Jack polynomials. For example the $(k,r,N)$-admissible representations (labeled by certain restricted Young diagrams) is related to the filling number $\nu=\frac{k}{r}$ FQHE ground state[@azuma1994explicit; @bernevig2008generalized; @bernevig2008properties; @estienne2010clustering; @estienne2010electron; @estienne2009relating; @feigin2002differential; @ha1994exact; @iso1995collective; @lee2014construction]. From our viewpoint the basic ingredients are Jack polynomials and additional restrictions, mostly from the fusion rule (which we do not explore in this article), will rule out some Jack polynomials systematically which results in the admissible representations.

The Halperin state, corresponding to the two-layer FQHE, shows a secret integrability dominated by also the triangularization, which says the number of boxes in Young diagram for the first layer always decreases while the one for the second layer increases and the total boxes of these two layers remain the same. The triangularization interaction, being triple in two kinds of bosonic operators, is quite complicated. It makes the explicit solution of the excitations slightly difficult. Nevertheless, since the triangularization is clear, we can give the explicit solution of the system in principle.

This article is organized as follows. In sec. 2, we review the famous Calogero-Sutherland model, its operator formalism, the CFT correspondence, the spectrum and eigenstates. In sec. 3 we obtain the fermionization of the CS theory followed by 