Algebraic combinatorics and representation theory

代数组合学和表示论

• 批准号: RGPIN-2018-05877
• 负责人: Stokke, Anna
• 金额: $ 1.31万
• 依托单位: University of Winnipeg
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676834
• 关键词: Algebraic; combinatorics; representation; theory

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.

该提案涉及代数组合学领域的项目。 组合数学中的许多问题都与计数对象有关。 代数学技术通常用于解决组合问题，相反，组合技术用于解决代数问题。我的提案中的许多项目都涉及一个称为循环筛选现象（CSP）的概念，该概念是由Reiner，Stanton和白色在2004年提出的，涉及使用多项式评估来解锁某些对象的对称性。 ** 在考虑握手模式时，会出现CSP的一个特定示例。 考虑一个圆形的桌子，上面坐着偶数个人。 握手模式是一种让所有坐在桌子旁的人握手而不交叉手臂的方式。 例如，如果6个人坐在一张圆桌旁，并且他们都被要求与另一个人握手而不交叉手臂，则有五种可能的方式来做到这一点，因此有五种握手模式。*如果我们把桌子上的位置编号为1，2，3，4，5，6，并从一个特定的握手模式开始，然后旋转桌子，它会给我们另一个握手模式。 我们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？能否产生一个公式来预测由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，握手模式中的两个是固定的，并且当桌子旋转三次时，握手模式中的三个是固定的。 当更多的人坐在桌子上时，很难计算可能性，这就是为什么需要制定这样做的公式。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是一个任意整数。 事实证明，握手模式的数量可以通过所谓的加泰罗尼亚数给出的漂亮公式来计算。 卡塔兰数可以用来给出多项式，当在某些点上求值时，给出通过循环表的特定旋转次数保持固定的握手模式的数量。 握手模式，连同表旋转和从Catalan数产生的多项式一起形成CSP。 我的研究涉及这类问题和对循环筛分现象的研究。