"Pushing" our way from the valley Delta to the generalised valley Delta
aa r X i v : . [ m a t h . C O ] J a n “PUSHING” OUR WAY FROM THE VALLEY DELTA TO THE GENERALISED VALLEYDELTA ALESSANDRO IRACI AND ANNA VANDEN WYNGAERD
Abstract.
In [Haglund, Remmel, Wilson 2018] the authors state two versions of the so called Deltaconjecture, the rise version and the valley version. Of the former, they also give a more general statementin which zero labels are also allowed. In [Qiu, Wilson 2020], the corresponding generalisation of thevalley version is also formulated.In [D’Adderio, Iraci, Vanden Wyngaerd 2020], the authors use a pushing algorithm to prove thegeneralised version of the shuffle theorem. An extension of that argument is used in [Iraci, VandenWyngaerd 2020] to formulate a valley version of the (generalised) Delta square conjecture, and tosuggest a symmetric function identity later stated and proved in [D’Adderio, Romero 2020].In this paper, we use the pushing algorithm together with the aforementioned symmetric functionidentity in order to prove that the valley version of the Delta conjecture implies the valley version ofthe generalised Delta conjecture, which means that they are actually equivalent.Combining this with the results in [Iraci, Vanden Wyngaerd 2020], we prove that the valley versionof the Delta conjecture also implies the corresponding generalised Delta square conjecture. Introduction
In [10], Haglund, Remmel and Wilson conjectured a combinatorial formula for ∆ ′ e n − k − e n in terms ofdecorated labelled Dyck paths, which they called Delta conjecture , after the so called delta operators ∆ ′ f introduced by Bergeron, Garsia, Haiman, and Tesler [2] for any symmetric function f . There are twoversions of the conjecture, referred to as the rise and the valley version.The case k = 0 of the Delta conjecture is the famous shuffle theorem which was proved by Carlssonand Mellit [3], using the compositional refinement formulated in [9]. The shuffle theorem, thanks to thefamous n ! conjecture , now n ! theorem of Haiman [11], gives a combinatorial formula for the Frobeniuscharacteristic of the S n -module of diagonal harmonics studied by Garsia and Haiman.Recently, a compositional refinement of the rise version of the Delta conjecture was announced in [5]and proved in [6]. These breakthroughs rely heavily on the novel Theta operators introduced in [5]. Thevalley version of the Delta conjecture remains an open problem today.The generalised Delta conjecture is a combinatorial formula for ∆ h m ∆ ′ e n − k − e n in terms of decoratedpartially labelled Dyck paths (the rise version first appeared in [10] and the valley version in [14]).Using the Theta operators, we conjectured a touching refinement (where the number of times the Dyckpath returns to the main diagonal is specified) of the valley version of the (generalised) Delta conjecture[12].In this paper, we prove that the touching refinement of the valley version of the Delta conjecture impliesthe touching refinement of the valley version of the generalised Delta conjecture. Our proof will rely ona new symmetric function identity proved in [7], which was suggested by a combinatorial argument wecall the pushing algorithm first described in [5] for paths with no decorations and then extended in [12]to paths with decorated contractible valleys.Combining this result with the results in [12], we obtain that, if the valley version of the Delta conjectureis true, then the valley version of the generalised Delta square conjecture is also true. Thus, the mainconjecture implies three other statements: the generalised version, the square version, and the generalisedsquare version. 2. Symmetric functions
For all the undefined notations and the unproven identities, we refer to [4, Section 1], where definitions,proofs and/or references can be found. e denote by Λ the graded algebra of symmetric functions with coefficients in Q ( q, t ) , and by h , i the Hall scalar product on Λ , defined by declaring that the Schur functions form an orthonormal basis.The standard bases of the symmetric functions that will appear in our calculations are the monomial { m λ } λ , complete { h λ } λ , elementary { e λ } λ , power { p λ } λ and Schur { s λ } λ bases.For a partition µ ⊢ n , we denote by e H µ := e H µ [ X ] = e H µ [ X ; q, t ] = X λ ⊢ n e K λµ ( q, t ) s λ the (modified) Macdonald polynomials , where e K λµ := e K λµ ( q, t ) = K λµ ( q, /t ) t n ( µ ) are the (modified) Kostka coefficients (see [8, Chapter 2] for more details).Macdonald polynomials form a basis of the ring of symmetric functions Λ . This is a modification of thebasis introduced by Macdonald [13].If we identify the partition µ with its Ferrer diagram, i.e. with the collection of cells { ( i, j ) | ≤ i ≤ µ i , ≤ j ≤ ℓ ( µ ) } , then for each cell c ∈ µ we refer to the arm , leg , co-arm and co-leg (denoted respectivelyas a µ ( c ) , l µ ( c ) , a µ ( c ) ′ , l µ ( c ) ′ ) as the number of cells in µ that are strictly to the right, above, to the leftand below c in µ , respectively.Let M := (1 − q )(1 − t ) . For every partition µ , we define the following constants: B µ := B µ ( q, t ) = X c ∈ µ q a ′ µ ( c ) t l ′ µ ( c ) , Π µ := Π µ ( q, t ) = Y c ∈ µ/ (1 , (1 − q a ′ µ ( c ) t l ′ µ ( c ) ) . We will make extensive use of the plethystic notation (cf. [8, Chapter 1]).We need to introduce several linear operators on Λ . Definition 2.1 ([1, [3.11]]) . We define the linear operator ∇ : Λ → Λ on the eigenbasis of Macdonaldpolynomials as ∇ e H µ = e | µ | [ B µ ] e H µ . Definition 2.2.
We define the linear operator Π : Λ → Λ on the eigenbasis of Macdonald polynomials as Π e H µ = Π µ e H µ where we conventionally set Π ∅ := 1 . Definition 2.3.
For f ∈ Λ , we define the linear operators ∆ f , ∆ ′ f : Λ → Λ on the eigenbasis of Macdonaldpolynomials as ∆ f e H µ = f [ B µ ] e H µ , ∆ ′ f e H µ = f [ B µ − e H µ . Observe that on the vector space of symmetric functions homogeneous of degree n , denoted by Λ ( n ) , theoperator ∇ equals ∆ e n .We also introduce the Theta operators, first defined in [5] Definition 2.4.
For any symmetric function f ∈ Λ ( n ) we introduce the following Theta operators on Λ :for every F ∈ Λ ( m ) we set Θ f F := if n ≥ and m = 0 f · F if n = 0 and m = 0 Π f ∗ Π − F otherwise , and we extend by linearly the definition to any f, F ∈ Λ .It is clear that Θ f is linear, and moreover, if f is homogenous of degree k , then so is Θ f , i.e. Θ f Λ ( n ) ⊆ Λ ( n + k ) for f ∈ Λ ( k ) . It is convenient to introduce the so called q -notation. In general, a q -analogue of an expression is ageneralisation involving a parameter q that reduces to the original one for q → . efinition 2.5. For a natural number n ∈ N , we define its q -analogue as [ n ] q := 1 − q n − q = 1 + q + q + · · · + q n − . Given this definition, one can define the q -factorial and the q -binomial as follows. Definition 2.6.
We define [ n ] q ! := n Y k =1 [ k ] q and (cid:20) nk (cid:21) q := [ n ] q ![ k ] q ![ n − k ] q ! Definition 2.7.
For x any variable and n ∈ N ∪ {∞} , we define the q -Pochhammer symbol as ( x ; q ) n := n − Y k =0 (1 − xq k ) = (1 − x )(1 − xq )(1 − xq ) · · · (1 − xq n − ) . We can now introduce yet another family of symmetric functions.
Definition 2.8.
For ≤ k ≤ n , we define the symmetric function E n,k by the expansion e n (cid:20) X − z − q (cid:21) = n X k =0 ( z ; q ) k ( q ; q ) k E n,k . Notice that E n, = δ n, . Setting z = q j we get e n (cid:20) X − q j − q (cid:21) = n X k =0 ( q j ; q ) k ( q ; q ) k E n,k = n X k =0 (cid:20) k + j − k (cid:21) q E n,k and in particular, for j = 1 , we get e n = E n, + E n, + E n, + · · · + E n,n , so these symmetric functions split e n , in some sense.The Theta operators will be useful to restate the Delta conjectures in a new fashion, thanks to thefollowing results. Theorem 2.9 ([5, Theorem 3.1] ) . Θ e k ∇ e n − k = ∆ ′ e n − k − e n The key symmetric function identity on which our proof relies is the following. We first formulated thisidentity by studying the combinatorics. Its proof is due to D’Adderio and Romero.
Theorem 2.10 ([7, Corollary 9.2]) . Given m, n, k, r ∈ N , we have h ⊥ m Θ e k ∇ E n − k,r = m X p =0 t m − p p X i =0 q ( i ) (cid:20) r − p + ii (cid:21) q (cid:20) rp − i (cid:21) q ∆ h m − p Θ e k − i ∇ E n − m − ( k − i ) ,r − p + i . We applied the change of variables j m, m k, p n − k − r, k r, s p, r p − i in order tomake it easier to interpret combinatorially and more consistent with the notation used in other papers.3. Combinatorial definitions
Definition 3.1. A Dyck path of size n is a lattice paths going from (0 , to ( n, n ) consisting of east ornorth unit steps, always ending with an east step and staying above the line x = y , called the maindiagonal . The set of such paths is denoted by D ( n ) . Definition 3.2.
Let π be a Dyck path of size n . We define its area word to be the sequence of integers a ( π ) = ( a ( π ) , a ( π ) , · · · , a n ( π )) such that the i -th vertical step of the path starts from the diagonal y = x + a i ( π ) .For example the path in Figure 1 has area word . Definition 3.3. A partial labelling of a square path π of size n is an element w ∈ N n such that if a i ( π ) > a i − ( π ) , then w i > w i − , • w > .i.e. if we label the i -th vertical step of π with w i , then the labels appearing in each column of π arestrictly increasing from bottom to top, with the additional restriction that the first label cannot be a .We omit the word partial if the labelling is composed of strictly positive labels only. Definition 3.4. A (partially) labelled Dyck path is a pair ( π, w ) where π is a Dyck path and w is a (partial)labelling of π . We denote by LD ( m, n ) the set of labelled Dyck path of size m + n with exactly n positivelabels, and thus exactly m labels equal to .Now we want to extend our sets introducing some decorations. Definition 3.5.
The contractible valleys of a labelled square path π are the indices ≤ i ≤ n such thatone either a i ( π ) < a i − ( π ) , or a i ( π ) = a i − ( π ) and w i > w i − .We define v ( π, w ) := { ≤ i ≤ n | i is a contractible valley } , corresponding to the set of vertical steps that are directly preceded by a horizontal step and, if we wereto remove that horizontal step and move it after the vertical step, we would still get a square path witha valid labelling. Definition 3.6. A valley-decorated (partially) labelled Dyck path is a triple ( π, w, dv ) where ( π, w ) is a(partially) labelled Dyck path and dv ⊆ v ( π, w ) .We denote by LD ( m, n ) • k the set of partially labelled valley-decorated Dyck paths of size m + n with n positive labels and k decorated contractible valleys.Finally, we sometimes omit writing m or k when they are equal to . Notice that, because of therestrictions we have on the labelling and the decorations, the only path with n = 0 is the empty path,for which also m = 0 and k = 0 .
12 4 056 0 3 • •
Figure 1.
Example of an element in LD (2 , • . Definition 3.7.
Let w be a labelling of Dyck path of size n . We define x w := Q ni =1 x w i | x =1 . For P := ( π, w, dv ) ∈ LD ( m, n ) • k we define x P := x w .The fact that we set x = 1 explains the use of the expression partially labelled , as the labels equal to do not contribute to the monomial. Definition 3.8.
Let P := ( π, w, dv ) ∈ LD ( m, n ) • k . Define a touching point of P to be the a point on themain diagonal that is the starting point of a non-decorated vertical step of P , labelled with a positivelabel. The touch of a path P , denoted touch ( P ) , is the number of touching points of P .For example the path in Figure 1 has touch .We define two statistics on this set. efinition 3.9. For ( π, w, dv ) ∈ LD ( m, n ) • k we define area ( π, w, dv ) := m + n X i =1 a i ( π ) , i.e. the number of whole squares between the path and the main diagonal.For example, the path in Figure 1 has area . Notice that the area does not depend on the labelling ordecorations. Definition 3.10.
Let ( π, w, dv ) ∈ LD ( m, n ) • k . For ≤ i < j ≤ n , the pair ( i, j ) is a diagonal inversion if • either a i ( π ) = a j ( π ) and w i < w j ( primary inversion ), • or a i ( π ) = a j ( π ) + 1 and w i > w j ( secondary inversion ),where w i denotes the i -th letter of w , i.e. the label of the vertical step in the i -th row. Then we define dinv ( π, w, dv ) := { ≤ i < j ≤ n | ( i, j ) diagonal inversion ∧ i dv } − dv. For example, the path in Figure has primary inversions ( (2 , , (2 , and (2 , ), secondary inversions( (2 , , (6 , and (6 , ) and decorated valleys. So its dinv equals − .It is easy to check that if j ∈ dv then either there exists some diagonal inversion ( i, j ) and so the dinv isalways non-negative (see [12, Proposition 1]).4. The valley generalised Delta conjecture
Now we have all the tools to state the conjectural formulas that are the object of this paper. Thefollowing conjecture was first stated in [10].
Conjecture 4.1 (Delta conjecture, valley version) . For n, k ∈ N with k < n ∆ ′ e n − k − e n = X P ∈ LD ( n ) • k q dinv ( P ) t area ( P ) x P . In [14], the authors proposed the following formula, containing the previous one as a special case ( m = 0 ). Conjecture 4.2 (Generalised Delta conjecture, valley version) . For m, n, k ∈ N with k < n ∆ h m ∆ ′ e n − k − e n = X P ∈ LD ( m,n ) • k q dinv ( P ) t area ( P ) x P . Recall that, using Theorem 2.9, the symmetric function can be reformulated using the Theta operatorsas follows: ∆ h m ∆ ′ e n − k − e n = ∆ h m Θ e k ∇ e n − k . We have the following refinements, first stated in [12].
Conjecture 4.3 (Touching Delta conjecture, valley version) . For n, k, r ∈ N with k < n Θ e k ∇ E n − k,r = X P ∈ LD ( n ) • k touch ( P )= r q dinv ( P ) t area ( P ) x P . Conjecture 4.4 (Touching generalised Delta conjecture, valley version) . For m, n, k, r ∈ N with k < n ∆ h m Θ e k ∇ E n − k,r = X P ∈ LD ( m,n ) • k touch ( P )= r q dinv ( P ) t area ( P ) x P . The goal of this paper is to prove that Conjecture 4.3 implies Conjecture 4.4.As a corollary, we obtain an analogous result for the corresponding square conjecture. We refer to [12] forthe missing definitions. In that paper, we introduced the aforementioned square analogue of the valleyversion of the Delta conjecture, and we showed that it is implied by Conjecture 4.4 onjecture 4.5 (Modified Delta square conjecture, valley version) . Θ e k ∇ ω ( p n − k ) = X P ∈ LSQ ′ ( n ) • k q dinv ( P ) t area ( P ) x P . This conjecture extends nicely to the m > case, and we also showed that the generalised valley Deltaconjecture implies the corresponding square analogue, that is, Conjecture 4.4 implies Conjecture 4.6. Conjecture 4.6 (Modified generalised Delta square conjecture, valley version) . ∆ h m Θ e k ∇ ω ( p n − k ) = X P ∈ LSQ ′ ( m,n ) • k q dinv ( P ) t area ( P ) x P . Combining these results with the main result of this paper, we get that, if Conjecture 4.3 holds, thenConjecture 4.4, Conjecture 4.5, and Conjecture 4.6 all hold.5.
The proof
Our proof comprises two steps: first, we will interpret 2.10 combinatorially. Then we will apply aninduction argument on the m variable of the same equation to conclude.The left hand side of 2.10, h ⊥ m Θ e k ∇ E n − k,r , coincides with h ⊥ m applied to the left hand side of 4.3.Applying h ⊥ m to the right hand side of 4.3 has the effect of selecting all the paths that have exactly m maximal labels, and setting the variable of that label equal to . In other words, if for a path P oflabelling w we define max( P ) := max( w ) , 4.3 implies h ⊥ m Θ e k ∇ E n − k,r = X P ∈ LD ( n ) • k touch ( P )= rP has m maximal labels q dinv ( P ) t area ( P ) x P | x max( P ) =1 . (1)Thus the combinatorial counterpart of 2.10 is the following. Theorem 5.1.
For all m, n, r, k ∈ N we have X P ∈ LD ( n ) • k touch ( P )= rP has m maximal labels q dinv ( P ) t area ( P ) x P | x max( P ) =1 = m X p =0 t m − p p X i =0 q ( i ) (cid:20) r − p + ii (cid:21) q (cid:20) rp − i (cid:21) q X P ∈ LD ( m − p,n − m ) • k − i touch ( P )= r − p + i q dinv ( P ) t area ( P ) x P Proof.
Start from an element enumerated in the left hand side of the equation: a Dyck path P of size n , with k decorations on valleys, touch r and m maximal labels. We apply what we call the pushingalgorithm , which comprises two operations. Note that any vertical step v labelled with a maximal labelmust be followed by a horizontal step h . Let v be any such step.(1) If the starting point of v lies on the main diagonal, delete v and h . If v was a decorated valley,this decoration also gets deleted.(2) If the starting point of v does not lie on the main diagonal, replace vh by hv and change the labelof v (which was a maximal label) to . If v was a decorated valley, it remains so. See Figure 2.This operation yields a valid Dyck path since v did not touch the main diagonal. The labellingalso stays valid as is smaller than any label of P . M → M →• • Figure 2. “Pushing” a step labelled with a maximal label M .We apply this procedure to all m steps labelled with a maximal label. See Figure 3 for an example. Let p be the number of vertical steps starting from the main diagonal with a maximal label. Let i be the Indeed, by definition h h ⊥ m Θ e k ∇ E n − k,r , h µ i = h Θ e k ∇ E n − k,r , h m h µ i and the homogeneous basis is dual to the mono-mial basis with respect to the Hall scalar product. umber of such steps that are decorated valleys. It follows that after applying the pushing algorithm, weobtain a path ˜ P of size n − p , with k − i decorations and m − p zero labels. Thus, ˜ P ∈ LD ( m − p, n − m ) • k − i .Since the touch does not take into account steps labelled or decorated steps starting from the maindiagonal, the touch of ˜ P is r − ( p − i ) .
23 4 4 13 4 2 34 • • •
23 0 13 2 3 0 • •
Figure 3.
The pushing algorithm.Clearly, performing (1) does not change the area and performing (2) reduces the area by one unit. Sincewe apply (2) m − p times, we have area ( P ) = area ( ˜ P ) + m − p ; which explains the factor t m − p .Let us now study what happens to the dinv. Performing (2) does not alter the dinv since any primarydinv pair involving v becomes a secondary dinv pair and vice versa. For (1), let us distinguish threetypes of steps on the main diagonal of P :(a) non-decorated steps with a maximal label, of which there are p − i ;(b) decorated steps with a maximal label, of which there are i ;(c) non-decorated steps with a non-maximal label, of which there are r − ( p − i ) .The steps of type (a) and (b) get deleted by the algorithm, so we must determine how they contributeto the dinv. The only dinv created by steps of type (a) is primary dinv with steps of type (c). So thecontribution to the dinv for the steps of type (a) depends on the interlacing of these two types of stepsand is q -counted by (cid:2) rp − i (cid:3) q . Similarly, the only dinv created by steps of type (b) is primary dinv withsteps of type (c). The contractibility of the decorated valleys implies that • there must be a step of type (c) before the first occurrence of a step of type (b); • between two steps of type (b), there must be a step of type (c);However, there may be a step of type (b) after all the steps of type (c). Thus, the contribution to thedinv for the steps of type (b) is q -counted q ( i ) (cid:2) r − ( p − i ) i (cid:3) q .Taking the sum over all the possible p ’s and i ’s, we get the announced formula. , Theorem 5.2 (Conditional generalised Delta conjecture, valley version) . If for n, k, r ∈ N the identity Θ e k ∇ E n − k,r = X P ∈ LD ( n ) • k touch ( P )= r q dinv ( P ) t area ( P ) x P holds, then for m, n, k, r ∈ N , the identity ∆ h m Θ e k ∇ E n − k,r = X P ∈ LD ( m,n ) • k touch ( P )= r q dinv ( P ) t area ( P ) x P lso holds. Proof.
We proceed by induction on m . For m = 0 , the statement is exactly the valley version of theDelta conjecture, which we are assuming to hold.For m > , by Theorem 2.10 we have h ⊥ m Θ e k ∇ E n − k,r = m X p =0 t m − p p X i =0 q ( i ) (cid:20) r − p + ii (cid:21) q (cid:20) rp − i (cid:21) q ∆ h m − p Θ e k − i ∇ E n − m − ( k − i ) ,r − p + i . By Theorem 5.1, we can rewrite the statement of Theorem 2.10 as m X p =0 t m − p p X i =0 q ( i ) (cid:20) r − p + ii (cid:21) q (cid:20) rp − i (cid:21) q X P ∈ LD ( m − p,n − m ) • k − i touch ( P )= r − p + i q dinv ( P ) t area ( P ) x P = m X p =0 t m − p p X i =0 q ( i ) (cid:20) r − p + ii (cid:21) q (cid:20) rp − i (cid:21) q ∆ h m − p Θ e k − i ∇ E n − m − ( k − i ) ,r − p + i . By induction hypothesis, whenever p > we have ∆ h m − p Θ e k − i ∇ E n − m − ( k − i ) ,r − p + i = X P ∈ LD ( m − p,n − m ) • k − i touch ( P )= r − p + i q dinv ( P ) t area ( P ) x P so all the terms of the sum except the one when p = 0 cancel out, and we are left with t m X P ∈ LD ( m,n − m ) • k touch ( P )= r q dinv ( P ) t area ( P ) x P = t m ∆ h m Θ e k ∇ E n − m − k,r which is, up to the substitution n n + m and a division by t m , exactly what we wanted to show. , Acknowledgements
The authors would like to thank Michele D’Adderio for the many interesting discussions on the topic.
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