A Q-analog of Foulkes' Conjecture

We propose a q-analog of classical plethystic conjectures due to Foulkes. In our conjectures, a divided difference of plethysms of Hall-Littlewood polynomials H n (x; q) replaces the analogous difference of plethysms of complete homogeneous symmetric functions h n (x) in Foulkes' conjecture. At q = 0, we get back the original statement of Foulkes, and we show that our version holds at q = 1. We discuss further supporting evidence, as well as various generalizations, including a (q, t)-version.


Introduction
The Foulkes' conjecture, which dates back to 1950 (see [10]), has a long and interesting history.Some headway has been made on it, but it remains open in general.A survey of the current state of affairs can be found in [6,14], and related papers include [3,5,7,17,18,19,25].In its original form, the conjecture states that, for any positive integers a and b, with a < b, the difference of plethysms1 of complete homogeneous symmetric polynomials expands with positive (integer) coefficients in the Schur basis {s µ } µ n of symmetric polynomials (here µ runs over the set of partitions of n = a b).Instances of this positivity are From the point of view of representation theory, one may interpret Foulkes' conjecture as saying that there is a GL(V )-module inclusion of the composite of symmetric powers S a (S b (V )) inside S b (S a (V )).Therefore each GL(V )-irreducible occurs with smaller multiplicity in S a (S b (V )) than it does in S b (S a (V )), and the conjecture reflects this at the level of the corresponding characters (with Schur polynomials appearing as characters of irreducible representations).Many interesting point of view may be considered, and some of these are nicely discussed in [14] both with an historical perspective2 , and explanations of ties with Geometric Complexity Theory.
Here, we consider symmetric "polynomials" (we will often say function) in a denumerable set of variables x = x 1 , x 2 , x 3 , . .., which are typically not mentioned explicitly.This makes it so that our statements hold irrespective of the number of variables occurring in the symmetric functions considered (ergo the dimension of the vector space V ).Notice that the case of Foulke's conjecture when this dimension is 2 is a theorem.Indeed, it corresponds to Hermite's Law of reciprocity (see [11]), which says that the modules S a (S b (V )) and S b (S a (V )) coincide when V has dimension 2 (over C).Another fact that is worth recalling is that Brion [4] has shown that (1) holds if b is large enough with respect to a.
Our q-analog replaces the relevant homogeneous symmetric function h n = h n (x) by the Macdonald (Hall-Littlewood) polynomial H n (x; q) := µ n K µ (q)s µ (x), where K λ (q) = τ q c(τ ) , with τ running through the set of standard tableaux of shape λ, and c(τ ) standing for the charge statistic.As we will recall further below, one has H n (x; 0) = h n (x).Hence we are considering a slightly different notion of q-analog, in which it is the specialization at q = 0 (rather than q = 1) that gives back the original statement.
Conjecture 1 (q-Foulkes).For any integers 0 < a b, the Schur function expansion of the divided difference For short, one says that F a,b (x; q) is Schur positive.This has been checked to hold whenever ab 25, and we will prove in the sequel that the conjecture is true at q = 1.
This does specialize, at q = 0, to the corresponding case of Foulkes' conjecture: A second part of Foulkes' conjecture, shown to be true by Brion [4], concerns the stability of coefficients as b grows while a remains fixed.To simplify its statement, we consider the linear operator which sends a Schur function s µ (x) to s µ (x), where µ is the partition obtained by removing the largest part in µ.Let us write f for the effect of this operator on a symmetric function f .For example, we get s 622 + s 442 + s 4222 + s 22222 = s 22 + s 42 + s 222 + s 2222 , which is clearly not homogeneous.Using this notation convention, the second part of Foulkes' conjecture states that, for all a b, the Schur expansion of also affords positive integers polynomials as coefficients.Observe that the "Bar" operator allows the comparison of homogeneous functions of different degrees, namely f a,b+1 of degree a(b + 1) with f a,b of degree ab.For instances of, one calculates f where the sum λ + µ of two partitions λ and µ, is the partition whose parts are λ i + µ i (with the convention that λ i = 0 if i greater than the number of parts of λ).More results along these lines may be found in [12,19].A similar phenomenon also seems to hold in our context, leading us to state the following.

Extensions
In her thesis, partly presented in [24], Vessenes attributes 3 the following generalization of Foulkes' conjecture to Doran [8].As before, let a < b and consider two extra integers c and d, both lying between a and b, such that ab = cd (observe that we do not assume that c < d).Then Vessenes's extension of Foulkes' conjecture states that expands with positive coefficients in the Schur basis.Setting c = b and d = a clearly gives back the original statement of Foulkes.For example, one calculates that Vessenes proves that ( 5) is indeed Schur positive whenever a = 2, and more instances may be shown to hold using formulas of [13] for the plethysms s µ • h k , for partitions µ of 3, 4 and 5.We have also checked it explicitlly for all cases when the overall degree is less or equal to 36.Extensive computer algebra experiments4 suggest that there are two natural (new) extensions to this conjecture, namely Conjecture 3. Assume that a c, d b, with ab = cd, and k is any positive integer, then we have Schur positivity of both the differences where we consider the rectangular shape partitions More experiments (for all cases when the overall degree is less than 18) suggest that all the above conjectures are all encompassed in the following one, which involves the (combinatorial) two parameters Macdonald polynomials H µ (x; q, t) indexed by rectangular shape partitions.
Conjecture 4. Assume that a c, d b, with ab = cd, and k is any positive integer.Then, we have N[q]-Schur positivity of We will show why division by 1 − q makes sense and how it implies previous conjectures in the following section.It is interesting to observe that, at t = 1, formula (7) specializes to the nice q-analog which thus have to be Schur positive if Conjecture 4 is to hold.

Supporting facts and implications
To further discuss and exploit the several implications of conjectures 1 and 4, and their ties to conjecture 3, we rapidly recall some properties and specializations of the symmetric polynomials H µ (x; q, t).

Combinatorial Macdonald polynomials
Recall that the polynomial H n (x; q) is a special instance of the "combinatorial" Macdonald polynomials H µ = H µ (x; q, t).Since the parameter t only appears in the H µ 's that are indexed by partitions having at least 2 parts, it makes sense to avoid its mention in H n .
For instance, we have The H µ are orthogonal with respect to the scalar product defined on the power-sum basis by the formula with p µ (x), p λ (x) q,t = 0 whenever µ = λ.Here, (µ) stands for the number of parts of µ, and k runs over the parts of µ.They afford the following specializations: Furthermore we have which may be written in the following form using plethystic rules of calculation, and formula (32): with = (µ) standing for the number of parts of µ.We also have the symmetry Hence it follows that Now, we may show that division by 1 − q makes sense in formulas ( 2) and (7).Indeed, for any a and b, formula (10) implies that we have (H a • H b )(x; 1) = h 1 (x) ab , since the the electronic journal of combinatorics 24(1) (2017), #P1.38 evaluation at q = 1 is compatible5 with plethysm.Hence, the numerator of the right-hand side of (2) vanishes at q = 1, and is thus divisible by 1 − q.More generally, using (13) with ab , it follows that the coefficients of each Schur function in the expression of ( 7) are polynomials in q.
Several specializations of ( 7) are of interest.In particular, at q = 0, we get where we write f s g if g − f is Schur positive.Comparing coefficients of the highest power of t on both sides of this gives which is precisely the second statement in (6).Likewise, setting t = 1 we get Specializing at q = 0 and t = 0 also gives the following instance of ( 5) We will also consider in the sequel the following6 q-analog of Schur functions: S µ (x; q) := ω q n(µ ) H µ (x; 1/q, 0), defined in terms of specialization at t = 0 of the combinatorial Macdonald polynomials7 H µ (x; q, t), with ω standing for the "usual" linear involution that sends s µ (x) to s µ (x).

Dimension Count
As mentioned previously, when a homogeneous degree n symmetric function f occurs as a (graded) Frobenius transform of the character of a S n -module, the dimension (Hilbert series) of this module may be readily calculated by taking its scalar product with h n 1 .On the other hand, general principles insure that there exists such a module (albeit not explicitly known) whenever f expands positively (with coefficients in N[q]) in the Schur function basis.Finding an explicit formula for this "dimension" may give a clue on what kind of module one should look for in order to prove the conjectures.With this in mind, let us set the notation dim(f ) := h n 1 , f .For instance, we may easily calculate that since h ab 1 may only occur in the plethysm In a classical combinatorial setup, formula ( 17) is easily interpreted as the number of partitions of a set of cardinality ab, into blocks each having size b.We say that this is a b a -partition.Indeed, using a general framework such as the Theory of Species (see [2]), it is well understood that h a • h b may be interpreted as the Polya cycle index enumerator of such partitions, i.e.: where n = ab, and d k denotes the number of cycles of size k in σ.Here, we further denote by fix σ the number of b a -partitions that are fixed by a permutation σ, of the underlying elements.It follows that is the difference between the number of a b -partitions and b a -partitions.Some authors have attempted to exploit this fact to prove Foulkes' conjecture (for positive and negative results along these lines see [20,21,22,23]).
It is interesting that we have the following very nice q-analog (at 0) of ( 18).
Proposition 5.For all a < b, we have and, letting q → 1, we find that the electronic journal of combinatorics 24(1) (2017), #P1.38 Proof.We first calculate dim(H a • H b ) directly as follows Now, exploiting classical properties of the logarithmic derivative D log f := f /f (with respect to q), we easily calculate that From this we may readily obtain that lim q→1 dim(F a,b (x; q)) gives (20).
3 The q-conjecture holds at q = 1 We start with an explicit formula that will be helpful in the sequel, setting the simplifying notation respectively for the even-part and odd-part of (h 2 + e 2 ) b in the "variables" h 2 and e 2 .We thing of these as the homogenous and elementary symmetric functions, for which we have the power-sum expansions p 2 1 = h 2 + e 2 and p 2 = h 2 − e 2 .
Lemma 6.For all a, b and k, we have the divided difference evaluation Proof.The limit on the left hand-side is the evaluation at 1 of the derivative of H a • H k b with respect to q.We use formula (14) to calculate this, exploiting the fact that the evaluation at 1 of the q-derivative of g(q) = (1 − q) m f (q) is Using (14) and the rules of plethysm to expand H a • H k b , and observing that the only partitions µ of a such that a − (µ) 1 are either µ = 1 a or µ = 21 a−2 , we find that we have the expansion with G(x; 1) = 0, and where [a]! q stands for the q-analog of a!.Thus, we get that the left the electronic journal of combinatorics 24(1) (2017), #P1.38 hand-side of ( 22) evaluates as recalling that H n (x; 1) = p n 1 and that p n this last expression is clearly equal to the right-hand side of equation (22).
It immediately follows that we have the following formula as a difference of two expressions obtained from the lemma.Proposition 7.For any 1 < a c, d b and k 1, with n := abk = cdk, we have Moreover, this a positive integer coefficient polynomial in h 1 , h 2 and e 2 ; hence, it expands positively in the Schur basis.
For instance, we get To make the positivity in the previous proposition more apparent, we exploit the following recursive approach to the calculation of F a,b = F a,b;b,a , as a polynomial in h 1 , h 2 and e 2 , together with Appendix 4.
with Θ a (b) defined recursively as the electronic journal of combinatorics 24(1) (2017), #P1.38 Using these calculation techniques, we find that In particular, for all b > a > 1, we have.
which implies the analog at q = 1 of the stability portion of Foulkes' conjecture, namely Proposition 9.For all a < b, and all partition λ, we have Proof.Indeed, using the classical Pieri rule for the calculation of h 1 s λ , it is easy to see that h It is interesting to calculate how F a,b (x; q) expands explicitly as a polynomial in q.Indeed, by a direct calculation, one gets with similar (but more intricate expressions as illustrated below) for higher degree terms.Hence, the conjectured Schur-positivity of F a,b (x; q) implies that we have Schur positivity of (( but we may show that in fact the electronic journal of combinatorics 24(1) (2017), #P1.38 Indeed, it follows readily from the definitions that In a sense this is because h ⊥ 1 acts as a derivation, sending h n to h n−1 , and this is a form of chain-rule.Hence the positivity of the coefficient of q in F a,b (x; q) is a consequence of the classical version Foulkes' conjecture, since Schur positivity is preserved by both operations of multiplication by h 1 and its adjoint8 h ⊥ 1 .However, for higher degree, it does not seem that we can calculate coefficients as easily.
To illustrate, we have calculated that the coefficient of q 2 in F a,b (x; q) is equal to Property (25) extends to the wider context of ( 5), so that the coefficient of q in the righthand side of (7) (for k = 1) is indeed Schur positive (assuming that (5) holds), since it is equal to 4 Expanding Foulkes' conjecture to more general diagrams For partitions α, β, γ, and δ, none of which equal to (1) and such that |α|•|β| = |γ|•|δ| = n, let us say that α, β, γ, δ is a Foulkes configuration for n, if and only if Clearly, for a < b, Foulkes' conjecture says that a, b, b, a is a Foulkes configuration.Likewise statement (5), under the conditions there specified, is equivalent to saying that a, b, c, d is a Foulkes configuration.Other cases are possible.Indeed, by direct explicit calculation we find the following: with the right-hand side having polynomial coefficients in q.In particular, this last condition requires that, at q = 1 we have the equality For instance, it is easy to check that this last equality holds when for any a, b, c, d, k in N, such that ab = cd, since both sides of (29) evaluate to e a+b k .Evidently, all q-Foulkes configurations are also Foulkes configurations, but most Foulkes configurations do not satisfy the extra requirement that (30) holds.Explicit calculations reveal that this condition significantly reduces the number of possibilities.
An intriguing development, explicitly checked out for all cases 9 with n up to 30, is that having both the necessary conditions ( 27) and ( 29) holding seems to be equivalent to having the full q-Schur positivity (28) holding too.In other words, we have the following general statement, which would reduce all q-versions to the q = 0 case.Conjecture 10.For partitions α, β, γ, and δ, such that e α • e β = e γ • e δ , we have Clearly, when both α and γ are one part partitions, respectively equal to a and c, the second condition in (31) is simply that e a β = e c δ .Observe that this implies that β = µ i and δ = µ j for some partition µ, and ai = cj.Only this simpler version is needed in all cases explicitly calculated.In other words, all configurations that we have found to satisfy e α • e β = e γ • e δ are such that α and γ are reduced to one part, and thus of the simple form stated.It seems that this should be easy to prove.

Appendix A: Background on symmetric functions and plethysm
Trying to make this text self-contained, we now rapidly recall most of the necessary background on symmetric functions.For more background, see [1,16].As is usual, we often write symmetric functions without explicit mention of the variables.Thus, we denote by p k (as in [16]) the power-sum symmetric functions using which, we can expand the complete homogeneous symmetric functions as where is the multiplicity of the part k in the partition µ of n, and z µ stands for the integer z µ := j d j d j !.
For instance, we have the very classical expansions As is also very well known, the homogeneous degree n component λ n of the graded ring Λ of symmetric functions, affords as a linear basis the set of Schur functions {s µ } µ n , indexed by partitions of n.Among the manifold interesting formulas regarding these, we will need the Cauchy-kernel identity.
with h n (xy) = h n (. . ., x i y j , . ..) corresponding to the evaluation of h n in the "variables" x i y j .Otherwise stated, we may express this by the generating function identity

Rules of plethysm
Plethysm is an associative operation on symmetric functions, characterized by the following properties.Let f 1 , f 2 , g 1 and g 2 be any symmetric functions, and α and β be in the electronic journal of combinatorics 24(1) (2017), #P1.38 The first four properties reduce any calculation of plethysm to instances of the fifth one.For a given symmetric function f , one may consider the plethysm f • (−) as an operator on symmetric functions.In fact, this operator may naturally be extended to any rational fraction in the underlying variables.It is sometimes more convenient to use the alternate notation f [−] for this operator and "add" the further rules Then, considering variable sets as sums x = x 1 + x 2 + x 3 + . .., one observes that f [x] corresponds to the evaluation of the symmetric function f in the variables x.Moreover, Cauchy's formula gives an explicit expression for the expansion of Likewise f [1/(1 − q)] = f [1 + q + q 2 + . ..], corresponds to the evaluation f (1, q, q 2 , . ..).
Another interesting classical property of Schur functions may be expressed as where s µ/ν stand for the skew Schur function characterized by writing ν ⊆ µ if ν i µ i for all i, and f ⊥ standing for the dual operator of multiplication by f for the usual scalar product −, − on symmetric functions (for which the Schur functions form an orthonormal basis).It is well known that s µ/ν is Schur positive, and s µ/0 = s µ .

Macdonald polynomials
With all this at hand, the polynomial H n (x; q) can be explicitly defined as as before [n] q !stands for classical the q-analog of n!: To get a Schur expansion for H n (x; q), we recall the hook length expression s µ [1/(1 − q)] = s µ (1, q, q 2 , q 3 , . ..) = q n(µ) the electronic journal of combinatorics 24(1) (2017), #P1.38 where h ij = h ij (µ) is the hook length of a cell (i, j) of the Ferrers diagram of µ, and n(µ) := (i,j) j.Now, using Cauchy's formula (33), with y = 1 + q + q 2 + . .., we find that It is well known that the coefficient of s µ (x) occurring here is a positive integer polynomial that q-enumerates standard tableaux with respect to the charge statistic.This is the qhook formula.Thus, we find the two expansions.
It is clear that H n (x; 0) = h n .The H n (x; q) function encodes, as a Frobenius transform, the character of several interesting isomorphic graded S n -modules such as: the coinvariant space of S n , the space of S n -harmonic polynomials, and the cohomology ring of the fullflag variety.More precisely, this makes explicit the graded decomposition into irreducibles of these spaces.Thus, the coefficient of s µ (x) in formula (38) corresponds to the Hilbert series10 of the isotropic component of type µ of this space.Using (34) to expand H n , the global Hilbert series of these modules can be simply obtained by computing the scalar product To see this, recall that p µ , p λ is zero if µ = λ, and p µ , p µ = z µ .To complete the picture, let us also recall that p µ , s λ is equal to the value, on the conjugacy class µ, of the character of the irreducible representation associated to λ.In particular, it follows that This is the Frobenius characteristic of the regular representation of S n , for which the multiplicities f µ are given by the number of standard Young tableaux of shape µ.Beside this notion of Frobenius transform that "formally" encodes S n -irreducibles as Schur function, another more direct interpretation of the above formulas is in terms of characters of polynomial representations of GL(V ), with V an N -dimensional space over C. Recall that the character, of a representation ρ : GL(V ) → GL(W ), is a symmetric function χ ρ (x 1 , x 2 , . . ., x N ) of the eigenvalues of operators in GL(V ).Through Schur-Weyl duality, out of any S n -module R and any GL(V )-module U , one may construct a representation of GL(V ): R(U ) := R ⊗ CSn U ⊗n , where S n acts on U ⊗n by permutation of components.This construction is functorial: R : GL(V )-Mod −→ GL(V )-Mod, and the character of R(U ) is the plethysm (f • g)(x 1 x 2 , . . ., x N ), where f is the Frobenius characteristic of R and g the character of U .Furthermore, under this construction, irreducible polynomial representations of GL(V ) correspond to irreducible S n -modules R.
If such is the case, one writes S λ (V ) when R is irreducible of type λ.The corresponding character is the Schur function s λ (x 1 , x 2 , . . ., x N ).For the special case λ = (n), we get the symmetric power S a (V ) whose character is h a (x 1 , x 2 , . . ., x N ), hence the character of S a (S b (V )) is the plethysm h a • h b .
Appendix B: N-positivity of Θ a (b) The proof of N-positivity of the solution of the recurrence occurring in Proposition 8 may be directly obtained as follows.Let us consider the positive integer coefficient series11 defined as:

a 1 F
a,b (x; 1)) − F a,b (x; 1) is Schur positive, since one of the terms in h a 1 s λ is the Schur function indexed by the partition obtained from λ by adding a boxes to its first line.Hence the lemma directly implies that F a,b+1 (x; 1) − F a,b (x; 1) = (F a,b+1 (x; 1) − h a 1 F a,b (x; 1)) + (h a 1 F a,b (x; 1)) − F a,b (x; 1)) is Schur positive.