# MATH 246: Algebraic Combinatorics Winter 2019, UC Davis

 Lectures: MWF 5:10-6:00pm in Veihmeyer 116 Office hours: You can talk to me after every class! Otherwise drop me an e-mail to set up a time. I will also be in my office Mondays 2-3pm. Instructor: Anne Schilling, MSB 3222, phone: 554-2326, anne@math.ucdavis.edu Text: Richard P. Stanley, "Enumerative Combinatorics, Volume II" Cambridge Studies in Advanced Mathematics 62, Cambridge University Press 1999. Other very useful texts: William Fulton, "Young tableaux", London Mathematical Society, Student Texts 35, Cambridge University Text 1997 Bruce E. Sagan, "The symmetric group, Representations, combinatorial algorithms, and symmetric functions", Springer, second edition, 2001 I.G. Macdonald, "Symmetric functions and Hall polynomials", Oxford Science Publication, second edition, 1995 Prerequisites: MAT 245; or permission by instructor Grading: There will be regularly assigned homeworks, which are due every second Friday in class. Please hand in at least one problem from each homework set. Also, every student should present at least 2 problems in class in the course of the entire quarter. Problems are discussed every second Friday. The only way to really learn and grasp the material is to play and work with it yourself! Web: http://www.math.ucdavis.edu/~anne/WQ2019/246.html

### Course description

Algebraic combinatorics at the graduate level, covering the following main topics:
(1) The ring of symmetric functions
(2) Various bases of symmetric functions
(3) Combinatorial definition of the Schur function
(4) RSK algorithm
(5) Littlewood-Richardson rule
(6) Characters of the symmetric group
(7) Further topics (time permitting)

### Lecture topics

• Ring of symmetric functions; orders on partitions (ch. 7.1, 7.2)
• Monomial symmetric functions; elementary symmetric functions (ch. 7.3, 7.4)
• Complete symmetric functions; involution omega (ch. 7.5, 7.6)
• Power sums (ch. 7.7)
• Scalar product (ch. 7.9)
• Schur functions (ch. 7.10)
• Various representations of tableaux/RSK algorithm (ch. 7.10 and 7.11)
• RSK, Cauchy identity (ch. 7.11 and 7.12)
• Dual RSK; classical definition of Schur function (ch. 7.14, 7.15)
• Viennot's geometric interpretation of RSK (Sagan ch. 3.6)
• Littlewood-Richardson coefficients; Pieri rule (ch. 7.15)
• Jacobi-Trudi determinants; nonintersecting lattice paths (ch. 7.16 and Sagan 4.5)
• Knuth relations; jeu de taquin (ch. A1.1, A1.2, Sagan 3.4, 3.7)
• Differential Posets (Sagan 5.1)
• Murnaghan-Nakayama rule (ch. 7.17)
• Characters of the symmetric group (ch. 7.18)
• Littlewood-Richardson rule (Sagan 3.8 and 4.9)
• Further topics

### Sage Code on Symmetric Functions

Symmetric functions and asymptotics of number of partitions
Number of partitions with an even number of even parts
Tableaux and RSK in Sage

### Homeworks

Homework 1 due Friday January 18, 2019
Homework 2 due Friday February 1, 2019
Homework 3 due Friday February 15, 2019
Homework 4 due Friday March 1, 2019
Homework 5 due Friday March 15, 2019