This course will focus on the representation theory of the symmetric group, most of which is highly combinatorial. After starting with basics of group representation theory, we will discuss combinatorial concepts such as Young tableaux, Specht modules, the Robinson--Schensted correspondence, jeu de taquin, the hook-length formula, and differential posets.
Meeting times: T & TH 12:00-1:15 PM
Contact Information:
Office: SC 235
Email: colindefant@gmail.com
Office Hours: TH 1:30-2:30 PM or by appointment
Textbook: The Symmetric Group (second edition) by Bruce Sagan
Problem Set 1 (Here is the Tex File.)
Problem Set 2 (Here is the Tex File.)
Problem Set 3 (Here is the Tex File.)
Problem Set 4 (TBA)
Problem Set 5 (TBA)
Potential Sources for Final Projects
(You are welcome to choose something that is not from this list!)
A few remarks on the octopus inequality and Aldous' spectral gap conjecture by Cesi
A new approach to the representation theory of the symmetric groups by Vershik and Okounkov
Generating a random permutation with random transpositions by Diaconis and Shashahani
Representation theory of the symmetric group in voting theory and game theory by Crisman and Orrison
Minimal skew semistandard tableaux and the Hillman–Grassl correspondence by Morales, Panova, and Park.
The Littlewood–Richardson rule and Gelfand–Tsetlin patterns by Doolan and Kim
Cyclic sieving, promotion, and representation theory by Rhoades
Promotion and evacuation on standard Young tableaux of rectangle and staircase shape by Pon and Wang
Growth diagrams, domino insertion and sign-imbalance by Lam
Rowmotion in slow motion by Thomas and Williams
Ornamentation lattices and intreeval hypergraphic lattices by Abram, Bastidas, Gelinas, Pilaud and Sack
Toric partial orders by Develin, Macauley, and Reiner
Two poset polytopes by Stanley
Permutrees by Pons and Pilaud
Promotion of Kreweras words by Hopkins
Differential posets by Stanley
d-Complete posets generalize Young diagrams for the jeu de taquin property by Proctor
Promotion and rowmotion by Striker and Williams