Coxeter groups and extreme monomial ideals (Spring 2024)


Coxeter groups are a special type of group with finitely many generators and relations that generalize both the group of permutations of a finite set of objects and the group of symmetries of a regular polygon. They have many interesting connections to Lie algebras, geometry, and combinatorics, and importantly, the Word Problem is completely solved for Coxeter groups; that is, there is an algorithm for deciding when two seemingly different expressions in their generators are equal.


In this project, we will explore a construction of Constantinescu, Kahle, and Varbaro connecting Coxeter groups to some other interesting algebraic objects, ideals of polynomials generated by quadratic monomials in several variables that exhibit quite extraordinary behavior. Computing many examples with the assistance of the computer algebra system Macaulay2 will play an important role in gathering evidence for any combinatorial formulas discovered in the project. One of our first goals will be to implement the constructions of Constantinescu et al. for producing extreme quadratic monomial ideals in Macaulay2. We will then investigate what else can be said about the Hilbert series, primary decompositions, and Alexander duals of these ideals. 



For more information contact Matthew Mastroeni (mmastro@iastate.edu)

People:


Pre-requisites: