#### Abstract

FJRW-theory is a recent advancement in singularity theory arising from physics. The FJRW-theory is a graded vector space constructed from a quasihomogeneous weighted polynomial and symmetry group, but it has been conjectured that the theory only depends on the weights of the polynomial and the group. In this thesis, I prove this conjecture using Poincaré polynomials and Koszul complexes. By constructing the Koszul complex of the state space, we have found an expression for the Poincaré polynomial of the state space for a given polynomial and associated group. This Poincaré polynomial is defined over the representation ring of a group in order for us to take G-invariants. It turns out that the construction of the Koszul complex is independent of the choice of polynomial, which proves our conjecture that two different polynomials with the same weights will have isomorphic FJRW rings as long as the associated groups are the same.

#### Degree

MS

#### College and Department

Physical and Mathematical Sciences; Mathematics

#### Rights

http://lib.byu.edu/about/copyright/

#### BYU ScholarsArchive Citation

Tay, Julian Boon Kai, "Poincaré Polynomial of FJRW Rings and the Group-Weights Conjecture" (2013). *All Theses and Dissertations*. 3604.

http://scholarsarchive.byu.edu/etd/3604

#### Date Submitted

2013-06-07

#### Document Type

Thesis

#### Handle

http://hdl.lib.byu.edu/1877/etd6228

#### Keywords

Poincaré polynomial, FJRW theory, Group-Weights conjecture, Koszul complex