Matrix factorizations for domestic triangle singularities

Abstract

Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are integers at least two, satisfying $1/a+1/b+1/c>1$. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type $(a,b,c)$. Equivalently, in a representation-theoretic context, we can work in the mesh category of $\mathbb{Z}\tilde{\Delta }$ over $k$, where $\tilde{\Delta }$ is the extended Dynkin diagram, corresponding to the Dynkin diagram $\Delta =[a,b,c]$. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the $\mathbb{Z}$-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from $\{0,\pm 1\}$.