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In this note (I guess presentation) which is about the relation between certain Jack polynomials and FQH states, Haldane states the following (paraphrasing slightly)

Generic incompressible FQHE droplets covering an area $2\pi N_\Phi$ are described by polynomial wavefunctions $f(z_1, \cdots, z_N)$ ($N$ number of particles) obeying both $$ \begin{aligned} &L^+f:=\left[\sum_{i=1}^N \frac{\partial}{\partial z_i}\right]f(z_1, \cdots, z_N)=0 && \text{highest weight condition}\\ &L^-f:=\left[N_\Phi(z_1+\cdots, z_N)-\sum_{i=1}^N z_i^2\frac{\partial}{\partial z_i}\right]f(z_1, \cdots, z_N)=0 && \text{lowest weight condition} \end{aligned} $$ This is the condition that the state is translationally-invariant when put on the Riemannn sphere.

In this paper section II, Haldane & Bernevig furthermore say that

Uniform states on the sphere satisfy the conditions $L^\pm \psi = 0$.

In the same paper the origin of why these condition are called highest/lowest becomes clear.

I'm looking for a reference for/proof of all of these facts:

  • Generic incompressible FQHE droplets covering an area $2\pi N_\Phi$ are described by polynomial wavefunctions $f(z_1, \cdots, z_N)$ ($N$ number of particles) obeying $L^\pm f = 0$.
  • The fact that the above condition is equivalent to saying the state is translationally-invariant when put on the Riemannn sphere.

  • And finally an explanation of what is meant by "Uniform states on the sphere satisfy the conditions $L^\pm \psi = 0$." (I'm guessing the single particle density obtained from the respective wavefunction is uniform on the sphere?) and furthermore why does $L^\pm \psi=0$ imply this assertion?

One thing that I know is a polynomial $f$ satisfies the highest weight condition if and only if for all $c\in \mathbb{C}$ we have $$ f(z_1+c, \cdots, z_N+c) = f(z_1, \cdots, z_N) $$ This is not hard to prove with a simple Taylor explansion and this implies translational invariance over $\mathbb{C}$. It is not exactly clear to me what's the lowest weight condition is about however. Neither I know how the rest of the conclusions are made.

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