Symmetries of Combinatorial Rings

组合环的对称性

• 批准号: 2053288
• 负责人: Victor Reiner
• 金额: $ 30万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2053288&HistoricalAwards=false
• 关键词: Symmetries; Combinatorial; Rings

This project investigates algebra that comes from objects with symmetry, such as the highly symmetric Platonic solids studied in antiquity: the cube, tetrahedron, octahedron, dodecahedron, and icosahedron. Over the centuries, people have understood that not only are these objects beautifully symmetric, they also give rise to new algebraic objects, called rings, inheriting the same symmetry. Understanding the symmetry of these rings has proven fruitful not only in many areas of mathematics, but also in applications to error-correcting codes, which allow us to send data over astronomical distances while removing corruptions from noisy transmission. The project will also involve undergraduate and graduate students in research.Specifically, the rings with symmetries studied in this proposal range from more commutative to less commutative rings. The commutative projects include invariant theory of polynomial rings, Stanley-Reisner rings, and Varchenko-Gelfand rings. The proposal also studies the representations of the symmetry groups on anti-commutative rings like exterior algebras and Orlik-Solomon algebras. Finally it studies group representations on some very non-commutative but still highly symmetric rings, such as Tits semigroup rings of hyperplane arrangements and Solomon descent algebras for reflection arrangements.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目研究来自对称物体的代数，例如古代研究的高度对称柏拉图立体：立方体、四面体、八面体、十二面体和二十面体。 几个世纪以来，人们已经认识到，这些物体不仅具有美丽的对称性，而且还产生了新的代数物体，称为环，继承了相同的对称性。 事实证明，理解这些环的对称性不仅在数学的许多领域，而且在纠错码的应用中都取得了丰硕的成果，纠错码使我们能够在天文距离上发送数据，同时消除噪声传输中的损坏。该项目还将让本科生和研究生参与研究。具体来说，本提案中研究的对称环范围从更多交换环到更少交换环。 交换项目包括多项式环不变理论、Stanley-Reisner 环和 Varchenko-Gelfand 环。 该提案还研究了反交换环上的对称群的表示，例如外代数和 Orlik-Solomon 代数。最后，它研究了一些非常不可交换但仍然高度对称的环上的群表示，例如超平面排列的 Tits 半群环和反射排列的所罗门下降代数。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

A Type B analog of the Whitehouse representation

白宫代表的 B 类类似物

• DOI: 10.1007/s00209-022-03200-7
• 发表时间: 2023
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Brauner, Sarah

Topology of Augmented Bergman Complexes

增强伯格曼复合体的拓扑

• DOI: 10.37236/10739
• 发表时间: 2022
• 期刊: The Electronic Journal of Combinatorics
• 作者: Bullock, Elisabeth;Kelley, Aidan;Reiner, Victor;Ren, Kevin;Shemy, Gahl;Shen, Dawei;Sun, Brian;Zhang, Zhichun Joy
• 通讯作者: Zhang, Zhichun Joy

Eulerian representations for real reflection groups

实反射群的欧拉表示

• DOI: 10.1112/jlms.12519
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: Brauner, Sarah