Algebraic combinatorics and representation theory

这项建议涉及代数组合领域的项目。组合学中的许多问题都与计算对象有关。代数技术通常用于解决组合问题，反过来，组合技术也用于解决代数问题。******我的提案中的许多项目都涉及到一个叫做循环筛分现象（CSP）的概念，它是由Reiner， Stanton和White在2004年引入的，涉及到使用多项式评估来解锁某些物体的对称性。******在考虑握手模式时，会出现CSP的一个特定示例。假设有一张双数人坐的圆桌。握手模式是指所有坐在桌边的人不交叉双臂握手的一种方式。例如，如果6个人坐在一张圆桌旁，他们都被要求与另一个人握手，但不交叉手臂，有五种可能的握手方式，所以有五种握手模式。******如果我们将表格中的位置编号为1,2,3,4,5,6，并从特定的握手模式开始，然后旋转表格，它将给出另一种握手模式。我们要旋转桌子多少次才能回到原来的握手方式？如果我们把桌子旋转一定的次数，有多少握手模式会保持不变？是否可以产生一个公式来预测特定旋转次数固定的握手模式的数量？对于6人的情况，当桌子旋转两次时，两种握手模式是固定的，当桌子旋转三次时，三种握手模式是固定的。当更多的人坐在桌子旁时，计算可能性就变得困难了，这就是为什么需要制定公式来计算可能性的原因。******假设我们推广这个问题，假设有2n个人坐在桌子旁，其中n是一个任意的整数。事实证明，握手模式的数量可以通过所谓的加泰罗尼亚数字给出的漂亮公式来计算。加泰罗尼亚数可以用来给出多项式，当在某些点求值时，给出握手模式的数量，这些模式通过圆形桌子的特定旋转次数保持固定。握手模式，与表旋转和由加泰罗尼亚数产生的多项式一起形成CSP。我的研究涉及这类问题和寻找循环筛分现象。

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.