# Canon in algebraic combinatorics and how to study

1) In subjects such as algebraic geometry, algebraic topology there is a very basic standard canonical syllabus of things one learns in order to get to reading research papers.

Is there a similar canon in algebraic combinatorics? (e.g., does someone working in matroids have knowledge of symmetric functions and vice versa?)

2) I want to know how much of EC 1,2 does a regular algebraic combinatorics researcher know? Do I try to solve the vast breadth of problems (at least the ones with difficulty level less equal 3- let's say) in those two books? How about attempting at reading and solving Bourbaki's Lie Groups and Lie Algebras chapters 4-6? This seems like the most read book of Bourbaki, and a treasure trove of Coxeter Group-Root System material. How do I go about studying Macdonald's Symmetric Functions and Hall Polynomials? I mention these books because they appear to be listed as a reference in many of the papers I see. But these are enormous, and reading and solving problems from cover to cover is probably impractical (Is it?).

I want to know how people tackle these kind of classic references. As well as how to practically study algebraic combinatorics.

3) Can someone point out if there is a list of topics-books/notes/videos (similar to https://math.stackexchange.com/questions/1454339/undergrad-level-combinatorics-texts-easier-than-stanleys-enumerative-combinator but with topics such as matroid theory, Coxeter groups, crystal bases included)?

• Can the criminality be defeated? No. Should we fight against the criminality? Yes. The same with exercises in EC 1,2. Jun 18, 2020 at 8:45
• There is no established concept of what algebraic combinatorics is (though this isn't much different from other subjects: e.g., do Gröbner bases belong to algebraic geometry?). At best you can try to cluster mathematicians according to their joint knowledge. EC1-2 form a major cluster in the sense that someone who knows the material of one chapter is rather likely to know that of another; still, very few are really deeply familiar with the whole territory. Jun 18, 2020 at 16:30
• I think algebraic topologists and algebraic geometers might dispute your first sentence. Jun 18, 2020 at 16:33
• The "canonical" reference on crystal bases is now the eponymous book by Bump and Schilling. On Coxeter groups, most tend to recommend Björner/Brenti for a first combinatorial introduction (there are also notes by Heckman geared more towards geometers). Jun 18, 2020 at 16:51
• Matroid theory now has several long monographs devoted to it: Oxley, Welsh, White, White again... Jun 18, 2020 at 16:56

It's a very strange though not unusual idea that one must study the subject before starting to work in it. No, you really don't, at least not in combinatorics.

Let me clarify how the process works. You study the subject to be smart, to learn basic ideas, tools and techniques. For that you take a class and do the homeworks, with exercises carefully chosen by the instructor. The EC1, EC2 is an opus magnum — part monograph, part textbook, and part reference source. I don't think it's meant to be read cover-to-cover with an attempt to solve all exercises (note aside - there are also supplementary exercises).

When you are mature and confident enough, you start working on a problem. The problem will guide you to tools, ideas and concepts that you don't yet know. If they are from a chapter of EC that you have not studied, you read it and go through some exercises which seem relevant. If they are from a different area altogether (say, from commutative algebra), you read up on that. Then go back to the problem and hope your newly acquired tools will prove helpful.

It can happen that once you learn the true nature of the new tools you realize that they are inapplicable or too weak/general to be used for your purposes. You are then back to square one, enriched with some new knowledge which you might find useful later in your work. But it doesn't mean you have to study the whole area before starting to work on a problem.

Let me leave you with a quote from Béla Bollobás on a related point, see here.

For me, the difference between combinatorics and the rest of mathematics is that in combinatorics we are terribly keen to solve one particular problem by whatever means we can find. So if you can point us in the direction of a tool that may be used to attack a problem, we shall be delighted and grateful, and we’ll try to use your tool. However, if there are no tools in sight then we don’t give up but we’ll try to use whatever we have access to: bare hands, ingenuity, and even the kitchen sink. Nevertheless, it is a big mistake to believe that in combinatorics we are against using tools — not at all. We much prefer to get help from “mainstream” mathematics rather than use “combinatorial” methods only, but this help is rarely forthcoming. However, I am happy to say that the landscape is changing.

• Dear Igor, I noticed you were a faculty name on a rectangular tiling thesis...You might enjoy (and know references) for a three-dimensional tiling problem (not mine...) math.stackexchange.com/questions/3772410/… that generalizes a Martin Gardner problem Aug 2, 2020 at 18:40

I would suggest that much of what I said in my answer to another MO question applies here, in spades. To a first approximation, the canon is the empty set. Start with a problem, and learn what you need as you go along.