Algebraic combinatorics and representation theory

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This proposal involves projects in the area of algebraic combinatorics. Many problems in combinatorics are concerned with counting objects. Algebraic techniques are often used to solve combinatorial problems and, conversely, combinatorial techniques are used to solve algebraic problems. Many of the projects in my proposal involve a notion called the cyclic sieving phenomenon (CSP), which was introduced in 2004 by Reiner, Stanton and White and involves using polynomial evaluations to unlock the symmetry properties of certain objects. A specific example of a CSP arises when considering handshake patterns. Consider a circular table at which an even number of people are seated. A handshake pattern is a way for all of the people seated at the table to shake hands without crossing arms. For instance, if 6 people are seated at a circular table, and they are all asked to shake hands with another person without crossing arms, there are five possible ways to do this, so there are five handshake patterns. If we number the places at the table 1,2,3,4,5,6, and start with a particular handshake pattern and then rotate the table, it will give us another handshake pattern. How many times would we have to rotate the table to return to the same handshake pattern? If we rotate the table a particular number of times, how many of the handshake patterns will remain fixed? Can a formula be produced that will predict the number of handshake patterns that will be fixed by a particular number of rotations? For the 6-person case, two of the handshake patterns are fixed when the table is rotated twice and three of the handshake patterns are fixed when the table is rotated three times. When more people are seated at the table it becomes difficult to count the possibilities, which is why it is desirable to produce formulae to do so. Suppose we generalize the problem and assume that there are 2n people seated at the table, where n is an arbitrary whole number. It turns out that the number of handshake patterns can be counted through nice formulae given by what are called Catalan numbers. The Catalan numbers can be used to give polynomials that, when evaluated at certain points, give the number of handshake patterns that remain fixed by a particular number of rotations of the circular table. The handshake patterns, together with the table rotation and the polynomial that arises from the Catalan number form a CSP. My research involves problems of this sort and the search for cyclic sieving phenomena.