Generalized Clustering Conditions of Jack Polynomials at Negative Jack Parameter α
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Generalized Clustering Conditions of Jack Polynomials at Negative Jack Parameter α B. Andrei Bernevig , and F.D.M. Haldane Princeton Center for Theoretical Physics, Princeton, NJ 08544 and Department of Physics, Princeton University, Princeton, NJ 08544 (Dated: October 30, 2018)We present several conjectures on the behavior and clustering properties of Jack polynomials at negative parameter α = − k +1 r − , of partitions that violate the ( k, r, N ) admissibility rule of Feigin et.al. [1]. We find that ”highest weight” Jack polynomials of specific partitions represent the minimumdegree polynomials in N variables that vanish when s distinct clusters of k + 1 particles are formed,with s and k positive integers. Explicit counting formulas are conjectured. The generalized clusteringconditions are useful in a forthcoming description of fractional quantum Hall quasiparticles. PACS numbers: 73.43.f, 11.25.Hf
I. INTRODUCTION
The Jack polynomials are a family of symmetric ho-mogenous multivariate polynomials characterized by adominant partition λ and a rational number parameter α .They appeared in physics in the context of the Calogero-Sutherland model [2] for positive coupling α .In a recent paper, Feigin et. al. [1] initiated thestudy of Jack polynomials (Jacks) in N variables, at neg-ative rational parameter α k,r = − k +1 r − ( k + 1, r − k , r , N )-admissible partitions λ : λ i − λ i + k ≥ r , by proving they form a basis of adifferential ideal I k,rN in the space of symmetric polyno-mials. They showed that the set of Jacks with parameter α k, and ( k ,2, N )-admissible λ are a basis for the space ofsymmetric homogeneous polynomials that vanish when k + 1 variables z i coincide. In [3], we found that theseJacks naturally implement a type of “generalized Pauliprinciple” on a generalization of Fock spaces for abelianand non-abelian fractional statistics [4]. We found that(bosonic) Laughlin, Moore-Read, and Read-Rezayi Frac-tional Quantum Hall (FQH) wavefunctions (as well asothers, such as the state Simon et al. [5] have calledthe “Gaffnian”) can be explicitly written as single Jacksymmetric polynomials. We identified r as the minimumpower (cluster angular momentum) with which the ad-missible Jacks vanish as a cluster of k + 1 particles cometogether.In [3] we adopted a physical perspective to the Jack problem and uniquely obtained the abelian and non-abelian FQH ground states and the admissibility ruleon partitions by imposing, on an arbitrary Jack poly-nomial, a highest and lowest weight condition commonin FQH studies on the sphere [6]. However, in imposingonly the highest weight condition on the Jacks we ob-tained another infinite series of polynomials of partitionswhich violate the Feigin et. al. admissibility rule. The( k, r, N )-admissible configurations of [1] do not exhaustthe space of well-behaved Jack polynomials at negative α k,r . We find an infinite series of Jack polynomials, withpartitions characterized by a different integer s which arestill well-behaved at negative rational α k,r . This paperis devoted to analyzing the clustering properties and thecounting of such polynomials. We note that these poly-nomials can be interpreted as lowest Landau level (LLL)many-body wavefunctions and have applications in theconstruction of FQH quasiparticle excitations [7].We obtain a characterization of the symmetric polyno-mials in N variables P ( z , z , ..., z N ) satisfying the follow-ing set of generalized clustering (vanishing) conditions:(i) P ( z , z , ..., z N ) vanishes when we form s distinct clus-ters, each of k + 1 particles, but remains finite when s − k + 1 particles are formed and (ii) P ( z , z , ..., z N ) does not vanish when a large cluster of s ( k +1) − s and k are integers greateror equal to 1 and the case s = 1 is the case described byFeigin et. al. [1]. More precisely, let F be the space ofall polynomials satisfying the clustering condition: P ( z = ... = z k +1 , z k +2 = ... = z k +1) , ..., z ( s − k +1)+1 = ... = z s ( k +1) , z s ( k +1)+1 , z s ( k +1)+2 , ..., z N ) = 0 (1)Let F be the space of all polynomials satisfying bothEq.(1), and the clustering condition P ( z = ... = z s ( k +1) − , z s ( k +1) , z s ( k +1)+1 , ..., z N ) = 0. In this paperwe look at the coset space F/F and focus on two prob- lems: (1) We find the generators of the ideal above, whichare the minimum degree polynomials (for N particles)satisfying the clustering conditions above, and (2) Wegive a ( conjectured ) analytic expression for the number FIG. 1: Examples of occupation to monomial basis conversion n ( λ ) → λ of linearly independent polynomials in N variables, ofmomentum (total degree of the polynomial) M and offlux (maximum power in each variable) N Φ that spanthe coset F/F . Our motivation is to find new propertiesof Jack polynomials at negative α which are applicable tothe study of FQH quasiparticles [7]. Our results are alsorelated to the Cayley-Sylvester problem of coincident loci[8,9]. II. PROPERTIES OF JACK POLYNOMIALS
The Jacks J αλ ( z ) are symmetric homogeneous polyno-mials in z ≡ { z , z , . . . , z N } , labeled by a partition λ with length ℓ λ ≤ N , and a parameter α ; the partition λ can be represented as a (bosonic) occupation-numberconfiguration n ( λ ) = { n m ( λ ) , m = 0 , , , . . . } of each ofthe lowest Landau level (LLL) orbitals with angular mo-mentum L z = m ~ (see Fig[1]), where, for m > n m ( λ )is the multiplicity of m in λ . When α → ∞ , J αλ → m λ ,which is the monomial wavefunction of the free bosonstate with occupation-number configuration n ( λ ); a keyproperty of the Jack J αλ is that its expansion in terms ofmonomials only contains terms m µ with µ ≤ λ , where µ <λ means the partition µ is dominated by λ [10]. Jacks arealso eigenstates of a Laplace-Beltrami operator H LB ( α )given by X i (cid:18) z i ∂∂z i (cid:19) + 1 α X i
1) = 0 , (4)where λ ℓ is the smallest (non-zero) part in λ . This im-poses the following two conditions: ( i ) α is a negative ra-tional, which we can choose to write as − ( k + 1) / ( r − k + 1) and ( r −
1) both positive, and relativelyprime; ( ii ) λ ℓ = ( r − s + 1, and n = ( k + 1) s − s > λ have multiplicity k , so that the orbital occupation partition is n ( λ k,r,s ) =[ n s ( r − k r − k r − k.... ], ( i.e , the ( k , r , N )-admissibilitycondition is satisfied as an equality for orbitals m ≥ λ ℓ ).We call these Jacks HW ( k, r, s, N ) states J α k,r λ k,r,s . Non-HW ( k, r, s, N ) states with n particles in the zeroth or-bital can be obtained by inserting zeroes (holes) in thepartition to the right of the λ ℓ ’s orbital. This definesa set of partitions whose Jacks satisfy the same cluster-ing properties as the HW Jacks J α k,r λ k,r,s (see Section III).These partitions are λ k,r,s : n ( λ k,r,s ) = [ n s ( r − n ( λ k,r )] FIG. 2: Solutions to L + J αλ = 0 are parametrized by oneinteger, s >
0. Only s = 1 states are both HW and LWstates on the sphere, and satisfy the clustering property thatthey vanish as the r ’th power of the distance between k +1-particles. The s > where n ( λ k,r ) is a ( k, r, N − n ) admissible configurationin the sense of Feigin et. al. . This set of partitions do not exhaust the number of polynomials with the cluster-ing property Eq.(1). The case s = 1 gives the generatorsof the ideals obtained by Feigin et. al and are relatedto FQH ground states and their quasihole excitations [3].The cases s > el. al. (see Fig.2). III. GENERALIZED CLUSTERINGCONDITIONS OF JACK POLYNOMIALS
We now present the two generalized clustering condi-tions satisfied by the (HW and non-HW) Jacks of the( k, r, s, N ) partitions J α k,r λ k,r,s . A. First Clustering Property
The Jacks J α k,r λ k,r,s allow s −
1, but not s , different clus-ters of k +1 particles. First form s − k +1 par-ticles z = ... = z k +1 (= Z ); z k +2 = ... = z k +1) (= Z ); ... ; z ( s − k +1)+1 = ... = z ( s − k +1) (= Z s − ), wherethe positions of the clusters: Z , ..., Z s − can be dif-ferent. Then, form a k ( not k + 1) particle cluster z ( s − k +1)+1 = ... = z s ( k +1) − (= Z F ) (a final s ’th clus-ter of k + 1 particles would make the polynomial vanish),see Fig[3]. With the above conditions on the particlecoordinates, the clustering condition reads: J α k,r λ k,r,s ( z , ..., z N ) ∝ N Y i = s ( k +1) ( Z F − z i ) r (5)Observe that when s = 1 Eq.(5) reduces to the usualclustering condition satisfied by the ( k, r ) sequence, given FIG. 3: Clustering and vanishing conditions of the polynomi-als defined by the HW and non-HW Jacks J α k,r λ k,r,s in [3]. B. Second Clustering Property
The Jacks J α k,r λ k,r,s allow a large cluster of n = ( k +1) s − n +1 particles to come at thesame point, as this would involve the formation of s clus-ters of k + 1 particles, which Eq.(5) forbids. Clustering n particles at the same point z = ... = z ( k +1) s − = Z we find the following property (see Fig.[3]): J α k,r λ k,r,s ( z , ..., z N ) ∝ N Y i = s ( k +1) ( Z − z i ) ( r − s +1 (6)The HW Jacks of partitions λ k,r,s satisfy an even morestringent property; with z = ... = z ( k +1) s − = Z , wefind: J α k,r λ k,r,s ( z , ..., z N ) = Q Ni = s ( k +1) ( Z − z i ) ( r − s +1 ×× J α k,r λ k,r ( z s ( k +1) , z s ( k +1)+1 , ..., z N ) (7)where n ( λ k,r ) = [ k r − k r − ...k ] is the maximum density( k, r, N − n )-admissible partition. For s = 1, Eq.(7)also reduces to the usual clustering condition satisfied bythe ( k, r )-admissible sequence [3]. We have performeda extensive numerical checks of the above conjecturedclustering conditions. We also note that the LHS andRHS of Eq.(7) match in both total momentum M (totaldegree of the polynomial): E J α k,r λ k,r,s ≡ M J α k,r λ k,r,s = ( N − ( k + 1) s + 1) ×× (cid:2) ( r − s + 1 + rk ( N − ( k + 1) s − k + 1) (cid:3) J α k,r λ k,r,s (8)and in flux (maximum degree in each variable) N : N = rk ( N − k − ( k + 1)( s − r − s − . (9) FIG. 4: The densest polynomials satisfying the clusteringEq.(5) and Eq.(6). These are polynomials of flux N andare obtained by applying L − on the HW Jack. The LW poly-nomial is also a Jack, but the polynomials in between cannotbe expanded in terms of only well-behaved Jack polynomialsat α = − k +1 r − . The superscript denotes the fact that we are consider-ing the momentum and flux of Jack polynomials of HWpartitions λ k,r,s . C. Additional Clustering Condition
We empirically find that the HW Jacks J α k,r λ k,r,s sat-isfy a third type of clustering which has no correspon- dence in the s = 1 case. Forming s − k + 1 particles together: z = ... = z k +1 (= Z ); z (2 k +1)+1 = ... = z k +1) (= Z );...; z ( s − k +1)+1 = ... = z ( s − k +1) = ( Z s − ) we find that the HW Jacks satisfy,up to a numerical proportionality constant, the cluster-ing: J α k,r λ k,r,s ( z , ..., z N ) = Q s − i The J α k,r λ k,r,s ( z , ..., z N ) are the HW states of an an-gular momentum multiplet of l ( λ k,r,s ) = l maxz and l z = l maxz , ..., − l maxz . The LW states of the multiplet( L − ) l maxz J α k,r λ k,r,s ( z , ..., z N ) are also single Jack polyno-mials of the “symmetric” partition to λ k,r,s in orbitalnotation (see Fig.[4]) The value of l maxz is: L z J α k,r λ k,r,s ≡ l maxz J α k,r λ k,r,s = 12 h (( r − s + 1)(2( k + 1) s − − N ) + rk (( k + 1) s − N − ( k + 1) s + 1 − k ) i J α k,r λ k,r,s (11)Most importantly, we find that powers of the operator L − = P i (cid:16) z i ∂∂z i − N Φ z i (cid:17) , acting on J α k,r λ k,r,s ( z , ..., z N )create linearly independent polynomials with thesame vanishing conditions Eq.(5), Eq.(6) as the HW J α k,r λ k,r,s ( z , ..., z N ). There are 2 l maxz + 1 such polynomi-als. IV. J α k,r λ k,r,s : SMALLEST DEGREEPOLYNOMIALS WITH GENERALIZEDCLUSTERING In the remainder of the paper we focus on the case r = 2. We empirically find the following property: given N particles (with N > n for the clustering conditionEq.(1) to be well defined), we find that the r = 2 HW Jacks J α k, λ k, ,s ( z , ..., z N ) are the smallest degree (smallestmomentum M - Eq.(8) - and flux N Φ - Eq.(9)) polyno-mials satisfying the clustering conditions Eq.(5) Eq.(6).There are exactly · l ( λ k, ,s ) + 1 polynomials in N vari-ables of N (Eq.(9)) and of unrestricted total dimensions,satisfying the clustering Eq.(5), Eq.(6). A basis for thisideal is, explicitly:( L − ) m J α k, λ k, ,s ; m = 0 , , ..., · l ( λ k, ,s ); (12)We find that ( L − ) · l ( λ k, ,s )+1 J α k, λ k, ,s = 0. One can eas-ily understand this counting by looking at the orbitaloccupation numbers of the relevant partitions. The oc-cupation number of the J α k, λ k, ,s is [ n s k k k...k k ]. Thisis the lowest weight partition (smallest degree polyno-mials) where the clustering conditions Eq.(5), Eq.(6) aresatisfied. Interpreting this as orbital occupation num- FIG. 5: Lowest degree polynomials (generators) of the clus-terings ( k, r, s ) = (1 , , s )FIG. 6: Lowest degree polynomials (generators) of the clus-terings ( k, r, s ) = (2 , , s ) ber, there exists a “symmetric” partition n ( λ maxk,r,s ) =[ k k...k k k s n ]. This is the highest total degree poly-nomial (bounded by the restriction that each variableseparately has degree at most N ) that satisfies thevanishing condition Eq.(1). It is also a Jack polyno-mial J α k,r λ maxk,r,s (See Fig[4]). Since the L − operator doesnot change the value of N Φ , maintains the clusteringproperty, and implements the angular momentum low-ering, we then easily count the polynomials as formingthe l = l maxz multiplet.For the case k = 1 , r = 2 , s = 2, this counting coincideswith the empirical counting observed by Kasatani et. al. [8].We remark that the expansion of ( L − ) m J α k,r λ k,r,s in Jackpolynomial basis J α k,r λ contains ill-behaved Jacks whichdiverge at the negative α k,r used. However, their coeffi-cients in the expansion also vanish to give an overall finitecontribution. These are the ”modified” Jacks introduced by Kasatani et. al. for the specific ( k, r, s ) = (1 , , 2) caseof the problem studied here. As we reach higher k and s integers, the number of ”modified” Jacks appearing inthe expansion of ( L − ) m J α k,r λ k,r,s grows large. We thereforeprefer to characterize the basis of these polynomials bythe HW Jack and the polynomials that result from it bysuccessive application of the L − operator.We now relax the constraint N Φ = N and focus on thecounting of the polynomials satisfying Eq.(5) and Eq.(6). V. COUNTING POLYNOMIALS We want to provide the counting of the number of lin-early independent polynomials in the ideal F/F . Westart by counting the s = 1 polynomials. These are re-lated to the admissible partitions of [1] or the generalizedPauli principle of [4, 3] A. Counting of ( k, r, s ) = (1 , r, Polynomials We first obtain a counting of linearly independent poly-nomials in N particle coordinates z , ..., z N , of total mo-mentum M , with the degree in each coordinate ≤ N Φ ,satisfying the condition P ( z , z , z , z , .. ) ∼ ( z i − z j ) r .We believe this result was previously known, althoughwe could not explicitly find it in the literature. Fromthe work of Feigin et. al. [1], this number is equal tothe number of ( k, r ) = (1 , r ) admissible partitions of N Φ orbitals, related by the squeezing rule [3] (so as tokeep the partition weight M constant). Call this num-ber p ,r, ( N, M, N Φ ). From the theory of partitions [12],such numbers are most easily obtained from a generatingfunction G ( q ), and we can analytically prove that: p ,r, ( N, M, N Φ ) = 1 M ! ∂ M G ,r, ( N, N Φ , q ) ∂q M | q =0 (13)where the generating function G ( q ) reads: G ,r, ( N, N Φ , q ) = q r N ( N − Q N Φ − r ( N − Ni =1 (1 − q i ) Q Ni =1 (1 − q i ) Q N Φ − r ( N − i =1 (1 − q i )if N Φ ≥ r ( N − 1) and p ,r, ( N, M, N Φ < r ( N − p ,r, ( N, M, N Φ ) represents a building block for futureresults. We have numerically checked, by building thenull-space of polynomials satisfying the clustering condi-tion P ( z , z , z , z , ... ) = 0, that p ,r, ( N, M, N Φ ) givesthe right polynomial counting. In the context of FQH,it reproduces the right counting for quasihole states. Forexample, it is known that the Laughlin state with x num-ber of quasiholes has ( N + x )! / ( N ! x !) independent statesand one can numerically check the identity: (cid:18) N + xx (cid:19) = r N ( N − xN X M = r N ( N − p ,r, ( N, M, N Φ = r ( N − x )We can now provide a formula for the number of poly-nomials in N variables, of total dimension M , of anymaximum power of each coordinate (any N Φ ), which sat-isfy the clustering condition P ( z , z .... ) ∼ ( z i − z j ) r . Wetake N Φ → ∞ to obtain the simpler expression: p ,r, ( N, M ) = 1 M ! ∂ M G ,r, ( N, q ) ∂q M | q =0 (14) G ,r, ( N, q ) = q r N ( N − Q Ni =1 (1 − q i )To find out the total number of symmetric polynomialssatisfying the vanishing P ( z , z , z , z , ..., z N ) = 0 wemust particularize to the lowest vanishing power possible, r = 2. In this case, Eq.(14) is identical to the formulaof Kasatani et. al. [8], although Eq.(13) represents amore comprehensive counting of the polynomials as itcontains information on the allowed maximum degree ineach variable, N Φ .Our aim to conjecture similar expressions for thecounting of the dimension space of the polynomials inthe coset space F/F . B. Counting of ( k, r, s ) = (1 , , s ) Polynomials Using p , , ( N, M, N Φ ), we now obtain the countingof polynomials in the ideal F/F with k = 1 , r = 2and s > 1. We first reproduce the result of Kasatani et. al. [8] which is the case ( k, r, s ) = (1 , , 2) of ourproblem. Kasatani et. al. [8] obtained the dimensionof the linear space of polynomials satisfying the clus-tering conditions P ( z , z , z , z , z , z , ..., z N ) = 0 and P ( z , z , z , z , z , ..., z N ) = 0, of total dimension M , ofany allowed maximum degree in each of the coordinates.We then derive the general case, which contains informa-tion about N Φ .Define p k =1 ,r =2 ,s =2 ( N, M ) = p , , ( N, M ) = p , , ( N, M, N Φ ≤ ∞ ) as the number of polynomialsin N variables of total momentum (degree) M , withany allowed N φ ( ≤ M ), satisfying the clustering con-dition Eq.(5), Eq.(6) (with k = 1 , r = 2 , s = 2). Thisnumber can be found as follows: start with the parti-tion n ( λ , , ), of total dimension (weight) M which has N − N ’th particle is pushed as far as neededto the right so that the polynomial has dimension M .This partition reads [30010101 .. 101 0 ... |{z} M − − N ( N − k, r, s, N ) admissibility, we cannot push thefirst N − p , , ( N, M ) is the sum of two terms:First, we can form ( k, r, s, N ) = (1 , , , N ) admissiblepartitions by keeping the occupancy of the zeroth or-bital to be 3 and by squeezing on the remainder partition[00010101 .. 101 0 ... |{z} M − − N ( N − 1] to form all the ( k, r ) = (1 , k, r ) = (1 , 2) admissible partitions of N − M − N − p , , ( N − , M − N − k, r, s, N ) = (1 , , , N ) clus-tering by taking some particles out of the zeroth-orbital,although these now involve divergent Jacks (with com-pensating vanishing coefficients). We can form all thepolynomials (of dimension M in N variables, with lessthan 3 particles in the zeroth orbital, and that satisfy theclustering conditions ( k, r, s, N ) = (1 , , , N )) by actingwith L − on all the polynomials of dimension M − 1, thatsatisfy the same clustering conditions. This number isthen p , , ( N, M − p , , ( N, M ) = p , , ( N, M − p , , ( N − , M − N − q M ,sum over M , re-shift variables in the sum and obtain: p , , ( N, M ) = 1 M ! ∂ M G , , ( N, q ) ∂q M | q =0 (16) G , , ( N, q ) = q ( N − N − (1 − q ) Q N − i =1 (1 − q i )This reproduces a formula obtained by Kasatani et. al. [8] through different methods.We now use the same resoning to count the dimensionof the ideal F/F with ( k = 1 , r = 2) and general s .The number of polynomials of N variables of momentum(total degree) M , with unrestricted N φ , which satisfy theclustering conditions Eq.(5) and Eq.(6) with k = 1 , r = 2and any s > p , ,s ( N, M ): p , ,s ( N, M ) = M ! ∂ M G , ,s ( N,q ) ∂q M | q =0 ; G , ,s ( N, q ) = q ( N − n N − n s ) (1 − q ) Q N − n i =1 (1 − q i ) (17)where n = 2 s − p k =1 ,r =2 ,s ( N, M, N Φ ) = p , ,s ( N, M, N Φ ) as the numberof polynomials in N variables of total momentum (de-gree) M , with flux ≤ N φ , satisfying the clustering con-ditions Eq.(5) and Eq.(6) with k = 1 and s = 2. Webriefly present the reasoning used to conjecture a countof these polynomials. The smallest dimension and fluxcorrespond to the partition [ n s ... ... n = 2 s − N Φ + 1. Some of the orbitals to the right might beunoccupied. This “padding” to the right has the ef-fect of allowing L − to move particles up to the right-most orbital. By symmetry in orbital space, the high-est partition corresponds to [00 ... ... s n ](with total number of orbital N Φ + 1). We can thenimmediately see that p , ,s ( N, M, N Φ ) = 0 for M < ( s + 1)( N − n ) + ( N − n )( N − n − 1) or for N φ 1) when the right-most orbital has been occupied by the maximum numberof particles possible, n . Then p , ,s ( N, M, N Φ ) reads: p , ,s ( N, M, N Φ ) == 0 if M < ( s + 1)( N − n ) + ( N − n )( N − n − or N φ < s + 1 + 2( N − n − M − ( s +1)( N − n ) X i =0 p , , ( N − n , i, N Φ − ( s + 1)) if M ≤ n N Φ + ( N − n )( N − n − NN Φ − ( N − n )( s +1) − M X i =0 p , , ( N − n , i, N Φ − ( s +1)) if n N Φ +( N − n )( N − n − < M ≤ N N Φ − ( N − n )( N − n + s )= 0 if M > N N Φ − ( N − n )( N − n + s ) p , , ( N, M, N Φ ) was explicitly given in a previous sub-section, and n = 2 s − M we canfind the number of polynomials of N variables, with de-gree in each variable N Φ and of unrestricted momentum(total degree) p , ,s ( N, N Φ ) = P ∞ M =0 p , ,s ( N, M, N Φ ),satisfying the clustering conditions Eq.(5) and Eq.(6) with k = 1 and s arbitrary integer. However, by ap-plying an empirical rule we observed, based on the mul-tiplet nature of these polynomials, we find an alter-nate simpler formula, which is not obviously equal to P ∞ M =0 p , ,s ( N, M, N Φ ); extensive numerical checks havehowever confirmed their equivalence: p , ,s ( N, N Φ ) == 0 if N φ < s + 1 + 2( N − n − N N Φ − s + 1) N + 2 n ( s + 1) + 1) NN Φ X i =0 p , , ( N − n , i, N Φ − ( s + 1)) − NN Φ X i =0 i · p , , ( N − n , i, N Φ − ( s + 1)) (18) p , , ( N, M, N Φ ) was explicitly given in subsection IV A.,and n = 2 s − C. Counting of ( k, r, s ) = ( k, , Polynomials We now move to the k > k, r ) = ( k, 2) statistics, i.e. of the Read-Rezayi Z k states. We want to countthe number of polynomials in N variables, of momentum(total degree) M with maximum flux (maximum degreein each coordinate) N Φ , that vanish when k + 1 particlescome together. We call this number p k, , ( N, M, N Φ ).As we know [1], this is equal to the number of ( k, M , made out of at most N parts, and with λ ≤ N Φ . We can derive this by perform-ing a slight modification of a formula due to Feigin andLoktev [13] (see also Andrews [12]). p k, , ( N, M, N Φ ) = 0for N Φ < k ( N − k ) or for M < k N ( N − k ) but otherwiseis: p k, , ( N, M, N Φ ) = M ! 1 N ! ∂ N ∂z N ∂ M ∂q M ×× (cid:0) q − N G k, , ( N φ , q, z ) (cid:1) | q =0 , z =0 where the generating function G k, , ( N φ , q, z ) is [13,12]: G k, , ( N φ , q, z ) = N Φ+12 X m ,n =0 m + n ≤ N Φ+12 z ( k +1)( m + n ) q k ( m − n + n ( N Φ +2))+ m m − + n ( N Φ +3 − n ) Q m i =1 (1 − q i )( zq i + m − − Q N Φ − n +1 i =2 m +1 (1 − zq i ) Q n i =1 ( q i − zq N Φ − i − n +3 − k = 1,with the simpler expression of p , , ( N, M, N Φ ) obtainedearlier. The formula above also correctly gives the dimen-sion of the quasihole Hilbert space in the Z k parafermionssequence. For one-quasihole, this is known to be: (cid:18) N/k + kk (cid:19) = N ( N − k ) k + N X i = N ( N − k ) k p k, , ( N, i, N φ = 2 k ( N − k )+1)Extensive numerical checks prove the above identity.Moreover, for the k = 2 Read-Moore state with 2 quasi-particles: (cid:18) N/ (cid:19) + (cid:18) ( N − / (cid:19) == P N ( N − k ) k +2 Ni = N ( N − k ) k p k, , ( N, i, N φ = k ( N − k ) + 2)Eq.(19) for ( k, 2) admissible partitions found by Fei-gin and Loktev [13] and prior to them by Andrews [12] gives the most information possible about the countingof the Read-Rezayi wavefunctions and quasiholes. It pro-vides information about the total degree of the polyno-mial (multiplet structure), which the usual counting [14]of quasiholes does not since it sums over all the possibletotal dimensions of the polynomials subject to a flux N Φ upper bound. D. Counting of the ( k, r, s ) = ( k, , s ) Polynomials Using p k, , ( N, M, N Φ ) we obtain the counting of poly-nomials in the F/F ideal with arbitrary k and s . Fol-lowing a line of reasoning similar to the one used in the k = 1 case, we find the number p k, ,s ( N, M, N Φ ) of poly-nomials in N variables, of momentum (total degree) M ,of flux N Φ that have the clustering conditions Eq.(5) andEq.(6) for general k and s > p k, ,s ( N, M, N Φ ) == 0; if M < ( s + 1) · ( N − n ) + 1 k ( N − n )( N − n − k ) or N Φ < s + 1 + 2 k ( N − n − k )= M − ( s +1)( N − n ) X i =0 p k, , ( N − n , i, N Φ − ( s + 1)); if 0 ≤ M ≤ n N Φ + 1 k ( N − n )( N − n − k )= NN Φ − ( N − n )( s +1) − M X i =0 p k, , ( N − n , i, N Φ − ( s +1)); n N Φ + 1 k ( N − n )( N − n − k ) < M ≤ N N Φ − ( N − n )( s + N − n k )(20)= 0 if M > N N Φ − ( N − n )( s + N − n k ) (21)where n = ( k + 1) s − M wecan find the number of polynomials of N variables, withflux N Φ and of unrestricted momentum (total degree) p k, ,s ( N, N Φ ) = P ∞ M =0 p k, ,s ( N, M, N Φ ), satisfying theclustering conditions Eq.(5) and Eq.(6) with k and s ar- bitrary integers. However, by using the angular momen-tum multiplet structure of these polynomials, we findan alternate formula which is not obviously equal to P ∞ M =0 p k, ,s ( N, M, N Φ ); extensive numerical checks havehowever confirmed their equivalence: p k, ,s ( N, N Φ ) == 0 if N φ < s + 1 + 2 k ( N − n − k )= ( N N Φ − s + 1) N + 2 n ( s + 1) + 1) NN Φ X i =0 p k, , ( N − n , i, N Φ − ( s + 1)) − NN Φ X i =0 i · p k, , ( N − n , i, N Φ − ( s + 1)) (22) p k, , ( N, M, N Φ ) is given in VC., and n = ( k + 1) s − VI. SUBIDEALS OF THE CAYLEY-SYLVESTERPROBLEM So far we have focused on the ideal F/F . We cansystematically characterize the ideal F of polynomials P ( z , z , ..., z N ) which vanish when we form s clusters of k + 1 particles in the following way: let the sub-ideals F i be the polynomials that satisfy (Eq.(1)) but that alsovanish when ( k + 1) s − i particles are brought at thesame point. Then F = S k +1 i =0 F i /F i +1 , where F = F and F k +1 is the ideal of polynomials that vanish when s − k + 1 particles are formed ( F k +2 = ∅ ). Hencethe polynomial ideal F i is defined by the two clusteringconditions: P ( z = ... = z k +1 , z k +2 = ... = z k +1) , ..., z ( s − k +1)+1 = ... = z s ( k +1) , z s ( k +1)+1 , z s ( k +1)+2 , ..., z N ) = 0 (23)and P ( z = ... = z s ( k +1) − i , z s ( k +1) − i +1 , z s ( k +1) − i +2 , ..., z N ) = 0 (24)We have not found the generators for the F i /F i +1 ideals,nor have were we able to find their counting rules for thegeneral case. However, we have solved the problem forseveral specific cases which we present below. A. Subideals of the ( k, r, s ) = ( k, , We now give the partitions for the generators (small-est degree highest weight polynomials) for the subideals F i /F i +1 for the infinite series ( k, r, s ) = ( k, , k + 1 particles are formed, but does notvanish when one large cluster of 2 k +1 particles is formed,is dominated by the root partition: | k + 100 k k k k...k k i : P ( z , ..., z | {z } k +1 , z , ..., z | {z } k +1 , z , z , .... ) = 0 P ( z , ..., z | {z } k +1 , z , z , .... ) = 0 (25)0 FIG. 7: Cayley-Sylvester subideals for polynomials satisfying P ( z , ..., z , z , ..., z , z , z , ... ) = 0 The polynomial above, as well as the ones we introducebelow, can be written as a linear combinations of mono-mials of partitions dominated by the root partition above,with coefficients that are uniquely defined by the HWand clustering conditions. These are of course, the Jacks.Then the smallest degree polynomial that vanishes wheneither 2 distinct clusters of k + 1 particles are formed orwhen a large single cluster of 2 k + 1 particles is formed,but does not vanish when one large cluster of 2 k particlesif formed, is dominated by the partition ( see Fig[7]): | k k − k − k − ... k − i : P ( z , ..., z | {z } k +1 , z , ..., z | {z } k +1 , z , z , ... ) = 0 (26) P ( z , ..., z | {z } k +1 , z , z , ... ) = 0 , P ( z , ..., z | {z } k , z , z ... ) = 0;The smallest degree polynomial that vanishes when ei-ther 2 distinct clusters of k + 1 particles are formed orwhen a large single cluster of 2 k particles is formed butdoes not vanish when one large cluster of 2 k − | k − k − k − k − ... k − i : P ( z , ..., z | {z } k +1 , z , ..., z | {z } k +1 , z , z , ... ) = 0 (27) P ( z , ..., z | {z } k , z , z , ... ) = 0 , P ( z , ..., z | {z } k − , z , z , ... ) = 0; and so on until. At last, the smallest degree polynomialthat vanishes when either 2 distinct clusters of k + 1 par-ticles are formed or when a single cluster of k +2 particlesis formed but does not vanish when one cluster of k + 1particles if formed, is dominated by the partition ( seeFig[7]): | k + 10 k k k...k i : P ( z , ..., z | {z } k +1 , z , ..., z | {z } k +1 , z , z , .... ) = 0 (28) P ( z , ..., z | {z } k +2 , z , z , ... ) = 0 , P ( z , ..., z | {z } k +1 , z , z , ... ) = 0;The polynomials of the last sub-ideal, Eq.(29) are re-lated to the quasiparticle excitations of abelian and non-abelian FQH states [7]. They perform well under k + 2-body repulsive interactions. We can also find the small-est weight partitions of polynomials that vanish when s clusters of k + 1 particles come together and when k + 2particles come together, but do not vanish when k + 1particles form a cluster: | k + 10 k + 10 ...k + 10 | {z } s − k k k... k i : P ( z , ..., z | {z } k +1 , ..., z s , ..., z s | {z } k +1 , z s +1 , z s +2 , ... ) = 0 (29) P ( z , ..., z | {z } k +2 , z , z , ... ) = 0 , P ( z , ..., z | {z } k +1 , z , z , ... ) = 0 B. Counting of the F k /F k +1 Subideal The counting of dimension of the subideals above is arather difficult (but tractable) problem. The “easy” ex-ceptions are the first subideal, whose generator is | k +100 k k k...k k i and whose counting formulas we havealready conjectured in the body of this manuscript, andthe last subideal whose generator is | k + 10 k k k...k k i ,and whose counting formula we conjecture below. Thenumber p F k /F k +1 k, , ( N, M, N Φ ) of polynomials, satisfyingthe clusterings of Eq.(29), of N variables, of momentum(total degree M ) and of flux (maximum separate degreein any variable) N Φ is: p F k /F k +1 k, , ( N, M, N Φ ) == 0; if M < · ( N − n ) + 1 k ( N − n )( N − n − k ) or N Φ < k ( N − n − k )1= M − N − n ) X i =0 p k, , ( N − n , i, N Φ − ≤ M ≤ n N Φ + 1 k ( N − n )( N − n − k )= NN Φ − ( N − n )2 − M X i =0 p k, , ( N − n , i, N Φ − n N Φ + 1 k ( N − n )( N − n − k ) < M ≤ N N Φ − ( N − n )(1 + N − n k )(30)= 0 if M > N N Φ − ( N − n )(1 + N − n k ) (31)where n = k + 1.By summing the previous expression over all M wecan find the number of polynomials of N variables,with degree in each variable at most N Φ and of unre-stricted momentum (total degree) p F k /F k +1 k, , ( N, N Φ ) = P ∞ M =0 p F k /F k +1 k, , ( N, M, N Φ ). By applying a rule based on the multiplet nature of these polynomials, we findan alternate formula which is not obviously equal to P ∞ M =0 p F k /F k +1 k, , ( N, M, N Φ ); extensive numerical checkshave however confirmed their equivalence: p subidealk, , ( N, N Φ ) == 0 if N φ < k ( N − n − k )= ( N N Φ − N + 4 n + 1) NN Φ X i =0 p k, , ( N − n , i, N Φ − − NN Φ X i =0 i · p k, , ( N − n , i, N Φ − 2) (32)where n = k + 1. VII. CONCLUSIONS In this paper we have made several new conjecturesabout the behavior of Jack polynomials at negative Jackparameter α . By applying a HW condition, we find thatthe ( k, r )-admissible partitions of Feigin et.al [1] do notexhaust the space of partitions for which the Jack poly-nomials are well-behaved. 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R.P. Stanley, Adv.Math. , 76 (1989). M. Lassalle, J. Func. Anal. , 289 (1998). G.E. Andrews, Theory of Partitions, Cambridge Univer-sity Press (July 28, 1998). B. Feigin and S. Loktev, arXiv:math/0006221v3. N. Read and E. Rezayi, Phys. Rev. B59N N Φ − ( N − n )( N − n + s ). There is also an ”intermedi- ate” total degree that is important in the counting, whichcorresponds to the partition [10101 ... .. n ] oftotal degree n N Φ + ( N − n )( N − n −