Syllabus MAT 246: Algebraic Combinatorics Winter 2010

 Lectures: MWF 1:00-2:00pm in PHYSIC 140 Office hours: TBA Instructor: Monica Vazirani, MSB 3224, phone: 554-2596, mjvazirani@ucdavis.edu Text: Richard P. Stanley, "Enumerative Combinatorics, Volume II" Cambridge Studies in Advanced Mathematics 62, Cambridge University Press 1999. an online version is available here (tba). 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 The only way to really learn and grasp the material is to play and work with it yourself! There will be one or two midterm exams: The problems on the midterm will be (partially) based on the homework problems. Web: http://www.math.ucdavis.edu/~vazirani/W11/246.html SmartSite Links for this CRN: MAT 246 001 WQ 2011

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

Homeworks

Homework 1 due January 14
Homework 2 due January 28
Homework 3 due February 11
Homework 4 due February 25
Homework 5 due March 11 (?)/ still under construction

Matthew Stamps is took notes in 2009 available here.