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
Course Notes (written up by Eliot Hodges)
Potential Sources for Final Projects:
The intermediate orders of a Coxeter group https://arxiv.org/abs/2203.00405
Lattices of acyclic pipe dreams https://arxiv.org/abs/2303.11025
Rowmotion in slow motion https://arxiv.org/abs/1712.10123
Acyclic reorientation lattices and their lattice quotients https://arxiv.org/abs/2111.12387
The canonical complex of the weak order https://arxiv.org/abs/2111.11553
Hopf dreams and diagonal harmonics https://arxiv.org/abs/1807.03044
Permutrees https://arxiv.org/abs/1606.09643
Lorentzian polynomials https://arxiv.org/abs/1902.03719
Faces of generalized permutohedra https://arxiv.org/abs/math/0609184
Positroids and noncrossing partitions https://arxiv.org/abs/1308.2698
Dyck paths and positroids from unit interval orders https://arxiv.org/abs/1611.09279
Balance constants for Coxeter groups https://arxiv.org/abs/2005.09719
Castelnuovo–Mumford regularity of matrix Schubert varieties https://arxiv.org/abs/2111.10681
Permutohedra, associahedra, and beyond https://arxiv.org/abs/math/0507163
Homomesy in products of two chains https://arxiv.org/abs/1310.5201
Bargain hunting in a Coxeter group https://arxiv.org/abs/2302.04404
Odd diagrams, Bruhat order, and pattern avoidance https://arxiv.org/abs/2009.08865
Rhombic tilings and Bott–Samelson varieties https://arxiv.org/abs/1605.05613
Homotopy type of the Boolean complex of a Coxeter system https://arxiv.org/abs/0806.0906
Combinatorial descriptions of biclosed sets in affine type https://arxiv.org/abs/2207.05998
Toric partial orders https://arxiv.org/abs/1211.4247
Two poset polytopes https://link.springer.com/content/pdf/10.1007/BF02187680.pdf