beyond https://arxiv.org/abs/math/0507163 

15. Homomesy in products of two chains https://arxiv.org/abs/1310.5201

16. Bargain hunting in a Coxeter group https://arxiv.org/abs/2302.04404

17. Odd diagrams, Bruhat order, and pattern avoidance https://arxiv.org/abs/2009.08865

18. Rhombic tilings and Bott–Samelson varieties https://arxiv.org/abs/1605.05613

19. Homotopy type of the Boolean complex of a Coxeter system https://arxiv.org/abs/0806.0906 

20.  Combinatorial descriptions of biclosed sets in affine type https://arxiv.org/abs/2207.05998

21. Toric partial orders https://arxiv.org/abs/1211.4247

22. Two poset polytopes https://link.springer.com/content/pdf/10.1007/BF02187680.pdf