2 These incompressible states occur at electron densities given in Eq. (1). Let us elaborate on the above conditions. From the definition given in Eq. 2, we have the ̂ sl (2)$_{k}$ WZW theory can be explicitly generated us- ing the U$_{q}$ ( sl (2)) quantum group at the root of unity q = e 2 πi/ ( k $^{+2)}$.18 It was also argued 18 that the braiding of the Read-Rezayi parafermions can be explicitly gen- erated using the U$_{q}$$_{1}$ ( sl (2)) ⊗$_{CZ}$$_{4}$$_{k,q}$$_{2}$ quantum group at the roots of unity q$_{1}$ = e 2 πi/ ( k +2) and q$_{2}$ = e − iπ/ 2 $^{k}$. For the Moore-Read sequence ( k =2), the braiding reduces to the spinor representation of the mirror reflections in the group SO(2 n ), as discovered long time ago by Nayak and Wilczek. 19 In all these algebraic derivations the anyons are as- sumed infinitely far apart from each other. In practice this will not be the case, and this is the main reason why the algebraic results need a direct confirmation showing that the braiding properties remain unchanged when fi- nite anyon densities are considered. So far, our numerical results confirm that this is the case. The general structure of Ψ ( k,m ) is where ˜ Ψ$_{k}$( z$_{1}$, . . . , z$_{N}$ ) is the correlation function of N number of Z$_{k}$ parafermion fields (we include in ˜ Ψ$_{k}$ the factors that make the correlation function non-singular). We shall see that the first condition of Eq. (3) relates to the part of the wavefunction contained in the second row of the above equation. Indeed, the right hand side of Eq. (4) reduces to The Hall sequences can be label by two integers: k =1,2,. . . , and m =0,1,. . . . The filling factor and the conformal charge for each sequence are given by: [ η (2 ,m $^{′}$) M Ψ ( k,m $^{)}$]( z$_{3}$, . . . , z$_{N}$ ) = ∫ d $^{2}$z$_{1}$ ∫ d $^{2}$z$_{2}$ × ( z ∗ 1 + z ∗ $_{2}$) $^{M}$( z ∗ 1 − z ∗ $_{2}$) m $^{′}$e − 1 $_{4}$( | z$_{1}$ | $^{2}$+ | z$_{2}$ | $^{2}$) × Ψ ( k,m $^{)}$( z$_{1}$, z$_{2}$, . . . , z$_{N}$ ) . (4) Ψ ( k,m $^{)}$( z$_{1}$, . . . , z$_{N}$ ) = ˜ Ψ$_{k}$( z$_{1}$, . . . , z$_{N}$ ) × ∏ i<j ≤ N ( z$_{i}$ − z$_{j}$ ) $^{m}$e − 1 4 ∑ N i =1 | z$_{i}$ | $^{2}$, (5) ∏ 2 <i<j ( z$_{i}$ − z$_{j}$ ) $^{m}$e − 1 4 ∑ N i =3 | z$_{i}$ | 2 ∫ d $^{2}$z$_{1}$ ∫ d $^{2}$z$_{2}$ × ( z ∗ 1 + z ∗ $_{2}$) $^{M}$| z$_{1}$ − z$_{2}$ | 2 m $^{′}$e − 1 $_{2}$( | z$_{1}$ | $^{2}$+ | z$_{2}$ | $^{2}$) × ( z$_{1}$ − z$_{2}$ ) m − m ′ N ∏ i =3 ( z$_{i}$ − z$_{1}$ ) $^{m}$( z$_{i}$ − z$_{2}$ ) m × ˜ Ψ$_{k}$( z$_{1}$, . . . , z$_{N}$ ) (6) ν = k km + 2 , c = 1 + 2( k − 1) k + 2 . (1) II. FRACTIONAL HALL SEQUENCES: GENERAL CONSIDERATIONS. The k =1, ν =1 / ( m + 2) sequence corresponds to the Laughlin state, 22 which has fractionally charged anyons carrying integer flux. Their braiding is Abelian. The k ≥ 2 sequences have fractionally charged anyons carrying fractional flux. Their braiding is believed to be Non- Abelian. The case k =2 corresponds to the Moore-Read sequence; 4 k =3 and greater correspond to the Read- Rezayi sequences. 5 Excepting k =1, 2 and 4, all sequences support universal quantum computation. We define the k -particle creation operators, η ( k,m ) † M , generating (from the vacuum) k -particles in a Laughlin state (we set the magnetic length to 1): The multiple integral projects the part of the integrand that is invariant to 4-dimensional rotations in the ( r$_{1}$ , r$_{2}$ ) sub-space and we argue that this part is zero. We focus to rotations around the axis connecting r$_{1}$ + r$_{2}$ to the origin. The second row of the above equation is invariant to such rotations but the invariant part of the remaining rows is identically zero. Indeed, the third and fourth rows are analytic functions in all complex variables, hence their product admit an expansion like $^{∑}$$_{α}$c$_{α}$ ( z$_{1}$ − z$_{2}$ ) $^{α}$, where c$_{α}$ may depend on z$_{1}$ + z$_{2}$ , z$_{3}$ , . . . , z$_{N}$ . The invariant part of such expansion is equal to c$_{0}$ , but c$_{0}$ is identically zero when m $^{′}$<m . This proves the first condition of Eq. 3. We consider now the second condition, which says that where z = x + iy is the complex representation of the posi- tion r =( x, y ) in the 2-dimensional plane, and M and m are integers larger or equal to zero. The Fractional Hall sequences can be described as follows. Incompressible states. We claim that the incompress- ible Hall state for an arbitrary ( k , m ), originally defined in terms of certain correlators of the$_{Z}$$_{k}$ parafermion con- formal algebra, is the highest electron density state sat- isfying is identically zero. The right hand side of Eq. (7) is equal [ η ( k +1 ,m ) M Ψ ( k,m $^{)}$]( z$_{k}$$_{+2}$, . . . , z$_{N}$ ) = ∫ d $^{2}$z$_{1}$ . . . ∫ d $^{2}$z$_{k}$$_{+1}$ × ( z ∗ 1 + . . . , + z ∗ k $_{+1}$) M ∏ i<j ≤ k +1 ( z ∗ i − z ∗ $_{j}$)$^{m}$e$^{−}$$^{1}$ 4 k +1 ∑ i =1 | z$_{i}$ | 2 × Ψ ( k,m $^{)}$( z$_{1}$, . . . , z$_{N}$ ) . (7) η (2 ,m $^{′}$) M | Ψ ( k,m $^{)}$〉 = 0 , ∀ M ≥ 0 and m ′ < m, η ( k +1 ,m ) M | Ψ ( k,m $^{)}$〉 = 0 , ∀ M ≥ 0 . (3) 〈 r$_{1}$ , . . . , r$_{k}$ | η ( k,m ) † M | 0 〉 = ( z$_{1}$ + . . . + z$_{k}$ ) M × ∏ i<j ≤ k ( z$_{i}$ − z$_{j}$ ) $^{m}$e − 1 4 ∑ k i =1 | z$_{i}$ | $^{2}$, (2)