Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity.

1. Introduction 1.1. Schur polynomials. The simplest definition of the ubiquitous Schur polynomials is combinatorial [13, I.3] (and also [14, chap.4]). For a partition λ, the Schur polynomial s λ (x), where x stands for the set (x 1 , . . . , x N ), is the sum over monomials weighted by the content of semi-standard tableaux T with shape λ (whose set is denoted T (λ)): with x T ≡ x α = x α1 1 x α2 2 · · · x αN N where α is a composition of weight |λ| (the sum of all the parts of λ) and represents the content of T , namely T contains α i copies of i. For instance, in three variables, where each term is in correspondence with a semi-standard tableau: 3) The number of semi-standard tableaux of shape λ and content µ (where µ is also a partition) is the Kostka number K λµ . The above relation can thus be rewritten where µ = α + , the partition obtained by ordering the parts of the composition α in weakly decreasing order. In the last equality, the symmetric character of the Schur polynomial is used to reexpress them in terms of the monomial symmetric polynomials m µ (x), m µ (x) = x µ1 1 · · · x µN N + distinct permutations, (1.5) and the order in the summation is the dominance order: λ ≥ µ if and only if |λ| = |µ| and k i=1 (λ i − µ i ) ≥ 0 for all k.
A closely related approach for constructing the Schur polynomials is based on the Pieri formula. The latter refers to the decomposition in Schur polynomials of the product of two Schur polynomials, one of which being indexed by a single-row partition, say s (k) . Schur polynomials can thus be constructed from the identity by multiplying successive s (k) 's. Recall that s (k) = h k , where h k stands for the completely homogeneous symmetric polynomials: (1.6) to which is associated the multiplicative basis h µ = h µ1 h µ2 · · · h µN (with h 0 = 1). The link between the Pieri rule and the above definition of the Schurs is In other words, by filling the rows µ i with number i and applying the Pieri rule from left to right to evaluate the multiple product of the s µi , amounts to construct semi-standard tableaux. Then, isolating the term s λ from the complete product is equivalent to enumerate the semi-standard tableaux of shape λ and content µ.
The objective of this work is to present conjectured combinatorial definitions of the Schur superpolynomials and their dual. In addition, we present the conjectural version of the Pieri rule for Schur superpolynomials. By duality, the resulting tableaux are not those appearing in the combinatorial description of the Schur superpolynomials but rather those of their dual version.
1.2. Schur superpolynomials. Superpolynomials refer to polynomials in commuting and anticommuting variables (respectively denoted x i and θ i ) and their symmetric form entails invariance under the simultaneous permutation of the two types of variables. The classical symmetric functions are generalized in [7]. Super-analogues of the Jack and Macdonald symmetric polynomials appear as eigenfunctions of the Hamiltonian of the supersymmetric extension of the Calogero-Sutherland (cf. [5,6]) and the Ruijsenaars-Schneider (cf. [3]) models respectively. Denote the Macdonald superpolynomials by P Λ ≡ P Λ (x, θ; q, t) where Λ is a superpartition (cf. Sect. A.1), (x, θ) denotes the 2N variables (x 1 , . . . , x N , θ 1 , . . . , θ N ) and q, t are two extra arbitrary parameters. P Λ is defined from [1]: (1). P Λ = m Λ + lower terms, (1.8) where lower terms are w.r.t. the dominance order between superpartitions (cf. Def. A.4), m Ω stands for the super-monomial (cf. Def. A.5) and the scalar product is defined in terms of the power-sum basis p Λ (cf. Def. A.6, Eq.(A.14)) with where we used the representation Λ = (Λ a ; Λ s ) with m being the number of parts of Λ a , and z λ = i i ni(λ) n i (λ)! where n i (λ) is the number of parts equal to i in λ. The Schur superpolynomials s Λ and their dual,s Λ , are defined as limiting cases of P Λ (x, θ; q, t): That these are different objects follows from the fact that for some monomial f (q, t). Both s Λ ands Λ have integral super-monomial decompositions (see below, Conjecture 1.3), which hints for an underlying combinatorial description for each of them.
That s Λ is the most natural extension of the standard Schur polynomial is supported by the following conjectural result. Let J Λ denote the integral form of the Macdonald superpolynomials and φ be the homomorphism defined on power-sums as φ : (1.13) Set M Λ := φ(J Λ ). Then we have the following generalization of the Macdonald positivity conjecture (see e.g., [13,VI (8.18?)], proved in [10] 14) are polynomials in q, t with nonnegative integer coefficients.
The polynomials K ΩΛ (q, t) generalize the Kostka polynomials of the usual Macdonald case, which are recovered when m = 0 (that is, for Λ a = ∅). Interestingly, the m = 1 Kostka polynomials provide a refinement of the m = 0 ones (see [2,Conj. 25]).
In contradistinction with the usual case, the K ΩΛ (q, t) is not the coefficient of m Λ in the expansion of s Ω in the monomial basis when q = 0 and t = 1; these are rather those ofs Λ .
The expansion coefficients of s Λ ands Λ in the monomial basis, i.e., There is no relation between the numbersK ΛΩ and K ΛΩ except that they both reduce to ordinary Kostka numbers for m = 0. In addition, no relation has been found betweenK ΛΩ and K ΛΩ (q, t) for particular values of q and t. In contrast, as indicated above, K ΩΛ has the following relation with the generic K ΩΛ (q, t): We have K ΩΛ = K ΩΛ (0, 1). Now, back to (1.11). A closer look shows that these are not at once sound definitions since in both limits, the scalar product (1.10) is ill-defined. However, a deeper investigation relying on [9,11] reveals that both limiting forms of the Macdonald superpolynomials turn out to be well behaved and this results in the expression of s Λ and s Λ in terms of key polynomials (see [2, App. A]).
Here we present an alternative and direct -albeit conjectural -combinatorial definition for both s Λ ands Λ that specifies their expansion in the super-monomial basis. The expansion coefficients are obtained by enumerating appropriate generalization of semi-standard tableaux.
The article is organized as follows. We first introduce the notion of super semi-standard tableaux in Section 2.
The link between such tableaux of shape Λ and the Schur superpolynomial s Λ is presented in Section 3. The construction readily implies the symmetric nature of s Λ and the triangular character of the monomial expansion.
The Pieri rule for the s Λ is given in Section 4. In Section 5 we introduce the dual super semi-standard tableaux whose enumeration describes (conjecturally) the dual versions Λ . The link with Pieri tableaux that arise in the multiple application of the Pieri rule for Schur superpolynomials is spelled out in Section 6.
Two appendices complete this article. The first is a review of the necessary tools concerning superpartitions and the classical bases in superspace. In appendix B, we present a collection of generalized bilinear identities for Schur superpolynomials.

Super semi-standard tableaux
In this section, we generalize semi-standard tableaux to diagrams associated with superpartitions. By a super tableau, we refer to the filling of all the boxes and the circles of the diagram of a superpartition with numbers from a given set. A super semi-standard tableau T • of shape Λ of degree (n|m) (cf. Sect. A.1), is a filling of each of the n boxes and m circles in the diagram of Λ with number from the set I = {1, 2, . . . , N }, for N ≥ n, and subject to the following rules. (The • upper-script in T • reminds that these are tableaux containing circles.) The numbers in the set I m are considered to be the m largest numbers of the set I and are ordered as i 1 > i 2 > · · · > i m . In addition i m > j ∀j ∈ I c m . In I c m , the ordering is the natural one.
Definition 2.1. (Fermionic doublets). Let i k and i k+1 be two consecutive fermionic numbers, k ∈ {1, . . . , m − 1}, lying in distinct bosonic rows. We say that i k+1 and i k belong to a doublet if i k+1 can be paired with an i k in a lower row. Other occurrences of i k+1 are said to be singlets.
Consider, for example, the following partially filled tableaux (where only fermionic numbers are considered): For the tableau on the left, the number 2 in box (1, 5) belongs to a doublet since it can be paired with the 1 in position (2,5); the remaining two 2 are singlets. (The 2 in the doublet could as well be taken to be the one in position (1,6).) Note that the three 1 in the third row cannot be parts of doublets being in a fermionic row. In the second tableau, the three 2 are singlets. For instance, the tableau at the right in (2.3) does not satisfy this rule since there can be at most two singlets 2 given that c 2,1 = 2. As a further example, notice that are not allowed but Indeed, since c 3,2 = 0, the 3 cannot be a singlet. In the second tableau the 3 is paired but now c 2,1 = 0 requires the 2 to be paired with a 1.
Rule 2.5. (Semi-standard filling). The bosonic rows are filled with numbers in I using the ordering defined in Rule 2.2 and such that numbers in rows are weakly increasing and strictly increasing in columns (disregarding the frozen numbers in fermionic rows).

Definition 2.2.
A tableau T • satisfying rules 2.1-2.5 is called a super semi-standard tableau.
Here is an example (2.5) When fermionic numbers appear in bosonic rows, being largest than the bosonic numbers, they occupy the rightmost positions (and by consequence, the downmost positions in columns). Once all fermionic numbers have been inserted, it is convenient to work with a reduced diagram, or, if filled, a reduced tableau. The numbers in boxes of the reduced tableau satisfy the ordinary semi-standard-tableau conditions for the numbers in the set I c m . In other words, T • red is a genuine semi-standard tableau. For the example (2.5), T • red is simply 6 6 7 8 7 7 8 . (2.6)

The combinatorial Schur superpolynomials
In this section, we present the conjectural combinatorial definition of the Schur superpolynomials. It is formulated in terms of the tableaux introduced in the previous section.
Let Λ be of degree (n|m). We defined a monomial ζ T • in the variables (x, θ) associated to the tableau T • by introducing a factor x i for each number i appearing in a box and a θ j for a circled j, the circle content being read from top to bottom: with θ Im = θ i1 · · · θ im and T * denotes the box content of T • . For instance, the monomial corresponding to the tableau (2.5) is where T • (Λ) denotes the set of super semi-standard tableaux of shape Λ.
The upper script c refer to the combinatorial definition: s c Λ might differ in principle from s Λ defined in (1.11). For example, s c (0;2,1) is obtained by summing the contribution of the tableaux: whose variable transcription reads In order to show that the superpolynomials s c Λ can be written in the basis of the monomial superpolynomials, their symmetric character must first be established.
Proof. It suffices to show that s c Λ is invariant under elementary transpositions, e.g.,: where (i, i + 1) is the permutation that exchanges simultaneously x i ↔ x i+1 and θ i ↔ θ i+1 . This action is lifted to tableaux as follows. Consider the involution : T • →T • such that the numbers of i's and (i + 1)'s are exchanged from T • toT • . Two cases need to be described.
(1) If i and i + 1 are bosonic numbers (i.e. ∈ I c m ), the involution only transforms the reduced tableaux, which are semi-standard. In that case, the involution is taken to be the usual one (see e.g., [14,Prop. 4.4.2]). From T • red , we identify all fixed pairs i, i + 1 (meaning: adjacent in the same column); other occurrence of i or i + 1 are said to be free. Then, in each row, the k free i and the l free i + 1 are replaced by l free i and k free i + 1. Here is an example to illustrate this with i = 4: 3 4 4 4 5 5 5 2  4 4 5 6 6 7 7 7 1  1  (2) If i and/or i + 1 ∈ I m , the operation simply amounts to interchange all the numbers i and i + 1 in T • . Here is an example for which both i = 1 and i + 1 belong to I m : 3 3 4 4 4 5 5 5 2  4 4 5 6 6 7 7 7 1  1 3 4 4 4 5 5 5 1  4 4 5 6 6 7 7 7 2  2 In both cases, the resulting tableau is still an element of T • (Λ) and clearly this transformation is an involution.
To substantiate the conjectural equivalence of s c Λ and s Λ (cf. Conjecture 3.1 below), we demonstrate the unitriangularity of s c Λ in its super-monomial expansion.
whereK c ΛΩ is the number of super semi-standard tableaux of shape Λ and content Ω, with Proof. Since s c Λ is symmetric, the decomposition (3.3) can be rewritten as an expansion in terms of the monomials m Ω (which form a basis for symmetric superpolynomials [7]). As a result, the expression (3.10) follows directly from the definition ofK c ΛΩ as the number of elements of T • (Λ) having content Ω. What has to be shown then is the statement (3.11), i.e., that the expansion is unitriangular.
At first, observe that one can focus on tableaux for which I m = {1, . . . , m} and with box content (1 Ω1 , 2 Ω2 , . . . , N ΩN ) in order to identify the multiplicity of the monomial m Ω since: Now, consider all the different monomials m Ω appearing in the expansion of s c Λ , cf. eq.(3.10). First, consider those fillings of Λ such that no fermionic numbers appear in bosonic rows. In this case, the content Ω = (Ω a ; Ω s ) is necessarily such that Ω a = Λ a . We are left with the filling of reduced diagrams of shape Λ s and content Ω s . In other words, we haveK where K Λ s ,Ω s , being indexed by two ordinary partitions, refers the usual Kostka numbers. The unitriangularity of the K Λ s ,Ω s 's proves the triangularity ofK c ΛΩ for the special case where Ω a = Λ a and it implies thatK c ΛΛ = 1. Next, consider the case where Ω a = Λ a . By construction, we have Ω a > Λ a (with respect to the dominance ordering for partitions but relaxing the constraint |Ω a | = |Λ a |). Indeed, the numbers k within a row, say i, ending with the circle k are frozen and some extra k may be inserted in the upper-right part of the tableau; therefore Ω a i ≥ Λ a i . Suppose that one fermionic number k is introduced in row i. Focussing on the tails of the two rows concerned here, we have: · · · · · · ❦ k part of Λ with a marked circle → · · · k · · · ❦ k part of content Ω → · · · · · · k ❦ k part of shape Ω (3.14) Clearly, the relation between Λ and Ω is as follows: the diagram of Ω is obtained from that of Λ by moving a box downward from a bosonic to a fermionic row. This operation satisfies Λ * ≥ Ω * and Λ ⊛ ≥ Ω ⊛ so that Λ ≥ Ω (cf. Def. A.4). Indeed, it suffices to compare the superpartitions composed of the two concerned rows: (b; a) ∈ Λ and (b + 1; a − 1) ∈ Ω, with a > b, and testing successively the truncated version of Λ * ≥ Ω * and Λ ⊛ ≥ Ω ⊛ : Now consider inserting several k in the same row (here 3) · · · · · · ❦ k part of Λ with a marked circle → · · · k k k · · · ❦ k part of content Ω → · · · k k k ❦ k · · · part of shape Ω from which it follows that Λ * ≥ Ω * and Λ ⊛ ≥ Ω ⊛ . Since these two processes can be done iteratively, the action of filling several boxes of bosonic rows with fermionic numbers always produce terms of lower degree with respect to the dominance ordering.
We now end this section with the announced conjecture that identifies the Schur superpolynomials defined combinatorially, namely s c Λ , with those obtained from Macdonald superpolynomials for q = t = 0, denoted s Λ : We Here the bosonic rows of Λ s all lie below the fermionic ones. Since the fermionic number k cannot appear in rows below the one ending with circled k, there cannot be any occurrence of the fermionic numbers in the bosonic rows. These bosonic boxes are then filled with numbers in the set {m + 1, · · · , N }, generating usual semi-standard tableaux enumerated by the usual Kostka coefficients.

The Pieri rule
We now present a conjectural version of the Pieri rule for the Schur superpolynomials s Λ defined by (1.11). In order to formulate the rule, we must recall the notion of horizontal and vertical k-strips: an horizontal (reps. vertical) k-strip has at most one square in each column (reps. row). In the following, we need an extension that includes circles.  representing respectively, an horizontal ❦ 3 -strip, a vertical ❦ 4 -strip and a 3-strip that is both horizontal and vertical. The Pieri rule relies on a specific rule, spelled out in the following, for the multiplication of a row or a column diagram with a generic diagram. I. Row multiplication: the squares and the circle (if the row is fermionic) that are added to the diagram Λ must form an horizontal strip, in addition to generate an admissible resulting diagram (i.e., rows are weakly decreasing and there can be at most one circle per row and column). Moreover, when inserting the squares of a row into the diagram of Λ, the circles of Λ can be displaced subject to the following restrictions: (i) a circle in the first row can be moved horizontally without restrictions; (ii) a circle not in the first row can be moved horizontally as long as there is a square in the row just above it in the original diagram Λ (i.e., the circle in row i can be displaced by at most Λ * i−1 − Λ * i − 1 columns); (iii) a circle can be displaced vertically in the same column by at most one row. II. Column multiplication: Interchange 'row' and 'column', 'horizontal' and vertical', 'above' and 'at the left' in I.

Definition 4.2. (Pieri diagrams)
. Let Λ and Γ be two superpartitions, with Γ a row or a column diagram (bosonic or fermionic). We denote by Λ ⊗ Γ the set of all admissibles diagrams, called Pieri diagrams, obtained by the multiplication of Γ with the diagram of Λ using the rule 4.1.
Note that a strip of either type needs not to be located completely on the exterior (or right) boundary of the larger diagram. However, when the circles are erased, the strip is indeed at the exterior frontier of the diagram, which is clear from the examples (4.1). Now, consider for example the multiplication of a bosonic row of length 3 with the diagram of Λ = (2, 0; 1), the operation being denoted by: where the boxes of the diagram ( ; 3) are marked by 1. Using rule 4.1, the resulting Pieri diagrams (or tableaux, the diagrams being partially filled) are: If, instead, we multiply a fermionic row, we obtain: (4.4) A similar example illustrating the multiplication of a bosonic column, namely (; 1, 1) (whose squares are marked 1 and 2) into Λ = (2, 0; 1), yields : while turning the column into a fermionic one by the addition of a circle marked 3, leads to a single configuration, namely (4.6) We are now in position to formulate the conjectural form of the Pieri rules. s Ω , and s Λ s (0; 1 r ) = The symbol #ℓ ⊙ stands for the number of circles in the diagram of Ω that lie below the one which has been added.
Here are some examples illustrating these Pieri formulas.
Example 4.1. Consider the product s (4,0;3) s (3;) . Using the multiplication rule 4.1, we find the following diagrammatic Schur superpolynomials expansion: Note that the last tableau appears with a minus sign since there is one circle below the added one (marked with 1). 1 Written in terms of the Schurs, this reads: s (4,0;3) s (3;) = s (6,4,0;) + s (5,4,1;) + s (5,4,0;1) − s (4,3,0;3) . (4.10) Remark: Here one might wonder why no tableau appears where the bottom unmarked circle is moved horizontally by two units (which is permitted by the length of the row just above) in (4.9). The only option would be (4.11) But the building strip is not an horizontal ❦ 3 -strip: its upper-right component is a square and not a circle.
Example 4.2. Consider next the product s (1;2,1) s (0;1,1,1) and fill the column of the second diagram with numbers 1 to 4. Using the Pieri formula, we have: (4.12) The third and fourth tableaux illustrate the vertical motion of the circle. The fifth and sixth tableaux exemplify the allowed horizontal move by one column even if it exceeds the number of squares of the previous row in the original tableau as long as this upper slot is occupied by a square of the strip (here the square marked 1). Note that in all cases, the circle marked 4 is below the unmarked one: the factor #ℓ ⊙ is 0 in all diagrams, so that we have: Remark: Note that the following tableau is not allowed in the decomposition (4.12) since the top circle has move vertically by two units. This is forbidden because that gives the circle a position exceeding the number of squares in the first column (cf. the point II (ii) of the rule 4.1). (4.15) 1 The origin of the relative sign is clear from the algebraic point of view: since a circle is associated with a factor θ (the anticommuting variables), we see that, compared with the first three diagrams, the last one is associated with a different ordering of first two θ factors. Upon reordering, this yields the minus sign.
Let us compare the tableaux resulting from the successive applications of the row-version of the Pieri rule with the super semi-standard tableaux described in Section 2. Consider for instance a complete filling of the tableau at the left in (2.3) for which the ordering is 1 > 2 > 5 > 4 > 3. Clearly, there is no way to remove successively horizontal strips composed of 1, and then of 2, 5, etc. Therefore, although the Pieri rule builds up Schur superpolynomials, there is no relation between the Kostka numbersK ΛΩ and this Pieri rule. It turns out that the latter is directly related to the dual semi-standard tableaux enumerated by the Kostka numbers K ΛΩ , namely, the expansion coefficients of the dual Schur superpolynomialss Λ in the monomial basis. These dual tableaux are introduced in Sect. 5.

The combinatorial dual Schur superpolynomials
In this section, we introduce a conjectural combinatorial definition for the second family of Schur superpolynomials, the dual versions Λ . It is formulated in terms of a dual counterpart of the super semi-standard tableaux.

5.1.
Dual super semi-standard tableaux. We thus first introduce these dual tableaux. The qualitative dual refers to the way the fermionic numbers are ordered and the rules for their insertion in the tableau. The rule 2.1 still holds and the set I m is defined as before, as the ordered set of the labels in circles read from top to bottom. But the rule 2.2 is replaced by its dual: Rule 5.1. (Dual ordering in the set I). The numbers in the set I m are still considered to be the m largest numbers of the set I but they are now ordered as i m > i m−1 > · · · > i 1 ( and i 1 > j ∀j ∈ I c m ). In I c m , the ordering is the natural one.
As already indicated, the rules for the filling of the fermionic numbers are also modified. At first, the numbers in the fermionic rows are no longer frozen. In addition, the fermionic number i k , when appearing in a bosonic row, is forbidden in the squares at the upper-right of the circled i k ; it can only appear in the dual region, at its lower-left. This is made precise in the following: a. the number i k can only appear at the left of the column ending with circled i k ; b. the fermionic number i k−ℓ , with ℓ ≥ 1, must appear above or at the right of the circled i k at least ℓ times; c. counting the boxes from right to left and top to bottom, the number of boxes marked i k must always be strictly greater than those marked i k+1 .
For instance, the fermionic filling (with I 2 = {1, 2}) of the shape (5, 3; 1) and content (3, 2; 1 4 ) are where K c ΛΩ is the number of elements ofT • (Λ) with content Ω and it satisfies This proof will also be omitted.
The multiplicity of each tableau is equal to the multiplicity of the corresponding reduced standard (i.e., with all entries being distinct) tableaux. This gives multiplicity 4 for the first two tableaux, 5 to the following two and 6 for the remaining ones which gives a total of 30, i.e., K c (8,4;1,1),(6,3;1 5 ) = 30. Conjecture 5.1. We haves c Λ =s Λ . Equivalently, K c ΛΩ = K ΛΩ .
Although there is no relation between the two versions of the Schur superpolynomials, the next section reveals a striking indirect connection. 6. Relating the Pieri rule to dual semi-standard tableaux As indicated at the end of Section 4, the Pieri tableaux obtained by successive row multiplications do not correspond to super semi-standard tableaux. They are rather related to their dual versions. This is a consequence of the nontrivial duality property of the Schur superpolynomials. Let ·, · be the scalar product (1.10) with q = t = 1: where s * Λ := (−1) ( m 2 ) ω(s Λ ′ ) and ω being the involution defined as [2,Corr.29]: Now, if we define a new basis H Λ given by H Λ :=p Λ a h Λ s , we observe that: Setting Λ = (n; ) in the previous relation implies thatp n = s (n;) . Indeed, (n; ) is the largest superpartition with degree (n|1) so that there is a single contributing term in the sum, and its coefficient is K ΛΛ = 1. The equation (6.5) can thus be written as: definition for the dual super semi-standard tableaux: K ΩΛ is given by the cardinality of the setP • (Λ, Ω). For the above example, Λ = (2, 1; 1 3 ) and Ω = (3, 2; 1), after the first step one has which gives forP • ((2, 1; 1 3 ), (3, 2; 1)): (6.14) This ordering could be used to formulate new rules for the construction of dual super semi-standard tableaux (rules which we have not found however). But we then loose the connection with the usual Kostka numbers that enumerate the filling of the reduced tableaux obtained once the fermionic numbers have been introduced (cf. the example 5.1 and (3.24) for a similar result in the non-dual context), whose connection is a great computational advantage.

Appendix A. Superpartitions and symmetric superpolynomials
In this first appendix, we summarize the basic notions and definitions pertaining to symmetric superpolynomials.
A.1. Superpartitions. Superpartitions are generalization of regular partitions and are the combinatorial objets used to label symmetric superpolynomials. We first give the following definition. such that We stress that Λ a has distinct parts and the last part is allowed to be 0. The number m is called the fermionic degree of Λ and n = |Λ| = i Λ i is the bosonic degree. Such superpartition Λ is said to be of degree (n|m), which is denoted as Λ ⊢ (n|m).
The diagrammatic representation of superpartitions is very similar to the usual Young diagrams representing partitions. By removing the semi-coma and reordering the parts of Λ, we obtain an ordinary partition that we denote Λ * . The diagram of Λ is that of Λ * with circles added to the rows corresponding to the parts of Λ a and ordered in length as if a circle was a half-box [7]. Here is an example of a superpartition with degree (27|5): which corresponds to Λ = (8, 6, 3, 2, 0; 5, 3). Each box and circle in the diagram of Λ can be identified by its position s = (i, j), where i denotes the row, numbered from top to bottom, and j denotes the column, numbered from left to right. A row or column ending with a circle is dubbed fermionic. Other rows and columns are said to be bosonic. In the above diagram, the box with a ⋆ has position s = (4, 3); it belongs to the third fermionic row and the second fermionic column. With this diagrammatic representation, it is simple to define the conjugate operation.  We now introduce the version of the dominance ordering that applies to superpartitions; it relies on the Λ * , Λ ⊛ representation.
Pictorially, Ω < Λ if Ω can be obtained from Λ successively by moving down a box or a circle. For example, In the latter two cases, the superpartitions are non-comparable.
A.2. Symmetric superpolynomials. Superpolynomials are polynomials in the usual commuting N variables x 1 , . . . , x N and the N anticommuting variables θ 1 , . . . , θ N . Symmetric superpolynomials are invariant with respect to the interchange of (x i , θ i ) ↔ (x j , θ j ) for any i, j [5]. The space of symmetric superpolynomials, denoted R SN = F [x, θ] SN where F is some field, is naturally graded : where R SN (n|m) is the space of homogeneous symmetric superpolynomials of degree n in the x variables and degree m in the θ variables. Bases of R SN (n|m) are labelled by superpartitions of degree (n|m). We now present the superpolynomial version of the classical bases [7].

Appendix B. Bilinear identities in superspace
We end this article with the presentation of a series of bilinear identities for Schur superpolynomials. They generalize the bilinear identity for the standard rectangular-type Schur polynomials found in [12]: where as usual k n means that the part k is repeated n times. This identity can be represented diagrammatically as (with k = 3 and n = 2) This (remarkable) identity is equivalent to the Dodgson's condensation formula (also know as the Desnanot-Jacobi matrix theorem), which reads (see e.g., [4] We now consider generalizations of (B.1) for the Schur superpolynomials. Note that tentative proofs along the above lines are bound to fail due to the absence of determinantal formula for superpolynomials. Here, a rectangular diagram can be extended in different ways by the adjunction of circles. For example for the above diagram (3 2 ) is generalized by the following four super-diagram (for which either Λ * or Λ ⊛ is rectangular): Conjecture B.1. Let k, n be integers with k > 1 and n > 1. Let r ′ , r ∈ {k, k − 1, 0} with r ′ > r and ǫ = δ r,0 . We have s (r;k n−1+ǫ ) s (k n ) = s (r+1−ǫ;(k+1) n−1+ǫ ) s ((k−1) n ) + s (r;k n+ǫ ) s (k n−1 ) (B.5) s (r ′ ;k n−1 ) s (r;k n−1+ǫ ) = s (r ′ +1,r+1−ǫ;(k+1) n−2+ǫ ) s ((k−1) n ) + s (r ′ ,r;k n−1+ǫ ) s (k n−1 ) (B.6) s (k,0;k n−1 ) s (k n ) = s (k+1,0;(k+1) n−1 ) s ((k−1) n ) + s (k,0;k n ) s (k n−1 ) (B.7) Note that in the first two cases, each identity is a compact formulation for thee identities as there are three ways of selecting the pair (r, r ′ ). For example, the identity (B.6) with k = 3, n = 2 and r ′ = 3, r = 2 reads For the same values of k and n, the diagrammatic form of (B.7) is: In addition, similar bilinear identities have been found for almost rectangular-type super-diagrams, namely for which neither Λ * nor Λ ⊛ rectangular.