website (on UPenn's CANVAS)
Overview: This course is a graduate-level topics course in Combinatorics, this semester will focus on Algebraic Combinatorics. The course will start with the representation theory of the symmetric group and get into the theory of symmetric functions from combinatorial perspective, introducing objects like standard Young tableaux, Schur functions and bijections like the RSK algorithm etc. We will then cover more recently developed topics like Schubert polynomials and some applications to statistical mechanics.
Prerequisites: linear and abstract algebra (representation theory of finite groups), complex analysis, combinatorics (undergraduate level and familiarity with generating functions). This course is NOT a continuation of Math 580 and can be taken without having taken Math 580 first.
B. Sagan "The symmetric group" (main text)
R. Stanley "Enumerative combinatorics, volume 2" (specifically, Chapter 7)
L. Manivel "Symmetric functions, Schubert polynomials and degeneracy loci"
W. Fulton "Young tableaux"
Lecture 1-2: Review of representation theory of finite groups.
Lecture 3-5: Representations of the symmetric group.
Lecture 6--: Symmetric functions, combinatorial algorithms etc...