Combinatorics and Algebra

Alexander Miller

In the talk we will introduce the basics of representation theory of finite groups, and then look at Sn the symmetric group. We will find that one can answer combinatorial questions or hint at combinatorial phenomena by using a bit of algebra. This talk will be tremendously accessible, assuming only basic algebra and counting ability:-) Although the talk will start and build up in an accessible fashion, our destination will present a hard open problem that illustrates the power of using various algebraic/representation theoretic techniques. If you're at all considering the field of combinatorics, I strongly suggest you attend, because it contains key ideas that you'll undoubtedly run into later on. If you're not interested in combinatorics, I still strongly suggest you attend, for the mere fact that representation theory is a terribly powerful tool, used all over in math, including number theory (infinitely many primes in arithmetic progressions, say), group theory (Burnside's famous theorem that a group whose order is only divisible by two primes is solvable), combinatorics, physics, and even probability (see the work of Persi Diaconis, for example).

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