Symmetries of Combinatorial Rings

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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.