Course Syllabus

Course description:

In this course, we will study topics in combinatorics and representation theory. In particular, we will look into permutations, alternating sign matrices and relations to statistical mechanics, symmetric functions, and crystal bases.

Textbooks:

We will not follow a specific textbook, but here are some useful references:

• Bruce E. Sagan, "The symmetric group, Representations, combinatorial algorithms, and symmetric functions", Springer, second edition, 2001. 
• David M. Bressoud, "Proofs and confirmations", Cambridge University Press 1999
• William Fulton, "Young tableaux", London Mathematical Society, Student Texts 35, Cambridge University Text 1997
• Eric S. Egge, "An introduction to symmetric functions and their combinatorics", AMS 2019

Grading:

60% of the grade will consist of journaling, 40% of the grade will be based on the presentation.

• Journaling: Every Friday by 5pm, every student needs to hand in a journal entry about the topics discussed that week during lecture. During lecture some problems might be posed, which should be solved in the journal entry.
• Presentation: There will be projects to be presented in groups of about 3 students throughout the quarter. A list of topics will be made available.

If, for some reason, you cannot attend lecture on a particular day due to illness or other reasons, please provide a written note from your doctor or authority stating that you were unable to attend. Your journal entry for that lecture will then be excused and your grade will be based on the other journal entries.

Computing:

During class, I will illustrate some results using the open source computer algebra system SageMath. When you follow the link, you can try it out yourself using CoCalc. Or you can sign up for a Class Account with the math department. Log into fuzzy.math.ucdavis.edu and type the command `sage` to launch a Sage session in the terminal. 

Topics:

• Permutations, alternating sign matrices

• Six-vertex model , Gelfand-Tsetlin patterns and monotone triangles
• Strict Gelfand-Tsetlin patterns
• Partition function of the six-vertex model, Boltzmann weights, Vandermonde determinant
• Symmetric and alternating polynomials
• Symmetric functions and monomial symmetric functions
• Schur polynomials, semistandard Young tableaux
• Properties of the partition function
• q=1 gives determinantal definition of the Schur polynomial
• q=0 gives combinatorial definition of the Schur polynomial
• Crystal bases and crystal graphs

Resources for UC Davis students:

You can find a lot of University resources on this page.