MAT631

Algebraic Combinatorics

Information

Instructor: Dr Manjil P. Saikia

Contact: [email protected] (please only use this email address to send me emails related to this course)

Lecture Time & Place: Tuesdays 1600 to 1715 (Room 331, SAS) and Wednesdays 1315 to 1430 (Room 400, SAS)

Office Hours: Tuesdays 1715 to 1745 & Wednesdays 1430 to 1500 (both days in my office)

Attendance Policy: Attendance is mandatory, check institute’s attendance policy on AURIS (please note, we have a new policy from this semester). If you are low on attendance then you should talk to me (at least 3 weeks before the end-semester examination).

Course Details

Available on AURIS, we will follow the syllabus more or less, but there will be lot of digressions. This course is philosophically a sequel to MAT315/515 Combinatorial Enumeration, but do not worry if you have not taken that course as we will recall all the basic objects required for the present course. We will start with some basic combinatorial objects that will appear throughout the course, then move on to some tableau combinatorics. In particular, we will see a proof of the famed RSK correspondence and the hook length formula (two gems of modern combinatorics). The next portion of the course will cover symmetric functions in quite a lot of detail, and then if time permits we will do little bit of representation theory and commutative algebra.

Reading Materials

Textbook: The textbook that I will use is Algebraic Combinatorics and Coinvariant Spaces by Francois Bergeron (CRC Press, 2019). There is also a coursepack which will be distributed to the registered students, containing some other materials taken from standard textbooks. In addition, I will use material from other textbooks which I will list below as the course progresses.

  • For a nice proof of the generating function of major index, you can consult Chapter 1 of The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics by James Haglund (American Mathematical Society, 2008).
  • For the combinatorial proof of the generating function of major index, you can consult page 17 of The Art of Computer Programming, Volume 3 by Donald E. Knuth (Addison-Wesley, 1998).
  • For an introduction to posets, you can consult Chapter 3 of Enumerative Combinatorics, Vol. 1, 2nd. ed. by Richard P. Stanley (Cambridge University Press, 2011).
  • The tiling proof of Euler’s Pentagonal Number Theorem is due to Eichhorn, Nam, and Sohn (see this paper).

Grading

There will be regular problem sets assigned, which will contribute 60% of the total score. The problem sheets will be shared below. The end-semester examination will be 3 hour long written exam which will contribute 40% to the total score.

Grades of A and A- will be awarded based on the discretion of the instructor. If you solve all problems in the assignments and participate in the class then your grades will positively reflect that.

Lectures

W 06/01: Introduction to the course, permutation statistics
W 15/01: Monomials, Diagrams, Orders
T 21/01: Yound Diagrams, Partitions
W 22/01: Euler’s pentagonal number theorem, lattice paths

Problem Sets

  • The problem set will be updated over the course of the semester. I will announce in class when I post a set here.
  • You should only submit the designated problem(s), but are encouraged to try the rest as well.
  • To get the most out of this course, you are expected to spend atleast twice the amount of lecture hours on your own.

Problem Set 1
Problem Set 2
Problem Set 3

Submissions

  • Due time: 11:59pm of each due date
  • Must be typed in LaTeX and submitted as PDF or printed
  • Begin each solution on a new page
  • State your sources at the top of each problem (even if you worked independently)

Late Policy

  • Penalty: Late submissions will be penalized by 20% per each late day
  • Extensions: There will be no extensions given (unless you have a medical leave approved for the day before the due date and the day of the due date)

Collaborations

  • You are encouraged to first work on the problems independently before seeking collaboration.
  • Meaningful collaboration is allowed and is encouraged for this course.
  • You must write up your own solutions.

Acknowledging collaborators and sources

It is required to acknowledge your sources (even if you worked independently).

  • At the beginning of the submission for each problem, write Collaborators and sources: followed by a list of collaborators and sources consulted (people, books, papers, websites, software, etc.), or write none if you did not use any such resources.
  • Failure to acknowledge will result in an automatic 20% penalty per problem.
  • Acceptable uses of resources include: looking up a standard theorem/formula/technique; using Wolfram Alpha/Mathematica/Python for a calculation (no need to mention lectures or book draft)
  • Unacceptable uses of resources include: directly looking up the problem online or in the research literature for a solution. (Once you have solved a problem, it is fine to seek and learn alternate solutions.)

Intentional violations of the above policies may be considered academic dishonesty/misconduct.

(Thanks to Professor Yufei Zhao for his extensive course policy page, from which I have created my own policies.)