Theta operators, refined Delta conjectures, and coinvariants

Theta operators, refined Delta conjectures, and coinvariants

Michele D’Adderio Université Libre de Bruxelles (ULB)
Département de Mathématique
Boulevard du Triomphe, B-1050 Bruxelles
Belgium
mdadderi@ulb.ac.be
Alessandro Iraci Università di Pisa and Université Libre de Bruxelles (ULB)
Dipartimento di Matematica
Largo Bruno Pontecorvo 5, 56127 Pisa
Italia
iraci@student.dm.unipi.it
 and  Anna Vanden Wyngaerd Université Libre de Bruxelles (ULB)
Département de Mathématique
Boulevard du Triomphe, B-1050 Bruxelles
Belgium
anvdwyng@ulb.ac.be
Abstract.

We introduce the family of Theta operators indexed by symmetric functions that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson [Haglund-Remmel-Wilson-2015] for . We show that the -variable Catalan theorem of Zabrocki [Zabrocki-4Catalan-2016] is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit in [Carlsson-Mellit-ShuffleConj-2015], extending one of their results.

Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for , and prove the case , which was also conjectured in [Haglund-Remmel-Wilson-2015], extending the shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for . Moreover we show how this implies the case of our generalized Delta square conjecture for , extending the square theorem of Sergel [Leven-2016] to a generalized square theorem for .

Still the Theta operators will provide a conjectural formula for the Frobenius characteristic of super-diagonal coinvariants with two sets of Grassmanian variables, extending the one of Zabrocki in [Zabrocki_Delta_Module] for the case with one set of such variables. We propose a combinatorial interpretation of this last formula at , leaving open the problem of finding a statistic that gives the whole symmetric function.

1. Introduction

In the 90’s Garsia and Haiman introduced the -module of diagonal harmonics, i.e. the coinvariants of the diagonal action of on polynomials in two sets of variables, and they conjectured that its Frobenius characteristic was given by , where is the nabla operator on symmetric functions introduced in [Bergeron-Garsia-Haiman-Tesler-Positivity-1999]. In 2002 Haiman proved this conjecture (see [Haiman-Vanishing-2002]). Later the authors of [HHLRU-2005] formulated the so called shuffle conjecture, i.e. they predicted a combinatorial formula for in terms of labelled Dyck paths. Several years later in [Haglund-Morse-Zabrocki-2012] Haglund, Morse and Zabrocki conjectured a compositional refinement of the shuffle conjecture, which specified also the points where the Dyck paths touches the main diagonal. Recently Carlsson and Mellit in [Carlsson-Mellit-ShuffleConj-2015] proved precisely this refinement, thanks to the introduction of what they called the Dyck path algebra. See [Willigenburg_History_Shuffle] for more on this story.

In [Haglund-Remmel-Wilson-2015] Haglund, Remmel and Wilson formulated the Delta conjecture, which can be stated as

where the sum is over labelled Dyck paths of size with positive labels and decorated rises. It turns out that for this formula reduces to the shuffle conjecture. Recently this conjecture attracted quite a bit of interest: see [TheBible, Section 2] for the state of the art on this problem.

Even more recently Zabrocki in [Zabrocki_Delta_Module] conjectured that this formula gives the Frobenius characteristic of the submodule of the super-diagonal coinvariants of degree in the Grassmannian variables (see Section 8 for the missing definitions). It turns out that the submodule of degree in the Grassmannian variables is precisely the module of diagonal harmonics. So the whole framework of “diagonal harmonics shuffle conjecture” got generalized to this new setting.

In this work we add a new piece to the puzzle, by extending the compositional shuffle conjecture to a compositional Delta conjecture. In order to do so, we introduce a new family of Theta operators on symmetric functions, indexed by symmetric functions . In fact with this article we want to make the case for the Theta operators.

Here are the highlights of the present paper:

  • We state a compositional Delta conjecture, which will read as follows: for a composition

    where are the symmetric functions appearing in the compositional shuffle conjecture, and is the composition given by the distances between the consecutive points where the Dyck path touches the diagonal, ignoring the rows containing a decorated rise.

  • We show that Zabrocki’s -variable Catalan theorem [Zabrocki-4Catalan-2016] is actually the Schröder case of our compositional Delta conjecture.

  • We show that the combinatorial side satisfies a recursion that can be described by the Dyck path algebra in the same way as Carlsson and Mellit proved it for the compositional shuffle conjecture. In this case our theorem will read as

    where is defined recursively, and it coincides with in [Carlsson-Mellit-ShuffleConj-2015] for . This reduces our refinement of the Delta conjecture to an identity of operators.

  • We state a touching generalized Delta conjecture, which refines the generalized Delta conjecture in [Haglund-Remmel-Wilson-2015] for to

    where are the symmetric functions already appearing in the shuffle conjecture and the sum of the righthand side is over labelled Dyck paths with nonnegative labels of size , with zero labels, decorated rises and touching the diagonal times. Furthermore we state a touching generalized Delta square conjecture, which refines our generalized Delta square conjecture [DAdderio-Iraci-VandenWyngaerd-Delta-Square] for to

    where the sum is over labelled square paths ending east with nonnegative labels, decorated rises and touching the diagonal times.

  • We prove the case of our touching generalized Delta conjecture, which was already conjectured in [Haglund-Remmel-Wilson-2015, Conjecture 7.5]. This extends the shuffle theorem of Carlsson and Mellit [Carlsson-Mellit-ShuffleConj-2015] to a generalized shuffle theorem for .

  • We prove the case of our touching generalized Delta square conjecture. This extends the square theorem of Sergel [Leven-2016] to a generalized square theorem for .

  • We extend Zabrocki’s conjecture [Zabrocki_Delta_Module] to the module of super-diagonal coinvariants of the diagonal action of on polynomials in two sets of commutative variables and two sets of Grassmanian variables:

  • We conjecture a combinatorial interpretation for the formula in the previous item at :

    where is the set of labelled Dyck paths with positive labels of size with decorated rises and decorated contractible valleys. We leave open the outstanding problem of finding a statistic that gives the whole symmetric function .

The rest of this paper is organized in the following way. In Section 2 we introduce the basic ingredients of symmetric functions, that will be needed in Section 3 to introduce the Theta operators and to state the basic theorems that led us to their definition. In Section 4 we recall the combinatorial definitions needed in Section 5 to state our refined Delta conjectures. In Section 6 we prove our results about the compositional Delta conjecture, while in Section 7 we prove our results about the touching generalized Delta conjectures. In Section 8 we state our conjectures about the Frobenius characteristics of super-diagonal coinvariants, while in Section 9 we state our combinatorial interpretation of those at . In Section 10 we state more conjectures about the Theta operators. Finally in Section 11 we give the details of the technical proofs of symmetric function theory that we left out in the previous sections.

Acknowledgements

We are happy to thank Mike Zabrocki for providing us useful references, programs and information to check our conjecture on the Frobenius characteristic of super-diagonal coinvariants.

2. Symmetric functions: basics

In this section we limit ourselves to introduce the necessary notation to state our main theorems and conjectures. We refer to Section 11.1 for more on symmetric functions.

The main references that we will use for symmetric functions are [Macdonald-Book-1995], [Stanley-Book-1999] and [Haglund-Book-2008].

The standard bases of the symmetric functions that will appear in our calculations are the complete , elementary , power and Schur bases.

We will use the usual convention that and for .

The ring of symmetric functions can be thought of as the polynomial ring in the power sum generators . This ring has a grading given by assigning degree to for all . As we are working with Macdonald symmetric functions involving two parameters and , we will consider this polynomial ring over the field . We will make extensive use of the plethystic notation.

With this notation we will be able to add and subtract alphabets, which will be represented as sums of monomials . Then, given a symmetric function , and thinking of it as an element of , we denote by the expression with replaced by , for all . More generally, given any expression , we define the plethystic substitution to be with replaced by .

We denote by the Hall scalar product on symmetric functions, which can be defined by saying that the Schur functions form an orthonormal basis. We denote by the fundamental algebraic involution which sends to , to and to .

With the symbol “” we denote the operation of taking the adjoint of an operator with respect to the Hall scalar product, i.e.

(1)

For a partition , we denote by

(2)

the (modified) Macdonald polynomials, where

(3)

are the (modified) Kostka coefficients (see [Haglund-Book-2008, Chapter 2] for more details).

The set is a basis of the ring of symmetric functions with coefficients in . This is a modification of the basis introduced by Macdonald [Macdonald-Book-1995], and they are the Frobenius characteristic of the so called Garsia-Haiman modules (see [Garsia-Haiman-PNAS-1993]).

If we identify the partition with its Ferrers diagram, i.e. with the collection of cells , then for each cell we refer to the arm, leg, co-arm and co-leg (denoted respectively as ) as the number of cells in that are strictly to the right, above, to the left and below in , respectively (see Figure 1).

Arm

Leg

Co-leg

Co-arm

Figure 1.

We set

(4)

and we define for every partition

(5)
(6)
(7)
(8)
(9)

Notice that

(10)

For every symmetric function we set

(11)

The following linear operators were introduced in [Bergeron-Garsia-ScienceFiction-1999, Bergeron-Garsia-Haiman-Tesler-Positivity-1999], and they are at the basis of the conjectures relating symmetric function coefficients and -combinatorics in this area.

We define the nabla operator on by

(12)

and we define the Delta operators and on by

(13)

Observe that on the vector space of symmetric functions homogeneous of degree , denoted by , the operator equals . Moreover, for every ,

(14)

and for any , on , so that on .

Recall the standard notation for -analogues:

(15)
(16)

and

(17)

and also the standard notation for the -rising factorial

(18)

In [Haglund-Morse-Zabrocki-2012] the following operators were introduced: for any and any

(19)

and for any composition of , denoted , we set

(20)

The symmetric functions were introduced in [Garsia-Haglund-qtCatalan-2002] by means of the following expansion:

(21)

Notice that setting in (21) we get

(22)

In particular, for we get

(23)

The following identity is proved in [Haglund-Morse-Zabrocki-2012]*Section 5:

(24)

where denotes the length of the composition .

Together with (23) it gives immediately

(25)

The following identity is proved in [Can-Loehr-2006]*Theorem 4:

(26)

3. The Theta operators

In this section we introduce the family of operators and we state a few results about them: we will give the missing proofs in Section 11. See also Section 10 for some conjectures about them.

Recall the definition of the invertible linear operator on defined by , and for any non-empty partition

(27)

For any symmetric function we introduce the following Theta operators on : for every we set

(28)

It is clear that is linear, and moreover, if is homogenous of degree , then so is , i.e.

Since in the present article we will use mostly a special case of these operators, we introduced the following shorter notation: for , we set

(29)

Notice that is the identity operator on .

The following theorems, which we prove in Section 11, led to the definition of the Theta operators.

Theorem 3.1.

For and ,

(30)
Corollary 3.2.

For and ,

(31)
Proof.

Using (23), we have

(using (30))

Theorem 3.3.

For and ,

(32)
Corollary 3.4.

For and ,

(33)
Proof.

We have

(using (26))
(using (32))

4. Combinatorial definitions

Definition 4.1.

A square path ending east of size is a lattice paths going from to consisting of east or north unit steps, always ending with an east step. The set of such paths is denoted by . We call base diagonal of a square path the diagonal with the smallest value of that is touched by the path (so that ). The shift of the square path is the non-negative value . We refer to the line as the main diagonal. A Dyck path is a square path whose shift is , i.e. its base diagonal is the main diagonal. The set of Dyck paths is denoted by . Of course .

For example, the path in Figure 2 has shift .

Figure 2. Example of an element in with reading word .
Definition 4.2.

A labelling or word of a square path of size ending east is an element such that when we label the -th vertical step of with

  • the labels appearing in each column of are strictly increasing from bottom to top;

  • there is at least one nonzero label labelling a vertical step starting from the base diagonal of ;

  • if starts with a vertical step, then this first step has a nonzero label.

The set of such labellings with labels equal to is denoted by , and we set .

A partially labelled square path ending east is an element of

We also define the subset of labelled Dyck paths as

Definition 4.3.

Let be a square path ending east of size . We define its area word to be the sequence of integers such that the -th vertical step of the path starts from the diagonal . For example the path in Figure 2 has area word .

Definition 4.4.

We define for each a monomial in the variables : we set

where is the label of the -th vertical step of (the first being at the bottom), and where we conventionally set . The fact that does not appear in the monomial explains the word partially.

Definition 4.5.

The rises of a square path ending east are the indices

or the vertical steps that are directly preceded by another vertical step.

A decorated square path (respectively Dyck path) is a pair where is a square path (respectively Dyck path) and . We set

A partially labelled decorated square path (respectively Dyck path) is a triple where is a square path (respectively Dyck path), and is a labelling of . We set

We will also use the following natural identifications

Definition 4.6.

Given a partially labelled square path, we call zero valleys its vertical steps with label (which are necessarily preceded by a horizontal step, hence the name valleys).

Definition 4.7.

Given a partially labelled square path of shift , we define its reading word as the sequence of nonzero labels, read starting from the base diagonal going bottom left to top right, then moving to the next diagonal, again going bottom left to top right, and so on.

If the reading word of is then the reverse reading word of is .

See Figure 2 and Figure 3 for an example.

Figure 3. Example of an element in with reading word .

We define two statistics on this set that reduce to the same statistics as defined in [Loehr-Warrington-square-2007] when .

Definition 4.8.

Let and be its shift. Define

More visually, the area is the number of whole squares between the path and the base diagonal and not contained in rows containing a decorated rise.

If then we set . In other words, the area of a path does not depend on its labelling.

For example, the path in Figure 2 has area .

Definition 4.9.

Let . For , we say that the pair is an inversion if

  • either and (primary inversion),

  • or and (secondary inversion),

where denotes the -th letter of , i.e. the label of the vertical step in the -th row.

Then we define

This second term is referred to as bonus dinv.

For example, the path in Figure 2 has dinv : primary inversions, i.e. and , secondary inversion, i.e. , and bonus dinv, coming from the rows , and . Notice that Dyck paths coincide with the square paths with no bonus dinv.

Definition 4.10.

Given we define its diagonal composition to be the composition of whose -th part is the number of rows of without a label or a decoration that lie between the -th and the -th vertical step of on the base diagonal not labelled by a (or from the -th step onwards if it is the last such step). See Figure 4 for an example. If , its diagonal composition is defined identically, without the conditions concerning the labels (since there are none).

Definition 4.11.

Given we define a touching point of to be a starting point of a vertical step of that lies on the base diagonal, and whose label is not zero. The touching number of is defined as the number of touching points of or equivalently as the length of its diagonal composition. See Figure 4 for an example. For these definitions are the same, without the condition concerning the labels (since there are none).

Figure 4. A partially labelled Dyck path with diagonal composition and touching number .

5. Statements of refined Delta conjectures

In this section we state our refined conjectures.

The following conjecture is due to Haglund, Remmel and Wilson [Haglund-Remmel-Wilson-2015].

Conjecture 5.1 (Generalized Delta).

Given with ,

(34)

We state our “touching” refinement of this conjecture.

Conjecture 5.2 (Touching generalized Delta).

Given , and ,

(35)
Remark 5.3.

It follows immediately from Corollary 3.2 that our touching generalized Delta conjecture implies the generalized Delta conjecture.

We now state our compositional refinement of the Delta conjecture, i.e. of the case of Conjecture 5.1.

Conjecture 5.4 (Compositional Delta).

Given , and ,

(36)
Remark 5.5.

We observe immediately that the compositional Delta conjecture implies the touching Delta conjecture: combinatorially we clearly have , while on the symmetric function side we use (24).

Remark 5.6.

Notice that we have a compositional version only of the Delta conjecture, and not of the generalized Delta conjecture, i.e. only for the number of zero labels equal to . For the dinv seems to be off at the compositional level. This situation should be compared with the final comments of [DAdderio-Iraci-VandenWyngaerd-GenDeltaSchroeder].

Still, we can state the following conjecture.

Conjecture 5.7.

Given , , and ,

(37)

The following conjecture appeared in our work [DAdderio-Iraci-VandenWyngaerd-Delta-Square].

Conjecture 5.8 (Generalized Delta square).

Given with ,

(38)

We state our “touching” refinement of this conjecture.

Conjecture 5.9 (Touching generalized Delta square).

Given , and ,

(39)
Remark 5.10.

It follows immediately from Corollary 3.4 that our touching generalized Delta square conjecture implies the generalized Delta square conjecture.

6. About the compositional version

In this section we prove some results about our compositional Delta conjecture.

6.1. Relation to the -variable Catalan theorem

In [Zabrocki-4Catalan-2016] Zabrocki showed that for any composition

is the -enumerator of Dyck paths of size with decorated rises and decorated peaks with . So this should match the Schröder of our compositional Delta conjecture, i.e. it should be equal to

Indeed, this follows immediately from the following lemma, which we prove in Section 11.

Lemma 6.1.

For any ,

(40)

This shows that the -variable Catalan result of Zarbrocki is really the Schröder case of our compositional Delta conjecture.

6.2. Relation to the Dyck path algebra

In [Carlsson-Mellit-ShuffleConj-2015] the authors construct the Dyck path algebra by defining combinatorial operators that can be used to compute the -enumerators of specific subsets of labelled Dyck paths. The purpose of this subsection is to use these operators to extend their results to decorated labelled Dyck paths.

6.2.1. Combinatorial translation

The goal is to relate our compositional Delta conjecture to the operators of the Dyck path algebra from [Carlsson-Mellit-ShuffleConj-2015]. Following [Carlsson-Mellit-ShuffleConj-2015], first we need to translate our -enumerator of into one of another bistatistic .

In order to do this we need to extend some definitions and bijections in [Haglund_Loehr_conj_Hilbert] to the decorated setting (cf. also [Haglund-Xin_Lecture-Notes, Chapters 2 and 3]).

Definition 6.2.

Let . We define its bounce path as a lattice path from to computed in the following way: it starts in and travels north until it encounters the beginning of an east step of , then it turns east until it hits the main diagonal, then it turns north again, and so on; thus it continues until it reaches .

We label the vertical steps of the bounce path starting from and increasing the labels by every time the path hits the main diagonal (so the steps in the first vertical segment of the path are labelled with , the ones in the next vertical segment are labelled with , and so on). We define the bounce word of to be the string where is the label attached to the -th vertical step of the bounce path.

We define the statistic bounce on and as

Let us start by describing Haglund’s classical bijection (Theorem 3.15 in [Haglund-Book-2008])

that transforms into .

Take and rearrange its area word in ascending order. This new word, call it , will be the bounce word of . We construct as follows. First draw the bounce path corresponding to . The first vertical stretch and last horizontal stretch of are fixed by this path. For the section of the path between consecutive peaks of the bounce path we apply the following procedure: place a pen on the top of the -th peak of the bounce path and scan the area word of from left to right. Every time we encounter a letter equal to we draw an east step and when we encounter a letter equal to we draw a north step. By construction of the bounce path, we end up with our pen on top of the -th peak of the bounce path. Note that in an area word a letter equal to cannot appear unless it is preceded somewhere by a letter equal to . This means that starting from the -th peak, we always start with a horizontal step which explains why is indeed the bounce word of .

We want to extend to . For an element we apply to . We must now specify what happens to the decorated rises and the labelling of .

Definition 6.3.

A corner (or valley) of a Dyck path is one of the indices

Corners will often be identified with the vertical steps of that are directly preceded by a horizontal step.

Proposition 6.4.

If then there exists a bijection between and .

Proof.

Let . It follows that . Take such that . While scanning the area word to construct the path between the -th and -th peak of the bounce path, we will encounter , directly followed by . This will correspond to a horizontal step followed by a vertical step in and thus to an element of . Arguing backwards, it is easy to see that this gives the desired bijection. ∎

Definition 6.5.

Given , we define to be the elements such that for every , , where is the index of the column containing the horizontal step preceding the -th vertical step of .

Definition 6.6.

The set of pairs with , and will be denoted . The indices in will be referred to as decorated corners: we decorate the -th vertical step of with a for all .

The set of triples with and will be denoted by . In this set, we represent inside the squares containing the main diagonal , starting from the bottom. See Figure 5 on the right for an example.

The following result (without decorations) first appeared in [Haglund_Loehr_conj_Hilbert] (see also [Haglund-Book-2008, Chapter 5]). We sketch its proof for completeness.

Proposition 6.7.

There exists a bijection

Sketch of the proof.

Take . We want to define