The super nabla operator
Fran\c{c}ois Bergeron, Jim Haglund, Alessandro Iraci, Marino Romero
TL;DR
The paper introduces the super nabla operator, a unifying framework for Macdonald polynomial eigenoperators, revealing new identities and combinatorial interpretations in symmetric function theory.
Contribution
It defines the super nabla operator, unifies known Macdonald eigenoperators, and derives new identities and combinatorial insights.
Findings
Super nabla operator generalizes existing Macdonald eigenoperators.
New identities are established for special parameter values.
Unified combinatorial interpretations are provided.
Abstract
We consider here a new operator, called ``super nabla'', which is shown to be generic among operators for which the modified Macdonald polynomials are joint eigenfunctions. All previously known Macdonald eigenoperators can readily be obtained from super nabla, including the usual nabla operator, the Delta operators, and other operators that have appeared in the literature. Thus, the super nabla operator furnishes an overall unified viewpoint on this family of operators, as well as opening up new possibilities. We prove several new identities arising from specializations of the parameters and involved in the specification of these operators, as well as unifying combinatorial interpretations.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons
The Super Nabla Operator
François Bergeron
Université du Québec à Montréal
LACIM
,
Jim Haglund
University of Pennsylvania
Department of Mathematics
,
Alessandro Iraci
Università di Pisa
Dipartimento di Matematica
and
Marino Romero
University of Pennsylvania
Department of Mathematics
Abstract.
We study here the Super Nabla operator, which is shown to be generic among operators for which the modified Macdonald polynomials are joint eigenfunctions. All known such Macdonald eigenoperators are essentially included in Super Nabla, including the usual Nabla operator, Delta operators, and other operators that have appeared in the literature. All of these involve parameters and , and we give several identities for specializations of these parameters, as well as combinatorial interpretations. We also describe positivity conjectures for Schur and elementary expansions.
Contents
- 1 Introduction
- 2 Combinatorial definitions
- 3 Symmetric function manipulations
- 4 Different weights
- 5 Multi-labeled Dyck paths
- 6 On power sums
- 7 Further Specializations
- 8 More conjectures about -positivity
- 9 Multivariate lift of
- 10 Connections with other work
- 11 Acknowledgements
Keywords: Macdonald Operators; Catalan Combinatorics; Symmetric Functions.
2020 Mathematics Subject Classification: 05A17; 05E10; 05E05
1. Introduction
Recent years have seen an ever growing body of research on algebras constructed out of operators for which Macdonald polynomials form a joint basis of eigenfunctions. Related subjects range from Algebraic Combinatorics to Statistical Mechanics, going through Representation Theory, Algebraic Geometry, as well as Link Homology, to name but a few. We consider here the modified version of Macdonald polynomials, denoted , for ranging over all integer partitions. Among many nice properties, they are known (see [Haiman-nfactorial-2001]) to have coefficients in when expanded in the basis of Schur functions. For instance,
[TABLE]
It is usual to express this fact by saying that they are Schur positive111With coefficients that are -polynomials having positive integer coefficients.. This Schur positivity has been naturally explained by showing (see [Garsia-Haiman-qLagrange-1996]) that is the Frobenius characteristic of an -graded -module , with the parameters and serving to encode the grading, and the Schur functions encoding -irreducibles. These Garsia-Haiman modules appear as sub-modules of the Space of Diagonal Harmonics:
[TABLE]
under the -action that diagonally permutes both sets of variables: which is to say that and . Recall that the Frobenius characteristic of an -graded -module is the symmetric function (in the variables )
[TABLE]
assuming that each component decomposes as an -module as
[TABLE]
with denoting (representatives of) Young’s irreducible module indexed by , and denotes its multiplicity in . In other terms, the coefficient of a Schur function in is the graded Hilbert series of occurrences of the irreducible in .
One of the main conjectures in [Garsia-Haiman-qLagrange-1996], proven by Haiman in [Haiman-nfactorial-2001], states that
[TABLE]
where stands for the degree elementary symmetric function, and is the Nabla operator of [Bergeron-Garsia-ScienceFiction-1999] which has the (modified) Macdonald polynomials as eigenfunctions with eigenvalue
[TABLE]
Here, runs over the set of cells of (see Section 2). Among other similarly defined operators, we have
[TABLE]
The main contribution of this work concerns the following extension of , which both unifies many previous results, and opens new intriguing possibilities.
Definition 1.1**.**
Considering a second set of variables , we simply set
[TABLE]
To express its properties, we consider various denumerable sets of variables , for any , and write for collections of these sets of variables. This allows us to consider the compact operator notation
[TABLE]
so that we have
[TABLE]
For any symmetric function , it is handy to extend the multi-variate Hall inner product to act on operators, as well as it does on symmetric functions. This way, we write for the operator that sends to
[TABLE]
with standing for the usual version of the Hall scalar product. With this notation at hand, we have
Proposition 1.2**.**
For all , and any homogeneous symmetric function of degree ,
[TABLE]
We further write for the collection obtained by adding the variables to the collection . One of our main goals is to give combinatorial models for expressions of the form , for interesting symmetric functions . In particular, we consider the cases when222Recall that it is usual in the subject to denote by the product . , for which we give combinatorial expansions at . Note that the family of these is a basis of , and that in terms of Theta operators (see [DAdderio-Iraci-VandenWyngaerd-Theta-2021]). We will show that all of these cases are “contained” in the special case . More precisely, if stands for an extra set of variables, we will show that the symmetric function contains all the information about the effect of the operator on any symmetric function.
In general, we conjecture that there is “strong” Schur positivity of for many interesting symmetric functions , including the case (see also Equation 8.4 and subsection 8.2). Typically, we have expressions of the form
[TABLE]
where runs over length vectors of partitions:
[TABLE]
; with terms composed of coefficients and products of symmetric functions
[TABLE]
in the sets of variables . As these expressions arise from multiplication of Macdonald polynomials, we have whenever is a obtained by a permutation of the partitions in . Extensive explicit calculations suggest the following333See Table 2, Table 3, and Table 4 for examples..
Conjecture 1.3**.**
For all and any , the coefficient in Equation 1.8 expands as a linear combination of Schur polynomials with positive integer coefficients.
Observe that the Schur expansion of the only involve Schur functions indexed by partitions with at most two parts, since there are only two variables. Extensive calculations confirms that the above property also holds when is applied to a broad family of symmetric functions, including (see Section 8).
In this work, we extensively explore the special case when is specialized at , giving results such as the following.
Theorem 1.4**.**
At , we have the following monomial, elementary, and Schur function expansions, respectively:
[TABLE]
where the first sum is over multi-labeled Dyck paths , in the rectangle, to which we associate some monomial in the variables and (see Section 5 for details). The second sum is over multi-labeled Dyck paths , with some monomial ; and the last sum is over lattice multi-labeled Dyck paths.
In Section 3 we describe two approaches giving these expansions and elaborate on the general combinatorial case, from where we derive alternate expansions. This involves statistics that “look right” (see Definition 4.1), which are also useful for the “square case” expansions of Section 6. This so called square case gives an analogous result when (see Section 3 for the necessary definitions). In particular, for , we have , and we get
Theorem 1.5**.**
[TABLE]
where the first sum is over multi-labeled -Dyck paths with a marked return, and the second sum is over those multi-labeled paths whose labels give lattice words (see Section 6).
2. Combinatorial definitions
As usual we may represent partitions by their Ferrers diagram (using the French convention), so that the number of cells on row is the size of the part of . Cells have usual cartesian coordinates, hence the cells of the row have coordinates , with ; and we write when is a cell (of the diagram) of . For a partition and a cell , we let and respectively denote the arm and leg of the cell . These are, respectively, the number of cells in that lie strictly to the right of and the number of cells strictly above . See Figure 1 for an example.
We recall that we have the following notations in this context:
[TABLE]
and as is usual, stands for the product .
To avoid confusion, we make the following distinction between words and tuples. We use the term word for a sequence whose letters are tuples of nonnegative integers. Hence is the digit of the letter . For any word or tuple , we denote the number of its letters by . Hence and . The size of a tuple is given by . If each is nonzero, then is said to be a composition and we write . If is a composition whose entries are weakly decreasing, then is a partition, denoted by . To illustrate, the word
[TABLE]
has length . The first letter of is the composition whose length is and whose size is .
Recall that the descent set and ascent set of an -tuple of integers are respectively defined as
[TABLE]
Related to these are the statistics
[TABLE]
Note that and actually are the and of the reverse of the tuple, hence the name. As is classical, to every composition of , we may associate the partial sums set:
[TABLE]
which establishes a bijection between compositions and subsets of . For a composition and tuple , let
[TABLE]
Given a word of tuples , we say is an -descent if there are precisely indices such that . Writing for the word formed by the digit of each tuple, we may set
[TABLE]
As an example, we have
[TABLE]
Let be the number of indices such that ; that is, is the multiplicity of in . We denote the multiplicity type of as .
A lattice word is a sequence such that, for all , we have
[TABLE]
This means that is a tuple such that every prefix has at least as many ’s as ’s, at least as many ’s as ’s, and so on. Such a word encodes a standard tableau where gives the row in which is placed. If the shape of the resulting tableau is , then we say that is a lattice word of type and write . For instance is a lattice word of type .
Denote by the set of all tuples whose entries can be rearranged to give , so that, formulaically, . It is convenient to collect the multiplicities of a sequence of tuples , with , into a single word of multiplicities , which we call the multiplicity type of .
We define the set of partition vectors of size rearranging to as
[TABLE]
This is the set of sequences of partitions with respective sizes given by the entries of , and whose collective union of parts rearranges to .
3. Symmetric function manipulations
Our goal in this section is to recall some symmetric function manipulations needed to derive upcoming expansions.
3.1. Combinatorics of forgotten symmetric functions
For of length , the combinatorial formula for the forgotten symmetric function [Egecioglu-Remmel-Bricks] is given by
[TABLE]
Plethystically substituting for , we get the expansion
[TABLE]
Definition 3.1**.**
For a given partition of , a column-composition tableau of type is a pair where is a composition that rearranges to , and is a sequence such that
[TABLE]
We denote by the set of column-composition tableaux of type , and by the subset of those such that . For , we define the length of as and size of as . We will write for when we need to specify the size of column in the column-composition tableau .
We can depict the elements of as follows.
Here, , (is the number of green cells), and . Since (there are no green cells in the first column), we have Denote the generating function of column-composition tableaux of type by
[TABLE]
By construction, these series give the principal specialization of the forgotten basis (see [IraciRomero2022DeltaTheta]*Proposition 5.2).
Proposition 3.2**.**
For any partition ,
[TABLE]
We also have the following specialization. The set of column-composition tableaux of type are in bijection with partitions whose largest part is at most . This gives
Corollary 3.3**.**
For any ,
[TABLE]
3.2. Classical properties of the Macdonald polynomials
Coined in plethystic notation, the classical Cauchy (kernel) identities state that
[TABLE]
whenever and are dual bases under the Hall scalar product of symmetric functions. Here, stands for the linear and multiplicative involution that send to , as well as to . In particular, the Schur basis is self dual. The modified Macdonald polynomials afford the following -Cauchy identity:
[TABLE]
writing for F\left[{\boldsymbol{x}}/{M}\right]=F\Big{[}{\textstyle\frac{\boldsymbol{x}}{(1-q)(1-t)}}\Big{]}, and where stands for the polynomial
[TABLE]
Just as the Cauchy identity is related to the Hall scalar product, the -Cauchy identity relates to the following scalar product.
Definition 3.4**.**
The -scalar product is defined on the basis of power sum symmetric functions by setting
[TABLE]
where the factor comes from the usual Hall scalar product, with the the multiplicity of the part in .
It follows that, for any symmetric function and , we have
[TABLE]
Observe that Equation 3.7 implies that we have the identities
[TABLE]
Furthermore, from the evaluation
[TABLE]
and Equation 3.7, we get the well-known expansion
[TABLE]
Equivalently, in terms of the operators of Definition 1.2, we have
[TABLE]
3.3. Specializations
Some of the simplest (plethystic) specializations of the modified Macdonald polynomials are as follows:
[TABLE]
where is formally considered to be such that . Hence, on symmetric functions of homogeneous degree , the operator specializes to
[TABLE]
A richer, similar specialization, in terms of an extra variable , is given by
[TABLE]
At the Macdonald polynomial specializes to
[TABLE]
where, for any symmetric function , we write for F\big{[}{\textstyle\frac{\boldsymbol{x}}{1-q}}\big{]}. Moreover, at we get the similar expression
[TABLE]
It may be worth recalling that444Clearly is also equal to , where stands for the Pochammer symbol. We sometimes write for the product.
[TABLE]
and more generally,
[TABLE]
where stands for the hook length of in , and .
Well-known operators may be obtained as specializations of . Recall that, for any given symmetric function , one considers the -operators ([Bergeron-Garsia-Haiman-Tesler-Positivity-1999]) defined by
[TABLE]
We then observe that
[TABLE]
Applying (independently) two such specializations to , we clearly get
[TABLE]
It follows by iteration that we can obtain any of the operators , by taking suitable coefficients in compositions of operators . Thus, any of the operators may also be obtained by simply taking suitable linear combinations of these compositions. Observe that, on non-constant symmetric functions, we can also obtain the operator from the specialization . Another interesting observation along these lines concerns the operator defined as follows.
Definition 3.5**.**
Define the linear operator by setting, for :
[TABLE]
As we will now show, both of the symmetric functions and are contained in . Indeed, we first observe from Equation 3.14 that we have
[TABLE]
If we let stand for another set of variables, then one calculates that
[TABLE]
We have thus shown, by linearity and an application of , that
Proposition 3.6**.**
For any partition , we have
[TABLE]
3.4. Setting
Clearly, the above calculations may be specialized (see Equation 3.16) at . Observe that, applying the specialization of Equation 3.16, we directly obtain
[TABLE]
Definition 3.7**.**
The specialization at of the operator may thus be considered as an operator, here denoted by , which is defined by
[TABLE]
Then, for any symmetric function , with no poles at , we clearly have
[TABLE]
More generally:
[TABLE]
so that we may consider the operator . Applying Cauchy’s formula with the pair of dual basis , we get
[TABLE]
and therefore
[TABLE]
The monomial expansion of may be calculated in several ways. One is by specializing, at , Haglund, Haiman, and Loehr’s formula for the modified Macdonald polynomial [Haglund-Haiman-Loehr]:
[TABLE]
The product can then be interpreted as follows. For any composition ,
[TABLE]
where if with ,
[TABLE]
and
[TABLE]
The only term in Equation 3.25 which we must describe is the specialization . To see what this must be, we first see that
[TABLE]
It can be shown, as done in [IraciRomero2022DeltaTheta], that at one has
[TABLE]
meaning that
[TABLE]
as was also seen in [Hicks-Romero-2018] using summation formulas.
Lastly, from Cauchy’s formula, it follows that
[TABLE]
where, for convenience, we set
[TABLE]
In the end, we can combine all of these identities together to get our preliminary expansion.
Proposition 3.8**.**
[TABLE]
Given a set of words , a function (with some given ring), and a composition , we consider the rational function
[TABLE]
In Definition 4.2 we will consider specific choices for that will be said to“look right”. For these, we will show that the above rational function is not only a polynomial, but it is given by the area enumeration of a class of -compatible labeled parallelogram polyominoes.
3.5. Setting
Recalling Equation 3.17, we may introduce the following operator.
Definition 3.9**.**
The specialization at of the operator may be considered as an operator, here denoted by , which is defined by
[TABLE]
Then, for any symmetric functions with no poles at , we clearly have
[TABLE]
Applying Cauchy’s formula with respect to the self-dual basis of Schur functions, we get
[TABLE]
using Frobenius’ notation for the hook partition . It follows that
Proposition 3.10**.**
For all , the effect of on the symmetric function is given by the formula
[TABLE]
4. Different weights
In this section we extend the results of [IraciRomero2022DeltaTheta]*Section 7,10 to more general weights. Through this section, we will fix a set of words in the alphabet , and we let be some fixed ring.
Definition 4.1**.**
Let be any alphabet of variables, and let be a set of words in the alphabet . Let be any ring, we say that a function looks right if there exist a weight function and a local weight function giving
[TABLE]
where depends only on and .
For example, if and is the set of all words with letters in , then is a statistic which looks right with local weight function if has an -descent at . Note that, for , , , and if and [math] otherwise, up to taking the -logarithm, we get back [IraciRomero2022DeltaTheta]*Definition 10.1. From now on, we assume that is equipped with a function that looks right.
Definition 4.2**.**
A word vector is a sequence of words whose concatenation is in . If , then we write . If looks right on subwords of elements in , then we may set
[TABLE]
and say that looks right on .
For instance, if , then is a word vector whose entries are words of lengths and respectively; and the letters in each word are tuples of length . Here, we can choose to be .
Definition 4.3**.**
Let be a partition of , and be a set of words in the alphabet . A sequence of -labeled column-composition tableaux of type is a tuple of labeled column-composition tableaux such that, with and , we have:
- (1)
and for , for some ; 2. (2)
.
We denote by the set of sequences of -labeled column-composition tableaux of type .
In other words, a sequence of -labeled column-composition tableaux is a tuple of column-composition tableaux of sizes with , where under column of we place an element of , . When read from left to right, the labels under the column give a word . If the vertical bars in each gives a composition rearranging to , then has parts rearranging to .
For , we set . Given a function on word vectors from that looks right, for we set
[TABLE]
where stands for the number of vertical bars in .
We now consider, for a fixed partition , the rational function
[TABLE]
We have the following analogue of [IraciRomero2022DeltaTheta]*Definition 7.1, which generates a sign-reversing involution on the set
Definition 4.4**.**
Let be one of the possible labeled column-composition tableaux appearing in a sequence , and suppose that has at least one bar. Then we say that can split and define , where is the portion of occurring before the first vertical bar, and is obtained from the portion of after the first vertical bar by adding cells to each column. The split map is weight-preserving and sign-reversing.
Proposition 4.5**.**
If , then .
Proof.
Let , with . Suppose with and , and let .
By definition, the weight has two components, one coming from the total size, and one coming from . Let . By definition of , the number of cells above stays the same, while the number of cells above increases by , so the first component of the total weight increases by . By definition of , we have
[TABLE]
so the second component of the total weight decreases by . The two changes cancel out and so the weight is preserved, as desired. ∎
Lemma 4.6**.**
Let , be two -labeled column-composition tableaux. There exists an such that if and only if
[TABLE]
If such an exists, then it is unique; we say that can join and set .
Proof.
If such an exists, then Equation 4.1 holds by construction. Suppose that Equation 4.1 holds. Then we can define as the labeled column composition tableau obtained by decreasing the size of each column of by and then concatenating it to , also concatenating their words. Equation 4.1 ensures that the result is still a column-composition tableau. It is now immediate that and that such an is unique. ∎
The following lemma is crucial to ensure that our sign-reversing, weight-preserving involution is well-defined.
Lemma 4.7**.**
Let , be two -labeled column-composition tableaux, and let . Then can join if and only if it can join .
Proof.
By construction, , so Equation 4.1 holds for and if and only if it holds for and . ∎
Again, for a fixed set of words and a partition , we can now define our weight-preserving, sign-reversing involution as follows.
Definition 4.8**.**
Given , define by the following process:
- (1)
if , then ; 2. (2)
if can split, then ; 3. (3)
if cannot split and can join , then ; 4. (4)
otherwise we inductively define .
For with no vertical bars, we have that for any , does not depend on , as having no vertical bars implies that the number of cells above the base is constant. Let us write in this case. Note that by definition. We have the following analogue of [IraciRomero2022DeltaTheta]*Theorem 7.8.
Proposition 4.9**.**
Let be the set of such that has no vertical bars, and for all ,
[TABLE]
Then is given by the weight-sum over :
[TABLE]
Proof.
Same as [IraciRomero2022DeltaTheta]*Theorem 7.8. ∎
We can now describe the fixed points in terms of parallelogram polyominoes.
Definition 4.10**.**
For integers and , a parallelogram polyomino of size is a pair of lattice paths from to , consisting of unit north-steps and east-steps, such that (the top path) lies always strictly above (the bottom path), except on the endpoints.
The area of a parallelogram polyomino of size is defined as
[TABLE]
Since the two paths and do not touch between the endpoints, is the minimal number of unit cells between them. One also sees that counts the number of lattice cells between and which do not touch .
Definition 4.11**.**
An , -labeled parallelogram polyomino is a triple such that is an parallelogram polyomino, and is a word in of lenght .
Definition 4.12**.**
Given that looks right, so that
[TABLE]
for some appropriate weight functions and , we say that a parallelogram polyomino is -compatible if contains exactly east-steps on the line .
Definition 4.13**.**
Given , we say that a -labeled parallelogram polyomino has type if the lengths of the maximal streaks of north-steps of form a composition .
Let be the set of all -labeled parallelogram polyominoes of type that are -compatible. We have the following.
Proposition 4.14**.**
There is a bijection such that
[TABLE]
Proof.
For , we need to define a triple
[TABLE]
corresponding to the polyomino and its labels. For , let
[TABLE]
the latter is guaranteed to be strictly positive because . We define
- •
- •
- •
where the product denotes the ordered concatenation of strings. We claim that is the desired bijection.
First we need to show that it is well-defined, i.e. is a -labeled parallelogram polyomino of type that is -compatible. Indeed, the lengths of the maximal streaks of north-steps of form a composition , satisfies the -compatibility condition by construction, and is a -labeling. We need to show that and have the same size and that always lies strictly above .
The total height of is , and the total height of is , so they agree. The total width of is
[TABLE]
which is exactly the total width of , as expected. To show that lies always strictly above , it is enough to check that, for , we have
[TABLE]
But this is immediate because the difference of the two quantities is .
Finally, we have to show that
[TABLE]
Let (and ) be the number of whole cells between (respectively ) and the bottom-right semi-perimeter of the rectangle containing . Then , where is the width of the path.
We can compute the first terms factors as follows: for each north-step of the relevant path, we count the number of east-steps that follow it and take the sum of these quantities over all north-steps. In this way, we get
[TABLE]
and
[TABLE]
and finally
[TABLE]
Since , we have
[TABLE]
because whenever ; and when the corresponding summand above is [math]. Therefore, the equality holds and the thesis follows immediately. ∎
Corollary 4.15**.**
For any ,
[TABLE]
where if , we set . In particular, if
[TABLE]
then
[TABLE]
5. Multi-labeled Dyck paths
In order to give a combinatorial interpretation of , we must first identify the proper sets and statistics to which we can apply the results from the previous section.
Let , , and for convenience, we index elements in from [math] so that for , we have . Let , that is, the number of ascents occurring at the same position between the and the letter of , and let
[TABLE]
where . Now, recall that
[TABLE]
This gives one way of getting a combinatorial interpretation, but we can also translate this result in terms of what we will call multi-labeled -Dyck paths. The explicit bijection to multi-labeled Dyck paths will be given in Definition 5.6, below. But first, we will state the main results.
Definition 5.1**.**
For positive integers , and , a multilabeled Dyck path is a pair , where is a Dyck path (i.e. a lattice path that always lie weakly above the diagonal , called the main diagonal), and is a word of -tuples such that, if has weak descents at position , then the north-step is followed by at least east-steps. In other words, if the north-step of has -coordinate then
[TABLE]
for all . The area of a multi-labeled Dyck path is defined as is done for rectangular Dyck paths, so that is the number of whole lattice cells between and the main diagonal. (Note that the area does not depend on .) When , we denote the set by .
The notion of is illustrated in Figure 3. The whole -expansion of \widehat{\nabla}_{{\color[rgb]{0,0.7,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.7,0}\boldsymbol{z}}}\widehat{\nabla}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}}}(e_{3}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}})) may be found in Table 1.
Proposition 5.2**.**
For any and ,
[TABLE]
Say we are constructing an element in starting from a path and word with corresponding to the monomials in . This means that . We will now create with so that the pair is a multi-labeled Dyck path. We will then also write and .
Suppose there are weak descents in at position . Then must have at least east-steps succeeding its north-step. If there are precisely east-steps directly after the north-step, then we require . If there are more than east-steps instead, then we are free to choose any value for . This gives an expansion as linear combination of products of elementary symmetric functions (see Equation 5.2 below).
Definition 5.3**.**
For a multilabeled Dyck path , we denote by the unique composition of whose corresponding partial sums set is such that
[TABLE]
Corollary 5.4**.**
For any and , we have
[TABLE]
The content of a multi-labeled -Dyck path is the multiplicity type . We will then write if . Denote by the set of with .
Corollary 5.5**.**
For any and , we have
[TABLE]
This equality follows from the following bijection.
5.1. From polyominoes to Dyck paths
For , let be the set of words with and for . In this section we complete the construction by giving a weight-preserving bijection between and the set
[TABLE]
of multi-labeled -Dyck paths with content and -composition . It is slightly easier to describe the inverse, so that is what we will do.
Definition 5.6**.**
Let be a multi-labeled -Dyck path, let and . We define , where and are the paths obtained from and respectively by removing, for every , east-steps of on the line and the same amount of east-steps of on the line .
Proposition 5.7**.**
The map defines a weight-preserving bijection between and , meaning .
Proof.
First, notice that, since , there must be at least east-steps of on the line . Notice that by definition we have at least east-steps of on each line. Since the first east-step of on the line must necessarily be strictly left of the first east-step of along the line , the is necessarily a parallelogram polyomino.
Now, the number of east-steps of on the line is exactly
[TABLE]
Thus is -compatible. By construction, the word of is the same as the word of , and so their contents are the same. By Definition 5.3, the -composition of is exactly given by the lenghts of the maximal vertical segments of . It follows that .
Since the process that defines is invertible, the map is bijective. Finally, notice that the number of cells in the row decreases by , but so does the number of east-steps of the bottom path on the line . This means that the area of is the same as the area of . ∎
5.2. Schur expansions
The Schur expansion of the modified Macdonald polynomials at is given by
[TABLE]
where is the set of lattice words of type . Given a word of tuples of length , we will say that is lattice of type if is a lattice word of type . We note that
[TABLE]
where is the set of lattice words of type , and the local weights we are using are the same as in the beginning of this section, given by the number of local ascents. Let be the subset of for which is lattice. It follows immediately from the work developed in the previous sections that we have
Corollary 5.8**.**
[TABLE]
5.3. Notable special cases
5.3.1.
When , we get back regular Dyck paths, but with two distinct sets of labels. By taking scalar products appropriately, we can get back some well-known combinatorial expressions, such as the shuffle theorem or the Delta conjecture (both the rise and valley).
Indeed, we have . This corresponds to the case in which the -labels appear in increasing order in the reading word, so we can disregard them; we are left with “classical” labeled Dyck paths, which give us back the shuffle theorem.
If we take instead , this corresponds to the case in which the reading word of the -labels is a shuffle of and . As usual in this case, we can interpret this as decorating peaks of the path, with the condition that whenever there is a decorated peak and a north-step in the column immediately to its right, since the -labels decrease, the -labels must increase. This condition is exactly the same as saying that the valley following a decorated peak must be contractible, which is the condition given by the valley version of the Delta conjecture. Of course the topmost peak does not correspond to any valley, and in fact we get rather than .
In both cases, the valley version of the Delta conjecture from [Haglund-Remmel-Wilson-2018] gives us a -statistic on our two-labeled Dyck paths with certain -labelings. We hope to extend this statistic to the full set of multi-labeled Dyck paths and any labeling.
5.4. Setting
One of the interesting specializations is when . In general, one gets an -expansion of the form
[TABLE]
with . In other words, this symmetric function is simultaneaously -positive in each set of variables. In the special case when consists of a single set of new variables, we have the following expression:
[TABLE]
where we have applied the well-known -map on Dyck paths, which interchanges the statistics area and dinv to bounce and area. Specializing gives
[TABLE]
5.5. Tiered Trees
It is worth noticing that, thanks to the following identity, we have another interpretation for that extends a conjecture of [DAdderio-Iraci-LeBorgne-Romero-VandenWyngaerd-2022].
Proposition 5.9**.**
[TABLE]
Combining this with [DAdderio-Iraci-LeBorgne-Romero-VandenWyngaerd-2022]*Conjecture 6.4, we have a monomial expansion (in both and ) for in terms of rooted tiered trees and tree inversions.
Recall that . The coefficient of in is the Kac polynomial of certain dandelion quivers with a dimension vector depending on ; which is to say that it is the polynomial enumerating absolutely indecomposable representations of the quiver over [DAdderio-Iraci-LeBorgne-Romero-VandenWyngaerd-2022, Gunnells-Letellier-Rodriguez-Quivers-2018]. It would be interesting to find a similar interpretation for any , not just for .
6. On power sums
Our goal in this section is to show the following. As before, we use the notation: .
Proposition 6.1**.**
For any nonempty partition ,
[TABLE]
where the sums are respectively over multi-labeled -Dyck paths with a marked return, and lattice multi-labled -Dyck paths with a marked return.
Definitions for the relevant sets are given in the subsequent sections; and they come directly from the general framework of Section 4.
6.1. Newton’s Identity
The selected rearrangements of a partition are all possible pictures that may be obtained by the following procedure:
- (1)
Select a rearrangement of the parts of , drawn as a list of rows. 2. (2)
Select one of the cells in the first part, , and mark the cell with a circle.
For instance, to get an element of , we may rearrange the parts as , and one of its possible markings is
[TABLE]
In terms of these objects, one has the following.
Proposition 6.2** (Newton’s Identity).**
[TABLE]
It follows that
[TABLE]
or equivalently
[TABLE]
Applying successive operators at , we get
[TABLE]
Using the Cauchy identity to expand as was done before, we have
[TABLE]
We may interpret the product
[TABLE]
as the generating function of column-composition tableaux with such that for at least one of the . The monomial expansion of the product is given by a sum over all words with , where if , then contributes the term
[TABLE]
Each of the components in the summand can thus be given a combinatorial interpretation, and we are led to the following definition.
Let , be partitions of , and let be any nonempty partition.
Definition 6.3**.**
A sequence of Labeled Selected Column-Composition Tableaux of type , denoted by , is constructed by the following process:
- (1)
Select . 2. (2)
Select . 3. (3)
For select with the condition that for some , . The selection in is represented by placing the circle above its corresponding column in . 4. (4)
For each select a sequence of words with two extra conditions: For notation, the words will be indexed by ; the word in the sequence is denoted by ; the letter in the word will be denoted by ; and we denote the word by .
The first condition is that is nonzero for . The second condition is that . 5. (5)
Set .
We define
[TABLE]
with
[TABLE]
and set . The sign of :
[TABLE]
gives again the parity for the number of vertical bars appearing .
With the notation of the previous section, the statistic which looks right in this case is given by the reverse major index (and the product of the appropriate monomials) along the last entries in the tuples of , and by in the first entries of all the tuples.
For instance, suppose , , and , and condider the rearrangement . We select
[TABLE]
and choose the column-composition tableaux in :
[TABLE]
Note here that at least one column-composition tableau has no cells in the first column. Now place a circle above a column in the first component, and replace each cell in the base row with a column of numbers. From left to right, the first column has the tuple , the second column has the tuple , and so on until we reach column containing the tuple . The next column is in the second column composition tableau, and it contains the tuple . We continue in this way. The first row, or the row, has nonzero entries which rearrange to .
This is a selected, labeled column-composition tableau. We see that the first row of labels (from the top), has nonzero entries which rearrange to . The second row of entries is given by .
To compute the weight for this sequence we first count the number of cells above the base. This gives . We now calculate the reverse major index for each row after the row. If we read the second row of labels in the first component, we get the word , which has an ascent in the first and second positions. The number of cells to the right of the first ascent is , and the number of cells to the right of the second ascent is . This gives . Similarly, the last row in the first component gives the word , which has ascents in positions . This gives . Continuing in this fashion, the second labeled tableau gives ; the third gives ; and the last set of labels gives . Therefore, all the labels given by contribute a total of . If the label appears in row of labels, it contributes to the weight by . Thus, the second row of labels in the first component of the sequence contributes to the total weight by a factor of .
The row’s weight is calculated as follows: In the first component, there is a in the second column. Since there are columns to its right, this label contributes a weight of . The second component has a two in the last column and last row, which contributes a factor of . The third component gives , and the last component gives . The first row therefore contributes a factor of . In total, we have found that
[TABLE]
Proposition 6.4**.**
For any , , and nonempty partition
[TABLE]
6.2. A sign-reversing involution
We define a variant of the split map from [Romero-Thesis],[IraciRomero2022DeltaTheta],[Hicks-Romero-2018] that reduces the infinite, signed sum of selected, labeled column-composition tableaux to a finite number of fixed points. We start by defining the split map at at any vertical bar. To this end, we define the relative local weight of two sequences of words with equal lengths, as is done in Section 4.
Definition 6.5**.**
Specializing the definitions from the previous sections, we have the following relative statistics: let and be two words of tuples of length . Then the relative statistic between and is the quantity
[TABLE]
Definition 6.6**.**
Let be a selected, labeled column-composition tableau, let be the the first letters of , and let be the remaining portion of . Suppose there is a vertical bar in after column . Then we define the split of at to be by the following procedure.
- (1)
Set where consists of the first columns of . If one of the first columns of was selected, we keep the selection over the corresponding column in . 2. (2)
Set where is made by starting with the last columns of and adding cells to each column. If there was a selected column in in the last columns, then we keep the selection over the corresponding column in .
Definition 6.7**.**
Suppose and are the split of some selected column-composition tableau, . This means that for some . Then we say that and can join and we set . Such a is unique and exists if and only if the following inequality holds:
[TABLE]
Our previous work in Section 4 implies that the map is weight-preserving.
Definition 6.8**.**
Given , we say that the circle is leading in if there are no vertical bars to its left (so that it is in the first component of ). Define a map by the following process:
- (1)
If the circle is not leading in , then there is a right-most vertical bar to the left of the circle, say occurring after column . Then if , set 2. (2)
If the circle is leading and can join , then . 3. (3)
If the circle is leading and cannot join , then , as given in Definition 4.4.
For example, the configuration of Figure 5 maps to the configuration :
[TABLE]
Proposition 6.9**.**
For , the map defined above is a weight-preserving, sign-reversing involution.
Proof.
The proof follows just as in Section 4. Clearly, steps 1 and 2 are inverses. And we know in step 3 is a weight-preserving, sign-reversing involution. Thus, we need to show that if falls into cases 1 or 2, then also falls into case 1 or 2. And if falls into case 3, then so does . It therefore suffices to show that for in case 3, also falls into case 3.
Suppose the circle is leading in and cannot join . Then this means that . By definition of , we must always have . This is because if splits into , then the last column of increases by in order to create . In other words, . Therefore using that
[TABLE]
If instead joins to make , then this argument also works if we interchange by .
There is a problem with this argument when consists of a single part, since we would not have the condition that cannot join . Suppose the circle is leading in and . We have to make sure that does not join with ; otherwise, we would have something from case 3 landing in case 2. This is only true if is nonempty:
If there is a nonempty label in , then , since at least new cells were added to each column to form . In particular , while (from the definition of ), . Therefore, cannot join . On the other hand, if there is a nonzero label in , then to join and , we would need . But since , we have that this cannot be so. Therefore, if falls into case 3 then so will . ∎
Note that all the fixed points must be in for some . Let be the set of fixed points of of type . Then we have shown that for any nonempty ,
[TABLE]
The set is given as follows: An element is fixed by if
- (1)
For all , cannot split, meaning there are no bars. This means that for each , and we can therefore denote this quantity with no subscripts: . 2. (2)
For all , cannot join , and cannot join . 3. (3)
The circle is leading in .
So each consists of a single part from , , and also . For example, suppose and . Then to create an element , first pick a rearrangement of , say , and select a cell to circle in the first part. These will be the base rows of the tableaux.
[TABLE]
Choose a rearrangement of , say , to place in the base rows.
[TABLE]
Now place two more letters under each of the cells in the bottom row.
[TABLE]
To construct we now choose a value for so that and . We require for at least one .
[TABLE]
In this case, we have the following values. To better illustrate the relative statistics, we have written where is the sum of the labels in the row and is the contribution from the ascents in the lower rows.
[TABLE]
The weight of this sequence of labeled, selected column-composition tableaux is given by
[TABLE]
Definition 6.10**.**
Recall that a parallelogram polyomino consists of two paths that only touch at the beginning and end. If is the length of the first vertical segment of , then there must exist some for which the distance between and along the line is at most unit. The minimal such is called the return of and is denoted by . Let and let be any nonempty partition. Denote by the set of -compatible polyominoes with a marked return, given by a quadruple , where
- (1)
, with and for , 2. (2)
is a -compatible parallelogram polyomino of type with ending in two consecutive east-steps, 3. (3)
and .
Definition 6.11**.**
Define the map using from Section 4 as follows: For ,
- (1)
If , then set . Keep the circle from inscribed in its corresponding north-step of , as is associated by the map from Proposition 4.14. 2. (2)
If , then there is a largest such that . Set
[TABLE]
again keeping the circle from in its corresponding north-step.
For instance, to find the image under for the last example, we first rearrange the parts to give
[TABLE]
Applying then gives the selected polyomino of Figure 6.
Proposition 6.12**.**
The map is a weight-preserving bijection. In particular, we have
[TABLE]
Proof.
The bijection gives us an injection into a subset labeled parallelogram polyominoes with a marked return. The only condition we need to ensure is that if gives the polyomino , then ends with two east-steps. If is sent to with underlying polyomino , then we know that and (since no two consecutive can join). Thus , meaning one of these quantities is nonzero. If , then the last north-segment of must lie at least two units West of the last point of the path. If , then the number of cells between the last vertical segment of and the bottom path is at least . This gives that in either case, must end with two east-steps. ∎
We now apply the map of Definition 5.6 which sends -compatible polyominoes with a marked return to multi-labeled -Dyck paths with a marked return, defined as follows.
Definition 6.13**.**
Let be any nonempty partition. A -Dyck path is a polyomino where is a -staircase, meaning that with . Denote by the set of multi-labeled -Dyck paths with a marked return whose labels are given by words in of length . Recall that if has labels given by , and is an -descent, then the north-step of the top path must be followed by at least east-steps.
The map can be applied to the set as we did before, shifting both the path and by the number of non-ascents in the labels.
Proposition 6.14**.**
For any and nonempty partition , we have
[TABLE]
For instance, the previous example of a marked return polyomino maps to the multi-labeled -Dyck path with a marked return in Figure 7.
6.3. A Schur function expansion for applications to power sums
We can follow the same procedure outlined in the prior section, but instead use labels giving the Schur expansion in the variables. This corresponds to taking labels which form lattice words. We are immediately led to consider the following subset of multi-labeled -Dyck paths.
Definition 6.15**.**
Let be a nonempty partition. The collection of lattice multi-labeled -Dyck paths with a marked return, denoted by , is the subset of elements such that
- (1)
The labels are tuples of length . 2. (2)
The words are lattice words. We denote the partition associated to this lattice word by . 3. (3)
The top path ends with two east-steps. 4. (4)
There is a marked row before the first return of the paths .
We have
Proposition 6.16**.**
For any and nonempty partition ,
[TABLE]
6.4. Square Paths
A combinatorial description of the effect of on a power sum symmetric function is given by the Square Paths Conjecture of Loehr and Warrington [Loehr-Warrington-square-2007], proved by Emily Leven Sergel in her thesis [Leven-2016]. The conjecture states that we can write as a sum over labeled square paths. We now define these structures and show that when and our results coincide to the Square Paths Conjecture when .
A square path is a lattice path consisting of north-steps and east-steps from to that ends in an east-step. The set of labeled square paths is generated by placing positive integers along the north-steps of the square path so that the columns are increasing when read from bottom to top. The monomial of a labeled square path is given by where is the number of occurrences of the label in . For example,
[TABLE]
is a square path whose monomial weight is . If the north-step on the lowest diagonal starts on diagonal , then a north-step on diagonal will contribute units of area. The above example would then have area . The number of diagonal inversions is calculated in a similar way to Dyck paths. Then the Square Paths Theorem can be stated as
[TABLE]
If we circularly rearrange the north-steps and east-steps of the square path above, so that the path begins with the right-most, lowest vertical segment, we get
[TABLE]
We here circle a cell on the main diagonal to mark the last column of the original square path. This corresponds to marking a row before the Dyck path returns to the main diagonal. The area is simply the area of the Dyck path. Recall that
[TABLE]
The first thing to note is that for , is completely determined: The path must travel from to with only north-steps and east-steps; and every east-step must be followed by a north-step (since ). This means that is the path which begins with an east-step, then alternates “east, north, east, north”, and so on. On the other hand cannot touch , so it must remain weakly above the line until the last step. Therefore, is a Dyck path. The return of the polyomino is equal to the return of as a Dyck path, i.e. the first time returns to the diagonal . This equates the two interpretations when we set .
Another application of our formulas is a proof of the Delta Square Conjecture of [DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019] when . Here, there is a formula for in terms of (partially) labeled square paths. Our methods allow one to give an interpretation for this symmetric function in order to prove the conjecture when .
7. Further Specializations
Interesting identities arise when one specializes the parameters and .
7.1. Specialization at
The modified Hall-Littlewood symmetric functions are the specialization . The associated Cauchy identity is then
[TABLE]
Observe that
[TABLE]
where is the composition of formed by the lengths of the contiguous cells in which are the top cell in their column. In other words, . Then
[TABLE]
implying that
[TABLE]
with .
7.2. Setting and
In the case when and , is given by the sum of multi-labeled -Dyck paths with no area. The global dimension of this symmetric function is the coefficient of . There is only one -Dyck path with no area, the staircase path given by . The number of east-steps between two consecutive north-steps is always at least the number of non-ascents between the labels of these two north-steps. Since each label is a -tuple (given by ), we see that it we cannot have any -descents between two consecutive labels. We can state this as follows:
Proposition 7.1**.**
Let be the set of permutations in with such that no position is a descent for all . Then
[TABLE]
7.3. Setting and
Recall that at we have
[TABLE]
where runs over Dyck paths of length , is the “riser” composition of .
[TABLE]
where stands for the Zeta-map.
[TABLE]
where the sum is over sequences of partitions, all of which are equal to but at most of them. For any , at least one of the components of the ’s is different from . The positive integer coefficients do not depend on the order of the partitions occurring in , and their value only depend on components that are different from the partition . Thus the coefficients are finitely many in values, and stable as grows. Finally, when , we have
[TABLE]
where runs over Dyck paths of length , is the “riser” composition of , and stands for the “Zeta” map. In the general case, the above fixes the coefficients which contain at most two partitions not equal to .
We have
[TABLE]
where stands for the LLT-polynomial, and is the Zeta map on “Dyck paths” (here described as subpartitions of the staircase ).
[TABLE]
where runs over Dyck paths of length , is the “riser” composition of , and stands for the Zeta-map.
8. More conjectures about -positivity
One of the intriguing open question in the study of the super nabla operator is that it seems to exhibit -positivities when one replaces and . For instance, for the resulting expressions may be found below for small values of . Let be given positive integers. For a lexicographically ordered sequence of partitions of , all different from , we consider the expression
[TABLE]
where runs over permutations of such that if and either or both and are larger than ; with . For example,
[TABLE]
When the variable sets are clearly given, we may simply write for . With this notation at hand, we have the following values555The left-hand side of the equalities specifies the variable sets.:
\begin{aligned} &\nabla_{\boldsymbol{y}_{1}}(e_{2}(\boldsymbol{x}))\big{|}_{q+1,t+1}=E_{(2)}+(q+t)E_{(2,2)};\\ &\nabla_{\boldsymbol{y}_{1},\boldsymbol{y}_{2}}(e_{2}(\boldsymbol{x}))\big{|}_{q+1,t+1}=E_{(2)}+(q+t)E_{(2,2)}+(q^{2}+qt+t^{2})E_{(2,2,2)};\\ &\nabla_{\boldsymbol{y}_{1},\boldsymbol{y}_{2},\boldsymbol{y}_{3}}(e_{2}(\boldsymbol{x}))\big{|}_{q+1,t+1}=E_{(2)}+(q+t)E_{(2,2)}+(q^{2}+qt+t^{2})E_{(2,2,2)}\\ &\hskip 113.81102pt+(q^{3}+q^{2}t+qt^{2}+t^{3})E_{(2,2,2,2)}.\end{aligned}
Observe above a general phenomena that also occur in all other cases that we have calculated. As we increase in Equation 8.1, the terms indexed by a given are stable, and their -coefficients are Schur positive. In other terms, experiments suggest that there is a stable expression that gives values for any in the form
[TABLE]
where the are Schur positive symmetric functions of , , and . For instance, omitting to write variable sets on the right-hand side, we have
[TABLE]
Observe that, for a given , this expression only involves a finite number of terms in view of the [math] case of Equation 8.1. Other examples are as follows:
\begin{aligned} &\nabla_{\mathcal{Y}}(e_{3}(\boldsymbol{x}))\big{|}_{q+1,t+1}=E_{(3)}+(3+s_{1})\,E_{(21,21)}+(3s_{1}+s_{2})\,E_{(21,3)}+(s_{1}+2\,s_{2}+s_{3})\,E_{(3,3)}\\ &\hskip 88.2037pt+(s_{3}+7s_{2}+12s_{1}+s_{11})\,E_{(21,21,21)}\\ &\hskip 88.2037pt+(s_{4}+6s_{3}+9s_{2}+s_{21}+3s_{11})\,E_{(21,21,3)}\\ &\hskip 88.2037pt+(s_{5}+5s_{4}+6s_{3}+s_{31}+3s_{21})\,E_{(21,3,3)}\\ &\hskip 88.2037pt+(s_{6}+4s_{5}+4s_{4}+s_{41}+2s_{31}+s_{22})\,E_{(3,3,3)}\\ &\hskip 88.2037pt+(s_{5}+10s_{4}+34s_{3}+39s_{2}+s_{31}+8s_{21}+12s_{11})\,E_{(21,21,21,21)}\\ &\hskip 88.2037pt+(s_{6}+9s_{5}+27s_{4}+27s_{3}+s_{41}+7s_{31}+s_{22}+12s_{21})\,E_{(21,21,21,3)}\\ &\hskip 88.2037pt+(s_{7}+8s_{6}+21s_{5}+18s_{4}+s_{51}+6s_{41}+s_{32}+9s_{31}+3s_{22})\,E_{(21,21,3,3)}\\ &\hskip 88.2037pt+(s_{8}+7s_{7}+16s_{6}+12s_{5}+s_{61}+5s_{51}+s_{42}+6s_{41}+3s_{32})\,E_{(21,3,3,3)}\\ &\hskip 88.2037pt+(s_{9}+6s_{8}+12s_{7}+8s_{6}+s_{71}+4s_{61}+s_{52}+4s_{51}+2s_{42}+s_{33})\,E_{(3,3,3,3)}\\ &\hskip 88.2037pt+\ldots\end{aligned}
\begin{aligned} &\nabla_{\mathcal{Y}}(e_{4}(\boldsymbol{x}))\big{|}_{q+1,t+1}=E_{(4)}+(s_{1}+2)\,E_{(211,22)}+(s_{2}+3s_{1}+4)\,E_{(211,31)}\\ &\hskip 88.2037pt+(s_{3}+4s_{2}+6s_{1})\,E_{(211,4)}+(s_{2}+s_{1})\,E_{(22,22)}\\ &\hskip 88.2037pt+(s_{3}+3s_{2}+4s_{1}+s_{11})\,E_{(22,31)}\\ &\hskip 88.2037pt+(s_{4}+3s_{3}+3s_{2}+s_{21}+2s_{11})\,E_{(22,4)}\\ &\hskip 88.2037pt+(s_{4}+5s_{3}+10s_{2}+8s_{1}+s_{21}+2s_{11})\,E_{(31,31)}\\ &\hskip 88.2037pt+(s_{5}+5s_{4}+10s_{3}+8s_{2}+s_{31}+3s_{21}+4s_{11})\,E_{(31,4)}\\ &\hskip 88.2037pt+(s_{6}+5s_{5}+9s_{4}+6s_{3}+s_{41}+4s_{31}+5s_{21})\,E_{(4,4)}\\ &\hskip 88.2037pt+(s_{3}+6s_{2}+15s_{1}+s_{11}+16)\,E_{(211,211,211)}+\ldots\end{aligned}
Many other values have been explicitly calculated, but they are more awkward to display. Similar calculations666As , this extends our above observations. for \nabla_{\mathcal{Y}}(\operatorname{\Xi}e_{\mu}(\boldsymbol{x}))\big{|}_{q\to q+1,t\to t+1} suggest the following.
Conjecture 8.1**.**
For any partition , there exist (stable) coefficients that are Schur-positive polynomials such that
[TABLE]
where runs over lexicographically ordered sequences of partitions, all having first part larger than .
8.1. Simpler -positivity at
For sure we may consider the simpler -expansions obtained when both and are set to be equal to (or [math] in the expressions above). We then observe that the stable expressions become finite, with the indices of of length bounded by . The case are included in the above values, and we have the following (again not mentioning variables sets on the right-hand side). It would be nice to have a simple combinatorial description of these coefficients.
\begin{aligned} &\nabla_{\mathcal{Y}}(e_{2}(\boldsymbol{x}))\big{|}_{q=t=1}=E_{(2)};\\ &\nabla_{\mathcal{Y}}(e_{3}(\boldsymbol{x}))\big{|}_{q=t=1}=E_{(3)}+3\,E_{(21,21)};\\ &\nabla_{\mathcal{Y}}(e_{4}(\boldsymbol{x}))\big{|}_{q=t=1}=E_{(4)}+2\,E_{(211,22)}+4\,E_{(211,31)}+16\,E_{(211,211,211)};\\ &\nabla_{\mathcal{Y}}(e_{5}(\boldsymbol{x}))\big{|}_{q=t=1}=E_{(5)}+5\,E_{(2111,32)}+5\,E_{(2111,41)}+5\,E_{(221,311)}+5\,E_{(221,221)}+5\,E_{(311,311)}\\ &\hskip 88.2037pt+25\,E_{(2111,2111,221)}+25\,E_{(2111,2111,311)}+125\,E_{(2111,2111,2111)};\\ &\nabla_{\mathcal{Y}}(e_{6}(\boldsymbol{x}))\big{|}_{q=t=1}=E_{(5)}+2\,E_{(222,3111)}+3\,E_{(21111,33)}+3\,E_{(21111,222)}\\ &\hskip 88.2037pt+6\,E_{(21111,42)}+6\,E_{(21111,51)}+6\,E_{(3111,411)}+9\,E_{(2211,411)}\\ &\hskip 88.2037pt+12\,E_{(3111,321)}+18\,E_{(2211,321)}+12\,E_{(21111,3111,3111)}\\ &\hskip 88.2037pt+36\,E_{(21111,21111,411)}+36\,E_{(21111,21111,411)}\\ &\hskip 88.2037pt+54\,E_{(21111,2211,3111)}+72\,E_{(21111,21111,321)}\\ &\hskip 88.2037pt+81\,E_{(21111,2211,2211)}+216\,E_{(21111,21111,21111,3111)}\\ &\hskip 88.2037pt+324\,E_{(21111,21111,21111,2211)}+1296\,E_{(21111,21111,21111,21111,21111)}.\\ \end{aligned}
Likewise specializing at , we get further similar expansions from Equation 8.4. For instance, we have the following values for some small size partitions.
\begin{aligned} &\nabla_{\mathcal{Y}}(\operatorname{\Xi}e_{11}(\boldsymbol{x}))\big{|}_{q=t=1}=E_{(0)}+2\,E_{(2)};\\ &\nabla_{\mathcal{Y}}(\operatorname{\Xi}e_{21}(\boldsymbol{x}))\big{|}_{q=t=1}=3\,E_{(21)}+3\,E_{(3)}+9\,E_{(21,21)};\\ &\nabla_{\mathcal{Y}}(\operatorname{\Xi}e_{111}(\boldsymbol{x}))\big{|}_{q=t=1}=E_{(0)}+9\,E_{(21)}+6\,E_{(3)}+18\,E_{(21,21)};\\ &\nabla_{\mathcal{Y}}(\operatorname{\Xi}e_{31}(\boldsymbol{x}))\big{|}_{q=t=1}=2\,E_{(22)}+4\,E_{(4)}+4\,E_{(31)}+8\,E_{(211,22)}\\ &\hskip 113.81102pt+16\,E_{(211,31)}+16\,E_{(211,211)}+64\,E_{(211,211,211)};\\ \end{aligned}
8.2. More general positivity
For more on the origin of the following construction, see [GorskyHawkesSchillingRainbolt]. Denote by the rational fraction
[TABLE]
with the convention that any vanishing factor is to be replaced by . For a given partition of , a standard Young diagram of shape , and , consider the rational fraction
[TABLE]
where and run over cells of , and writing for if . Then, set
[TABLE]
Conjecture 8.2**.**
Whenever the vector is the step sequence of a triangular partition [BergeronMazin2022] (“under any line” in the terminology of [BlasiakHaimanMorsePunSeelinger2023ExtendedDelta]), then is -Schur positive. Moreover, the -expansion (in all of of the -variables) of \nabla_{\mathcal{Y}}(\mathcal{E}_{\boldsymbol{v}}(\boldsymbol{x}))\big{|}_{q\to q+1,t\to t+1} has Schur-positive coefficients in the -variables.
9. Multivariate lift of
We now discuss how, conjecturally, the operators admit an extension to several parameters (not only and ), for any . In order to do this, we build on experimental observations, illustrated by explicit data. As in Equation 1.8, for some symmetric function, we consider expansions of the following form which satisfy conditions described below.
[TABLE]
still with running through vectors of partitions of . However, the coefficients will now be symmetric functions involving many parameters , that are assumed to be -Schur positive in cases that are of interest here. We may drop the upper indexing when , writing simply . Again we have the symmetry property
[TABLE]
for all permutations of the entries of . It will be handy to use the special notation for the case when all the entries of are equal to .
As we expect to get an extension of , we require that
[TABLE]
When , writing when , we may describe the above expressions as tables, with rows indexed by the s_{\lambda}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}}) and columns indexed by the s_{\mu}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}}). Using techniques explained below, we calculate the entries of Table 2 and Table 3 which describe the coefficients of s_{\mu}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}})s_{\nu}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}}) in \mathcal{N}_{\boldsymbol{y}}(e_{n}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}})). These are of the form , with . Observe that we write symmetric functions of in a variable free manner.
As a matter of fact, these two cases introduce no new term when compared with the expansion of \nabla_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}}}(e_{n}(\boldsymbol{x})). It is only with that interesting changes occur, as shown in Table 3. Observe that there is a unique entry (the coefficient of s_{1111}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}})\,s_{1111}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}})) which involves a Schur function indexed by a partition with more than two parts, so that
[TABLE]
Although this has no impact when one specializes , we will see that many properties become much clearer if one allows these extra terms.
In Table 4 each of the coefficients s_{\lambda}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}}) in
[TABLE]
is presented as an individual table, with rows indexed by the s_{\mu}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}}) and column by s_{\nu}({\color[rgb]{0,0.7,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.7,0}\boldsymbol{z}}). As before, entries777Whose values are imposed by conditions of Equation 9.7 below. are expanded in the Schur function basis .
In general, the structure of the coefficients of depends (in part) on the first following principle.
Principle 9.1**.**
For any , the symmetric function is such
[TABLE]
where is the partition obtained, up to rearrangement and removing [math]’s, from the vector whose entry is one less than the first part of , if is of hook-shape, and [math] otherwise.
For example, . Observe that, for any permutation. It follows that.
Proposition 9.2**.**
For all , Equation 9.4 implies that , for , and we have .
When , Principle 9.1 forces the addition of “extra terms” to , besides those that come from . In general, we have
[TABLE]
One may calculate that we have the following values. Here the effect of Equation 9.4 is outlined with parentheses. Indeed sends all other terms to [math], so that only the outlined part contributes to the equality.
\begin{aligned} &\mathcal{A}^{(1)}_{4}=s_{6}+s_{41}+s_{31}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s_{111}},\\ &\mathcal{A}^{(2)}_{4}=s_{(12)}+s_{(10,1)}+s_{91}+s_{82}+s_{72}+s_{63}+s_{62}+s_{44}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(s_{711}+s_{521}+s_{421}+s_{222})},\\ &\mathcal{A}^{(3)}_{4}=s_{(18)}+s_{(16,1)}+s_{(15,1)}+s_{(14,2)}+s_{(13,2)}+s_{(12,2)}+s_{(12,3)}+s_{(11,3)}+s_{(10,3)}+s_{93}\\ &\qquad\qquad+s_{(10,4)}+s_{94}+s_{85}+s_{75}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(s_{(13,1,1)}+s_{(11,2,1)}+s_{(10,2,1)}+s_{931}+s_{831}+s_{731}}\\ &\qquad\qquad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+s_{741}+s_{551}+s_{822}+s_{632}+s_{532}+s_{333})},\\ \end{aligned}
\begin{aligned} &\mathcal{A}^{(1)}_{5}=s_{(10)}+s_{81}+s_{71}+s_{62}+s_{61}+s_{43}+s_{42}+s_{311}+s_{411}+s_{511}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s_{1111}},\\ &\mathcal{A}^{(2)}_{5}=s_{(20)}+s_{(18,1)}+s_{(17,1)}+s_{(16,1)}+s_{(16,2)}+s_{(15,2)}+2s_{(14,2)}+s_{(13,2)}+s_{(12,2)}\\ &\qquad\qquad+s_{(14,3)}+s_{(13,3)}+2s_{(12,3)}+s_{(11,3)}+s_{(10,3)}+s_{(12,4)}+s_{(11,4)}+2s_{(10,4)}+s_{94}+s_{84}\\ &\qquad\qquad+s_{(10,5)}+s_{95}+s_{85}+s_{86}+s_{76}+s_{66}+s_{(15,1,1)}+s_{(14,1,1)}+s_{(13,1,1)}\\ &\qquad\qquad+s_{(13,2,1)}+2s_{(12,2,1)}+2s_{(11,2,1)}+2s_{(10,2,1)}+s_{921}+s_{(11,3,1)}+2s_{(10,3,1)}\\ &\qquad\qquad+2s_{931}+s_{831}+s_{731}+s_{941}+2s_{841}+2s_{741}+s_{641}+s_{751}+s_{651}\\ &\qquad\qquad+s_{(10,2,2)}+s_{922}+2s_{822}+s_{722}+s_{622}+s_{832}+s_{732}+s_{632}+s_{642}+s_{542}+s_{442}\\ &\qquad\qquad+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(s_{(11,1,1,1)}+s_{9211}+s_{8211}+s_{7211}+s_{7311}+s_{5311}+s_{5411}+s_{6221}}\\ &\qquad\qquad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+s_{5221}+s_{4221}+s_{2222})}\\ \end{aligned}
\begin{aligned} &\mathcal{A}_{6}=s_{(15)}+s_{(13,1)}+s_{(12,1)}+s_{(11,1)}+s_{(10,1)}+s_{(11,2)}+s_{(10,2)}+s_{92}+s_{82}+s_{72}\\ &\qquad\qquad+s_{93}+s_{83}+s_{73}+s_{63}+s_{74}+s_{64}+s_{44}+s_{(10,1,1)}+s_{911}+2s_{811}+s_{711}+s_{611}\\ &\qquad\qquad+s_{821}+s_{721}+s_{621}+s_{521}+s_{421}+s_{631}+s_{531}+s_{431}+s_{441}\\ &\qquad\qquad+s_{6111}+s_{5111}+s_{4111}+s_{3111}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s_{11111}}\end{aligned}
Principle 9.3**.**
More generally, the coefficients satisfy the following requirement. We assume that the last entry of , is a hook shape with first part888In view of the symmetry, this could be any other component. equal to . Setting , we require that
[TABLE]
Equivalently, Equation 9.7 says that terms on both sides agree when restricted to Schur functions indexed by at most two parts. Applying the principles, and all requirements outlined above, we obtain formal expressions that “lift” to many parameters. In particular, from \mathcal{A}_{4}=s_{6}+s_{41}+s_{31}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s_{111}}, , and , we may efficiently describe \mathcal{N}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}}}(e_{4}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}})) as done in Table 5.
A similar description of \mathcal{N}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{y}}}(e_{5}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}})) is given by Table 6. Perforce, we have
[TABLE]
since Principle 9.3 requires that
, so that must appear in .
Moreover, as , we must also have sitting in , on top of which already explains the term .
Further coefficients on which no extra condition are imposed (besides the fact that they are given by the value of \nabla_{\boldsymbol{y}}(e_{5}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{x}}))), are:
[TABLE]
We will discuss further below why needs a supplemental term which is not imposed by the conditions of Equation 9.7. It is easy to see that this does not interfere with said conditions.
Observe in general that we have
Proposition 9.4**.**
For all , and , Equation 9.4 implies that
[TABLE]
where .
10. Connections with other work
Although we give explicit combinatorial models in the case for several general applications of operators, we have no such combinatorial descriptions for the general setup of parameters. It is possible that such descriptions may be found exploiting the work of Carlsson and Mellit (see [Mellit-Poincare, Carlsson-Mellit-Nabla]). Building on an expansion of Mellit for
[TABLE]
Carlsson and Mellit [Carlsson-Mellit-Nabla] obtain a new proof of the Shuffle theorem for . In their proof, they specialize to , and take the coefficient of to reduce Equation 10.1 to . With this specialization, they show that the infinite series involves nilpotent endomorphisms which allows a further reduction to a finite series of monomials in , with the correct positive -weights. An interesting byproduct of their method is that the statistic dinv pops out naturally, without having to know the Shuffle conjecture beforehand. Further study may yield a refinement of their argument giving a positive monomial expansion for .
11. Acknowledgements
M. Romero was partially supported by the NSF Mathematical Sciences Postdoctoral Research Fellowship DMS-1902731.
References