About two months ago, I wrote a study notes named *Symmetry and Harmonic Motion* on how to exploit symmetry of a system doing harmonic motion. I use a baby problem: two particles in 1D, to demonstrate it is possible to determine the eigenmodes from pure group theoretical analysis. The baby problem is too trivial to show the power of group theory, so today we continue with the kindergarten problem: three particles in 2D. To fully understand this study notes, it is recommended to go through my previous post about the baby problem, where various useful mathematical tools are introduced. Two of the tools are especially important for today’s discussion. The first is Schur’s lemma, discussed here, and the second is the projection operator into desired mode, discussed here.

I want to impose $Z_2$ global symmetry in the tensor network renormalization calculate. It will reduce the computational cost of the algorithm and will stabilize the renormalization group flow of tensor. The method is first discussed by Singh, Pfeifer and Vidal (2010) in a general context, and then later by Singh, Pfeifer and Vidal (2011) and by Singh, Vidal (2012), focused on abeliean and non-abeliean symmetry group respectively. I want to review the kindergarten problem here before diving into their discussion on tensor.

## The Kindergarten Problem: Three Particles in 2D

The problem is shown in the figure above. We have three particles connected by springs, moving in a plane. Imagine we start from equilibrium configuration, and focus on small deviation from it, so the dynamic variables are displacements from the equilibrium, as is shown in the figure. We can put them together to form a vector with $6$ components $\mathbf{x} = (x_a^1, x_a^2, x_b^1, x_b^2, x_c^1, x_c^2)$.