UC Davis

Given a simplicial complex $\Delta$ and its barycentric subdivision $\sdd$, we explore two homogeneous systems of parameters: one is the elementary symmetric functions inside the $\N$-graded Stanley-Reisner ring of $\Delta,$ $\k[\Delta],$ and the other is written based on a balanced coloring of $\sdd$ which lives inside the $\N^d$-graded Stanley-Reisner ring of $\sdd,$ namely, $\k[\sdd].$ The first is stable under symmetries and the other is stable under colorful automorphisms of $\sdd.$ In this paper, we develop methodology to explore comparing the resolutions of $\k[\Delta]$ and $\k[\sdd]$ over their respective parameter rings. We then prove it in the trivial case, which is simply the coinvariant algebra with alternate grading and one non-trivial case.