Combinatorics is the art of counting, how many such objects are there. Algebra deals with how objects can interact. Representation theory sits between the two. In particular, it uses combinatorial techniques to prove algebraic questions. Herein I use it to derive information about the symmetric group, S_n, by proving a combinatorial identity. In mathematics though, we always seek the strongest possible theorem, the broadest result. Thus it is natural to consider here not only S_n, but also several other related diagram algebras. The conjecture and part, but not all, of the proof will generalize.
After introducing the necessary definitions, background, and tools, the second chapter, the heart of the paper, contains a proof of the main conjecture. The two main objects of study are the Roichman and Saxl weights, which are functions on permutations. In particular, I prove that the sum of the two weights over all symmetric permutations, those which are their own inverse, is equal and give an explicit calculation for it. Since the weights are always +1, -1 or 0, much of the proof will be done by finding and cancelling pairs of elements with opposite sum. This part is aimed at a general audience and requires little background. Once the central result has been proven, I turn to the deeper algebraic background of the problem and explain why it is important. In particular, this result allows us to construct a model which tells us a great deal about S_n.
This leads us naturally to consider the general case, where we state the version of the conjecture. In order to deal with it, we will need several new concepts. These and their properties of which will be introduced next, comparing the general ones to how they specialize for the symmetric case; indeed these will often be identical. We can then give partial results toward its proof. While I am unable to prove the entire result, I solve a special case and point out possible approaches to finish the problem.
This work arose out of a summer research project with my advisor, Tom Halverson, and was continued during the 2005-06 academic year. I am indebted to him for everything, to the NSF for supplying funding, and to the entire math department at Macalester for instilling in me a passion for mathematics and the ability to pursue it.
Decker, Michael D., "A Model Representation for the Symmetric Group and the Partition Algebra" (2006). Mathematics, Statistics, and Computer Science Honors Projects. 2.
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