Observations on Symmetric Circuits

We study symmetric arithmetic circuits and improve on lower bounds given by Dawar and Wilsenach (ArXiv 2020). Their result showed an exponential lower bound of the permanent computed by symmetric circuits. We extend this result to show a simpler proof of the permanent lower bound and show that a large class of polynomials have exponential lower bounds in this model. In fact, we prove that all polynomials that contain at least one monomial of the permanent have exponential size lower bounds in the symmetric computation model. We also show super-polynomial lower bounds for smaller groups. We support our conclusion that the group is much more important than the polynomial by showing that on a random process of choosing polynomials, the probability of not encountering a super-polynomial lower bound is exponentially low.