Algebraic combinatorics and representation theory

这项提议涉及代数组合学领域的项目。组合学中的许多问题都与计数对象有关。代数技术经常被用来解决组合问题，反过来，组合技术也被用来解决代数问题。*我提案中的许多项目都涉及循环筛选现象(CSP)的概念，该概念是由Reiner，Stanton和White在2004年提出的，涉及使用多项式计算来解锁某些对象的对称属性。*在考虑握手模式时，会出现一个特定的CSP示例。考虑一张圆桌，上面坐着偶数个人。握手模式是所有坐在桌子旁的人握手的一种方式，而不是交叉手臂。例如，如果六个人坐在一张圆桌旁，他们都被要求在不交叉手臂的情况下与另一个人握手，那么有五种可能的握手模式，所以有五种握手模式。*如果我们把桌子上的位置编号为1，2，3，4，5，6，然后从特定的握手模式开始，然后旋转桌子，它会给我们另一种握手模式。我们需要旋转桌子多少次才能回到相同的握手模式？如果我们将桌子旋转特定的次数，有多少握手模式将保持不变？能否得出一个公式来预测通过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，两个握手模式是固定的，当桌子旋转三次时，三个握手模式是固定的。当有更多的人坐在桌子旁时，就很难计算出可能性，这就是为什么需要制定公式来计算的原因。*假设我们推广这个问题，并假设有2n个人坐在桌子旁，其中n是任意整数。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，当在某些点上求值时，该多项式给出了通过圆桌的特定旋转次数保持固定的握手模式的数目。握手模式与桌子旋转和从加泰罗尼亚数字产生的多项式一起形成CSP。我的研究涉及到这类问题和对循环筛选现象的探索。

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.