Symmetries of Combinatorial Rings

这个项目研究来自具有对称性的物体的代数，例如古代研究的高度对称的柏拉图固体：立方体、四面体、八面体、十二面体和二十面体。几个世纪以来，人们已经了解到，这些物体不仅具有美丽的对称性，它们还产生了新的代数物体，称为环，继承了同样的对称性。了解这些环的对称性不仅在许多数学领域被证明是卓有成效的，而且在纠错码的应用方面也是如此，纠错码允许我们在天文距离上发送数据，同时消除噪声传输中的破坏。该项目还将涉及本科生和研究生的研究。具体地说，在这个建议中研究的具有对称性的环的范围从更多的交换环到更少的交换环。交换方案包括多项式环、Stanley-Reisner环和Varchenko-Gelfand环的不变理论。该方案还研究了反对易环上对称群的表示，如外代数和Orlik-所罗门代数。最后，它研究了一些非常非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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.