This is the website and syllabus for the 2021 Semester 2 topics course Crystals: combinatorial algorithms and tensor categories.
Tensor categories (classes of mathematical objects with a notion of tensor product) have long been studies by mathematicians. In the 1980’s ideas from mathematical physics, in particular the search for solutions of the Yang-Baxter equation, motivated the desire to find interesting examples of such categories. The initial answer came from the Leningrad school of mathematicians working on quantum integrable systems who introduced quantum groups.
This class will be devoted to constructing a different (though very related) set of examples of interesting tensor categories, namely the category of crystals. A crystal is a purely combinatorial object and our attempt to understand them will give us the opportunity to learn about some of the jewels of algebraic combinatorics like the RSK algorithm, the Schützenberger involution, the Littlewood-Richardson coefficients and much more.
Officially: Algebra 1
Unofficially: The nature of the subject material means that strictly speaking, we will not assume very much background apart from some linear algebra and a modest amount of group theory. However, you will need a healthy dose of mathematical maturity to get something out of this course. We will be moving along quickly, and I will expect students to have the capacity to work out unfamiliar concepts using analogies to concepts you already know, or by looking them up for yourself. I highly recommend that you only take this class if you performed outstandingly in several HPC classes (the content of those courses is not so important). You can always discuss your situation with me.
There will be (roughly) fortnightly homeworks (%60 of your grade) and a (short) final exam (40% of your grade). Depending on interest, the final homework may involve a creative/artistic component.
Homework should be written up independantly, however collaboration is highly encouraged. We will frequently discuss homework problems in class and you are encouraged to ask me questions if something doesn’t make sense. Some problems will be difficult and/or open-ended and a complete solution isn’t necessary to acheive a perfect score. Effort and ingenuity will be rewarded!
Problem sets will be posted here:
The lecture notes will be updated regularly. We will follow a variety of sources:
The advertised topics below are liable to change.
I will collect here some reading that you can do on subjects realated to class material or extending class material