The super-Virasoro singular vectors and Jack superpolynomials relationship revisited

A recent novel derivation of the representation of Virasoro singular vectors in terms of Jack polynomials is extended to the supersymmetric case. The resulting expression of a generic super-Virasoro singular vector is given in terms of a simple differential operator (whose form is characteristic of the sector, Neveu-Schwarz or Ramond) acting on a Jack superpolynomial. The latter is indexed by a superpartition depending upon the two integers r,s that specify the reducible module under consideration. The corresponding singular vector (at grade rs/2), when expanded as a linear combination of Jack superpolynomials, results in an expression that (in addition to being proved) turns out to be more compact than those that have been previously conjectured. As an aside, in relation with the differential operator alluded to above, a remarkable property of the Jack superpolynomials at alpha=-3 is pointed out.

1. Introduction 1.1.Brief overview of past results in the superconformal case.The remarkable relationship between Virasoro singular vectors and Jack polynomials [2,16,17,19] turns out to have a surprising supersymmetric counterpart.Indeed, there are two completely different formulations of this representation of the super-Virasoro singular vectors in terms of symmetric polynomials.
The first one [9] relies on the extension of the Jack polynomials to superpolynomials, which are polynomials that depend upon additional anticommuting variables.In this context, the simple link which exists in the Virasoro case, namely that a singular vector is represented by a single Jack indexed by a rectangular partition, becomes somewhat more complicated.Indeed, instead of a single Jack superpolynomial (sJack for short) indexed by a superpartition of rectangular form, a singular vector is represented by a linear combination of sJacks indexed by the so-called self-complementary superpartitions which are controlled by the expected rectangle.A self-complementary superpartition is such that when glued in a specific manner with (a slightly modified version of) itself, it fills the defining rectangle.The explicit expression for the coefficients of these linear combinations has been conjectured (and heavily tested) in both the Neveu-Schwarz (NS) and Ramond (R) sectors [1,9].The second one [3], proposed almost simultaneously, bypasses the manipulation of fermionic variables by bosonising the free fermion modes in an intricate way.Then the symmetric polynomial representing the singular vector is a Uglov polynomial indexed by a rectangular partition, exactly as in the Virasoro case.The Uglov polynomials, like the Jacks, can be obtained from the Macdonald polynomials, which depend upon two parameters q and t, in a special limit.While the Jack case is recovered by taking q = t α and t → 1, the Uglov case corresponds to q = −|t| α and t → −1 (see e.g., [14,Chap. 10]).This connection, conjectured in [3], has recently been proved in [20].Notice however that this second construction is limited to the NS sector.
In the present work, we reconsider the first approach along the lines of the argument of [17] which is sketched in the following subsection.
1.2.Schematic version of the argument in the Virasoro case.The derivation of the relationship Virasoro-Jack proceeds via the following steps.

1
(1) Using the free boson representation of the Virasoro algebra, write the singular vector at grade rs in terms of a composite screening charge as |χ r,s = dz f (z, h − ) |η r,s (1.1) where f is a polynomial in the free boson negative modes (h − is the subalgebra of the negative modes of the Heisenberg algebra, see (2.13)) and in the z = (z 1 , . . ., z r ) variables.This function comes from the normally ordered product of r vertex operators together with the prefactor generated by this ordering.The vector |η r,s is a suitable Fock highest-weight state specified by two positive integers r and s, such that the resulting expression |χ r,s is a well-defined (nontrivial) singular vector.
(2) Identify, within f , a Jack polynomial evaluated at z −1 i , so that where (s r ) is the rectangular partition whose diagram has r rows all of length s and α is a free parameter -free because this Jack actually does not depend upon α, being equal to a single monomial.This is so because the number of variables is equal to the number of parts in (s r ) and because (s r ) is the lowest partition (with respect to the dominance ordering) of degree rs having this property.The term R(z, h − ) contains a factor w(z; κ) which is precisely the weight function defining the scalar product for the Jacks with a particular parameter κ: This suggests trying to write the integral over f (z, h − ) in terms of a scalar product for which, by setting α = κ, we already have the left component, namely P (s r ) (z −1 ), and the weight w(z; κ).
(3) Introduce an algebra isomorphism ̺ relating the operators a n , n < 0, of h − to symmetric polynomials: where γ is a constant to be fixed shortly and p m = ∞ i=1 y m i with the y i being new variables (in unlimited number).Under the inverse action of this isomorphism, R(z, h − ) gets transformed into R(z, y).
(4) Then observe that there is a particular value of γ, related to κ, for which R(z, y) can be decomposed as for some constant b ν (κ).We have thus (5) We next use the orthogonality property of the Jacks, namely: the sought-for result.
1.3.How the argument is modified in the superconformal case.Again we keep the presentation at a sketchy level.The free field representation of the superconformal algebra involves a free boson and a free fermion.
(1) A singular vector at grade rs/2 (in a reducible highest-weight module of the superconformal algebra) is written in terms of a composite screening charge which is expressed in terms of the variables z = (z 1 , . . ., z r ) and a set of anticommuting variables θ = (θ 1 , . . ., θ r ).The function F (•) is the product of r normally ordered super-vertex operators.The notation (h ⊗ C) − refers to the negative modes of the free boson and the free fermion (C stands for Clifford algebra, which is sector-dependent, NS or R).The highest weight |η r,s is the tensor product of suitable bosonic and fermionic highest-weight states.
(2) Appealing to the theory of symmetric superpolynomials, we identify within the function F (•) a Jack superpolynomial (1.10) The replacement of the variables (z, θ) by (z −1 , ∂ θ ) in a superpolynomial can be interpreted as the adjoint operation defined by the orthogonality relation of superpolynomials: The label Γ, called a superpartition, is a direct generalization of the rectangular partition (s r ).In addition, we obtain a residual (sector-dependent) factor B(z, θ) that is neither part of a sJack nor part of the weight function w(z; κ) defining the sJack scalar product.The expression for B(z, θ) can be transformed into an operator that acts on the Jack superpolynomial, inside the orthogonality relation: (1.12) B † is symmetric but it does not act diagonally on P (κ) Γ (z, θ).Therefore B † P (κ) Γ (z, θ) generates a linear combination of Jack superpolynomials: (1.13) (3) We then define an algebra isomorphism whose inverse maps the term R(•) into the power sum version of the Cauchy formula which can be transformed into a bilinear sum of sJacks: (4) The singular vector (1.9) becomes then (5) Finally, we use the orthogonality relation of the sJacks, dz dθ w(z; κ) P to conclude that the singular vector is given by a linear combination of the terms that appear in Γ (z, θ) but whose expansion coefficients are dressed by normalization factors n Ω , i.e., d Γ,Ω → d Γ,Ω n Ω .The resulting expression for the singular vector is thus of the form Now, the factors n Ω are known.However, in general, the coefficients d Γ,Ω are not known.We thus end up an implicit, albeit general, expression for the singular vectors.
How does this compare with the conjectured expressions in [1,9]?Note first that these latter results are presented in a different sJack basis, that is, for a different value of the Jack parameter, which explains the mismatch with the present results.In general, the expressions presented here are simpler in that they contain less terms.Another advantage is that here they are derived, and therefore proved, as opposed to being conjectured.But in [1,9], there is an explicit conjecture for the expression of all the coefficients for any singular vector, which is not the case here.
1.4.Organization of the article.In Section 2, we collect review material on the superconformal algebra, its free field representation and the definition of the screening charges from which the singular vectors are constructed.Sections 3 and 4 are devoted to the explicit derivation of the representation of the singular vectors in terms of symmetric superpolynomials, in the NS and R sectors respectively.These analyses rely on the theory of Jack superpolynomials, which is summarized in Appendix A. Finally, in Appendix B, which is somewhat off the main theme of the article, we display a remarkable formula for the action of the operator B, in the NS sector, on the sJacks P (−3) Γ .
2. Super-Virasoro singular vectors from screening charges 2.1.The superconformal algebra.The generators of the superconformal transformations are the stressenergy tensor T (z) and its superpartner G(z), which satisfy the following operator product expansions (OPEs): where c ∈ C is the central charge.Writing the mode decomposition of these fields as where ǫ specifies the sector the expressions (2.1)-(2.3)are equivalent to the (anti)commutation relations The relations (2.6)-(2.8),with n, m ∈ Z and k, l ∈ Z + ǫ, define the N = 1 superconformal algebra which is denoted by svir ǫ .
Let M c,h denote the Verma module freely generated from the highest-weight vector |h defined by h being the conformal dimension.A basis for the descendant vectors in M c,h is The grade of the vector (2.10) is its L 0 -eigenvalue relative to the highest-weight state, namely i n i + j m j .
For generic values of c and h the module M c,h is irreducible.But when c and h are related to each other in a special way the Verma module is reducible.More precisely, let c = c(t) and h = h r,s (t) be parametrized by the following expressions where t ∈ C and r, s ∈ Z (and rs ∈ Z + ).Then, the Verma module M c(t),hr,s(t) , for all t = 0, contains a singular vector at grade rs/2.We stress that the parity of r + s determines the sector of svir ǫ : 2ǫ = (r + s − 1) mod 2.

Free field representation.
In view of connecting the expression for the singular vectors of the superconformal algebra with symmetric polynomials, we first define an embedding of svir ǫ into a free field algebra, namely, the algebra of a free boson and a free fermion.Let a(z) = n∈Z a n z −n−1 be the field whose Laurent modes satisfy the commutation relations of the Heisenberg algebra h, The field a(z) is the derivative of the free bosonic field It is thus natural to extend h by the addition of the mode a * , which satisfies the commutation relation Verma modules for h are called Fock modules.A highest-weight state |λ of the Fock module F (λ) with λ ∈ C is defined by a 0 |λ = λ|λ , a n |λ = 0, ∀n > 0. (2.16) The Fock modules F (λ) are irreducible and, as vector spaces, they are isomorphic to where S(h − ) is the symmetric algebra of h − or, equivalently, the polynomial ring in the negative modes of h.Note that the exponential of the operator a * acts on Fock modules as follows: Since the superconformal algebra contains odd generators (G r , r ∈ Z + ǫ), we also need to introduce a Clifford algebra in the free field representation.Let b(z) = n∈Z+ǫ b n z −n− 1 2 be the free fermionic field whose Laurent modes satisfy the Clifford algebra C ǫ , where n, m ∈ Z + ǫ and ǫ = 0, 1 2 .There is a unique Fock (Verma) module over C ǫ , for each choice of ǫ, and it is likewise irreducible.In the Neveu-Schwarz sector (ǫ = 1 2 ), the fermionic Fock module is isomorphic, as a vector space, to the exterior algebra in the negative modes: (2.20) The Ramond sector presents a minor complication because the fermionic Fock module has two independent ground states that are interchanged by b 0 .As b 0 squares to 1  2 and not 0, the Ramond Fock module cannot Moreover, we may also impose {b ± 0 , b n } = 0 for all n = 0.With this decomposition, we realize the Ramond Fock module as being isomorphic, as a vector space, to the exterior algebra (2.23) The full free field algebra is the tensor product of h and C ǫ .The corresponding Fock modules, denoted by F ǫ (λ), are the tensor product of the Fock module F (λ) for h with either the Neveu-Schwarz Fock module (ǫ = 1 2 ) or the Ramond Fock module (ǫ = 0).They are characterized by their highest-weight states which satisfy (2.16) as well as and n i ∈ Z, m j ∈ Z + ǫ.As a vector space, it is clear that (2.26) The relations (2.13), (2.15) and (2.19) imply the following OPEs (2.28) The sought for realization of the superconformal generators is where α 0 is a free (complex) parameter (the background charge in the Coulomb gas formalism) related to the central charge by (2.30) In terms of modes, (2.29) becomes where, we recall, ǫ = 0 ( 1 2 ) in the R (NS) sector.
Using the above expression for L 0 , the conformal dimension of a highest-weight state in F ǫ (λ) is found to be where (2.33) 2.3.Vertex operators and screening charges.In terms of the free fields just introduced, we define the (super) vertex operator as (2.34) where ζ = (z, θ) and θ is a Grassmann variable.The normal ordering in the vertex operator means that the positive and negative free field modes are separated, the former being placed at the right.By a straightforward computation, the normal ordering for two vertex operators, with in the NS sector and in the R sector.
It is also simple to verify that V λ (ζ) is a primary superfield, whose decomposition into field components takes the form V λ (ζ) = : exp[λφ(z)] : + θ λ b(z) : exp[λφ(z)] : .(2.37)For Φ(ζ) to be a primary superfield of dimension h, with Φ(ζ) = ϕ(z) + θξ(z), the following conditions need to be satisfied: The conformal dimension of the field V λ (ζ) is given by h λ , see (2.33).This entails the correspondence We next introduce the screening charges as (with the convention d θ θ p = δ p,1 ) where the value of α ± is fixed by enforcing h α± = 1/2, which yields (2.41) The constraint on the conformal dimension ensures the commutativity of Q ± with both T (z) and G(z), which is manifest from the second and fourth OPEs in eq.(2.38).
The screening charges Q ± define thus intertwiners between representations of the superconformal algebra.As a result, Q ± |χ is a highest-weight state if |χ is itself a highest-weight state.More generally, we can define a new set of screening charges by composition of the screening charge Q ± .Let k be a positive integer and define where C k is a certain integration contour. 1   In order to describe the superconformal singular vectors in terms of a Fock-space construction, we need to introduce the following Fock highest-weight states |α r,s : with r, s ∈ Z.We will denote by F r,s = F ǫ (α r,s ) the Fock module associated with the highest-weight state |α r,s with the understanding that (2.44) The action of ± relates Fock modules as follows: (2.45) The following statement links the composite screening charges with the explicit expressions for the singular vectors in svir ǫ .Proposition 2.1 ( [13]).In the Verma module M c(t),hr,s(t) , rs ∈ Z + , which belongs to the NS sector when r + s is even and the R sector when r + s is odd, there exists a nonvanishing singular vector at grade rs/2 given either by with h r,s = (α 2 r,s − α 0 α r,s )/2.

The Neveu-Schwarz algebra
We first consider explicit expressions for the singular vectors in the NS sector, i.e. for the svir 1 2 algebra.3.1.Manipulating the integral representation of the singular vectors: identifying therein a Jack superpolynomial.Using the expression (2.35) for the normal ordering of k vertex operators, one obtains for the screening operator (2.42) 1 The description of the contour C k is somewhat subtle and we refer the reader to [12,13] for an explicit construction.For the present calculations, we will take for granted that, up to a certain normalization (which is irrelevant for our purpose), the integration contour C k is equivalent to that appearing in the scalar product of the Jack (super)polynomials -see Appendix A.
where p n (z) and pn (z, θ) are the power sum symmetric superpolynomials (defined in (A.8)).(Note that, since the normalization of singular vectors is arbitrary, we will not care about global phase factors.)Acting with Q [k] ± on a generic highest-weight state |α p,q of the Fock module F p,q , we have where all positive modes have annihilated the highest-weight state.Notice that this action preserves the sector: p + q = p + q + 2k mod 2.
(3.9) Indeed, the monomial expansion of this sJack contains a single term since all lower terms with respect to the dominance order contain more than r parts and hence must be zero when the number of variables of each type is fixed to r, that is, when restricted to the non-zero variables (z 1 , . . ., z r , θ 1 , . . ., θ r ) (see eqs (A.11) and (A.24)).Note that this truncated sJack is independent of the value of κ.
Thus, in the expression that appears in the first line of (3.6), one recognizes the adjoint expression of this Jack superpolynomial (3.9) (namely, (3.9) with the replacements z i → z −1 i and θ i → ∂ θi -cf. the definition (A.17)).A closer look shows that the resulting integral has a structure akin to the (integral) scalar product defined in Appendix A. In the next subsection this will be made explicit.

3.2.
Implementing the relationship between free fields and symmetric polynomials.The idea is now to make precise the above observation and rewrite the expression (3.6) in the form of a scalar product of symmetric superpolynomials, following the analysis of the Virasoro and sl(2) cases [17,18].Using equation (A.16), we can rewrite equation (3.6) in the form of a (h ⊗ C ǫ )-valued scalar product as follows where κ + is fixed in order for (1−z i /z j ) (α 2 + −1)/2 to be the kernel of the scalar product, namely (1−z i /z j ) 1/κ+ (cf.(A.16)): and F stands for By using the orthogonality property of the Jack superpolynomials, if we set κ = κ + , then the singular vector will be given by the coefficient of F that is proportional to P Γr,s (z 1 , . . ., z r , θ 1 , . . ., θ r ).
Let us now introduce a sector-dependent isomorphism ρ ǫ , which, in the NS sector (ǫ = 1 2 ) is defined as It thus relates the symmetric superpolynomials and the free field negative modes.For this purpose, we have introduced a new set of (infinitely many) indeterminates (y, φ) = (y 1 , y 2 , . . ., φ 1 , φ 2 , . ..)where y i and φ i are even and odd variables respectively.
The expression (3.10) becomes In this last expression, we have swapped the first product in F given in (3.12) to the left side of the scalar product, thereby replacing it by its adjoint (using the definition (A.17)): with κ − = 2/(α 0 α − ) and where the sum is over superpartitions Ω such that In this section we give some examples of singular vectors using the superpolynomial construction.We consider only cases with r ≤ s and, to lighten the notation, we set In terms of the parametrization (2.11), we have The |χ 1,s sequence.For s = 1, 3, 5, . . ., we have Recall that any singular vector is defined up to a (non-zero) global constant.

3.3.2.
The |χ 2,s sequence.For s = 2, 4, 6, . . ., we have where so that Recall that such expressions are easy to handle since we work with only two variables of each type: z 1 , z 2 and θ 1 , θ 2 (simply because r = 2).Hence, we can write Dividing this expression by 1, 1 κ N and using the definition of the coefficient c Λ (κ; N ) (cf. eq.(A.26)) then, up to an irrelevant global factor, we have It only remains to evaluate a ratio of coefficients c Λ .Using formula (A.21) and the trick mentioned at the end of Appendix A (see (A.30) and the example given there), we obtain After simplification, we finally have As a consistency check, we have verified explicitly the correctness of the resulting expression of the singular vector for the first few values of s.

The Ramond algebra
We now turn to the construction of singular vectors in the Ramond sector, i.e. for the svir 0 algebra.This is somewhat more complicated than for the NS case essentially because the fermionic field b(z) now decomposes into integer modes: this brings half-integer powers of z and zero modes.The first point spoils the single-valued character of the vertex operator defined in (2.34) (which is also reflected in the normal ordered relation (2.36)).We have already, in Section 2.2, dealt with the second point by decomposing the zero mode b 0 into a linear combination of two new modes b + 0 and b − 0 which we treat as positive and negative, respectively.In any case, the screening charge construction can still be applied to obtain a symmetric polynomial representation of the singular vectors.± .We first point out a simple trick to remove the half-integer powers of the z i variables.Since the variables θ i are being integrated, we can make the change of variables: θ i = √ z i η i with η i being new Grassmann variables.In this case, the Berezin integration (which, we recall, acts as a derivative) becomes This transformation removes all the square roots of the z i in the vertex operators themselves, since and in the prefactor resulting from their normal ordering (cf.eq.(2.36)), i.e.
leaving only the overall multiplying factor resulting from (4.1) that will enter in the construction of the symmetric superpolynomial.We then obtain, after relabeling η → θ, the following expression: 4.2.Constructing the singular vectors.Using Prop.2.1 and eq.(4.4), the expression for the singular vector |χ + r,s at level rs/2 is In this equation, we have taken into account that the positive modes annihilate the highest-weight state.Note that this integral is well-defined (single-valued) only if r + s is odd (thanks to the factor i z −1/2 i coming from (4.1)), a condition that is satisfied in the R sector.Using the notation (3.7), we introduce the superpartition Γ r,s+1 of degree ( 1 2 rs|r) given by for r < s.The corresponding sJack is whose adjoint version is recognized in the first square bracket of (4.5).Again, it is independent of κ.The strategy is the same as in the NS sector.We observe that the factor 1≤i =j≤r where we used κ + = 2/(α 2 + − 1) = 2/(α + α 0 ), can be interpreted as the kernel of a scalar product • , • κ+ r .To achieve this goal, we introduce the isomorphism ρ ǫ which, for ǫ = 0, is defined by The sum is over all superpartitions Ω such that and the operator D(r; α + ) is given by A similar analysis for |χ − r,s leads to the following expression with κ − = 2/α 0 α − and where the sum is now over superpartitions Ω such that Ω ⊢ ( 1 2 rs|m), m = 0, 1, . . ., s.
. (4.17) Using eqs (A.27) and (A.28), we have The singular vector can be written explicitly as and the relative coefficient can be simplified to ) we have thus (with the normalization fixed so that the coefficient of a 3 −1 is 1) Of course, in the above expression, b 0 could be replaced by b − 0 / √ 2. Using κ = 2/(α 0 α + ), α 0 = (t − 1)/ √ t and α + = √ t, we recover the correct expression -in terms of free field modes -for the operator Ξ 2,3 defining the singular vector |χ 2,3 : for a given a ≥ l.Using property (A.24) with k = a − l, we have Since the operator B(l; t) acts non-trivially only on the anticommuting variables, through their derivatives, it is convenient to expand its action on a specific monomial in the θ i as where we choose to write the expansion in powers of (−t).The term B k results from the action of 2k distinct derivatives on θ 1 • • • θ l , thus reducing the total fermionic degree by 2k.Because B(l; t) is symmetric (in superspace), in the expansion (B.6) we can restrict ourselves to the analysis of the coefficient that contains θ 1 • • • θ l−2k -the complete expression for B k then being recovered by an appropriate symmetrization.
The coefficients B k turn out to have remarkably simple compact expressions.These are for the NS sector and for the R sector, where pf(•) denotes the pfaffian.Using the identities [n] but with each part reduced by 1 (which is δ (2n) (0, 0) in the notation of [5]).
to simplify the expression.Recall that the full expression of B(l; t) mΓ is obtained by first symmetrizing (B.13) and then summing over k from 0 to ⌊l/2⌋.It follows from [11] that the Jack polynomials evaluated at α = −3 are well-defined for the partitions µ ǫ [k], (meaning that they have no poles), since these partitions satisfy the (2, 2, 2k)−admissibility condition.Recall that a partition λ is (2, 2, 2k)−admissible if it has 2k parts and satisfies λ i − λ i+2 ≥ 2, i = 1, . . ., 2k − 2. (B.15) Moreover, these Jack polynomials vanish if any three of their variables coincide (this corresponds to the so-called clustering property).Consequently, the right-hand side of eq.(B.13) vanishes if any three variables in the set {z l−2k+1 , . . ., z l } coincide.But this is also true for three z i in the set {z 1 , . . ., z l−2k } due to the antisymmetry of the Vandermonde determinant.Moreover, the last product in the first line in (B.13) also vanishes whenever two variables that belong to one set coincide with one variable of the other set.Therefore, we have (with Γ defined in (B.The superpolynomials that vanish when three (bosonic) variables coincide have been studied in [10].Let us denote their vector space by F (2) l , that is, (B.17) The analog of the admissibility condition (B.15) for superpartitions is as follows.A superpartition Λ is (2, 2, l)−admissible if it satisfies be identified with [b 0 , b −1 , . ..].It may be realized as a direct sum of two copies of [b −1 , b −2 , . ..], however we shall find it convenient to instead follow [9, App.B] and decompose the zero mode as highest-weight state of the Ramond Fock module and b − 0 maps it to the other independent ground state.Imposing (b ± 0 ) 2 = 0 and {b + 0 , b − 0 } = 1 ensures the validity of the identity b

4. 1 .
Treating the square roots.The first step in the construction of the singular vectors consists in evaluating the composite screening charge Q[k]

(B. 18 ) 20 ) 5 ❦ 4 ❦ 1 ,
Denote the set of (2, 2, l)−admissible superpartitions by π l and introduce the following vector space I This conjecture implies that B(l; t) mΓ ∈ I (2,2) l .Quite remarkably, in the NS sector, an explicit expression for the action of B(l; t) on mΓ in the sJack basis {P (−3) Λ } can be conjectured.Let Λ be a superpartition with exactly m circles.From left to right (equivalently, from bottom to top), mark each circle of Λ with 1, 2, . . ., m.For instance Λ = (4, 3, 2, 1, 0; ) : ⊤ i (Λ), for 1 ≤ i ≤ m − 1, be the operator that acts on the diagram of Λ by replacing the circle marked by i with a box and removing the one marked by i + 1.These operations may be composed, noting that we do not relabel the circles during the intermediate operations.For instance ⊤ 2 (4, 3, 2, 1, 0; ) = (4, 3, 0; 2, 2) : ❦ (B.22) This expression(3.16)captures the exact form the NS singular vector at grade rs/2 when r ≤ s.Making it fully explicit requires the expression for the operator E(r; α + ) on P (i.e., for the same value of the free parameter, here κ + ).This expression is not known.But strangely enough, this expansion in terms of the sJacks P Ω E(r; α + )P .21)We stress that this expression for |χ 2,s is much simpler than that conjectured in [9, eq.(B.23)] which contains (s + 1) terms, instead of only two here.4.3.3.A detailed evaluation of |χ 2,3 .Let us detail the case (r, s) = (2, 3) in order to further illustrate the manipulations in the R sector.The two contributing Jack superpolynomials are