Those who can do. Those who can’t…

…represent symmetric groups.

So, I’ve moaned before about my teaching load here (more than once in fact), but it’s not all bad.

First, a bit of a back story. In my dim and distant past I attended wonderful series of lectures delivered by the inimitable Stuart Martin on the representation theory of the symmetric group. At the time I really enjoyed it and come the examinations I even did pretty well in the subject.

When I dived into my PhD, however, I sadly drifted away from this beautiful and remarkable body of ideas.

Now, at about the time that I arrived in Colombia, one of my colleague was delivering a basic course on representation theory, you know, that one. Several of the students really enjoyed it and were asking to do a second course. The person who delivered the first one, however, was due to leave in September (remember the semester here starts at the beginning of August) meaning that if a second course was going to be delivered it would have to be given by…me.

So, there were students wanting to study some sort of second course on rep theory at exactly the time that I was trying to find something to think about/research/work on etc. Returning to those fondly remembered part III lectures seemed like an attractive ‘two birds with one stone’ idea (after all, if in trouble, getting back to basics is usually a good plan). After perusing the shelves of the library for a while, offering a reading course on the symmetric group seemed like a convenient and sensible idea.

Naturally, being a second course, it has attracted some of the better students. (As some of the indication of the quality of the students we are talking about I am teaching several these people.) Talking to them about the material has been really great fun and rewarding (and even pushed me into learning some really nice mathematics that I probably wouldn’t have met otherwise).

An example: one of the brightest students here, who happens to be doing my representation theory course, actually came to my office a couple of days ago asking about one of the exercises in the book regarding Catalan numbers. These are well known basically for counting things – lots of things. The particular question in the book in was asking the student to ‘use the results of this chapter’ (could they be much more vague?) to prove that the usual formular the nth Catalan number in fact counts the number of standard taleaux of shape (n,n). The chapter in question included the hook rule. Simply writing this rule in this special case on my white board immediately gave the answer. You could hear a pin drop – everyone in the room was blown away at how beautiful and simple this solution to the problem was.

A course that is a great pleasure to teach.

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2 Responses to “Those who can do. Those who can’t…”

  1. Colin Reid Says:

    I liked that course so much, I even approached Stuart Martin about becoming his PhD student. Unfortunately I didn’t get a good enough score in Part III to get funding at Cambridge. Life is full of what ifs…

  2. Peter Cameron Says:

    The hook formula is the quickest and nicest way to get the formula for the Catalan numbers. But you have to prove the hook rule first, and this is not so straightforward. Maybe you could turn this to advantage: find the formula for the Catalan numbers another way, and then use the above argument to motivate the hook formula before plunging into the proof.

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