aa r X i v : . [ m a t h . C O ] M a y NEW FORMULAS AND CONJECTURES FOR THE NABLAOPERATOR
FRANC¸ OIS BERGERON
Abstract.
The operator nabla, introduced by Garsia and the author, plays a crucialrole in many aspect of the study of diagonal harmonics. Besides giving several newformulas involving this operator, we show how one is lead to representation theoreticexplanations for conjectures about the effect of this operator on Schur functions.
Contents
1. Introduction 12. Macdonald polynomials and the ∇ operator 23. Representation theoretic conjecture for ∇ ( − S , n − ) 54. Operators 6References 12 Introduction
Denoting by ∆ n ( x ) the Vandermonde determinant in the variables x = x , . . . , x n , weconsider the spaces D n ; f := L ∂,E [ f ( x )∆ n ( x )] , (1)where f ( x ) is a symmetric polynomial in the x -variables. Here, for a polynomial p ( x ),we denote by L ∂,E [ p ( x ] the smallest vector space (say over Q ) that contains p ( x ) and isclosed under taking partial derivatives, as well as applications of the operators E ( k ) uv := n X i =1 u i ∂v ik , (2)where u and v stand for any two of r sets of n variables: x = x , . . . , x n , y = y , . . . , y n , z = z , . . . , z n , etc . The symmetric group S n acts on D n ; f by diagonal permutation of the various variablesets u , sending u i to u σ ( i ) , for σ ∈ S n . Although we restrict most of our discussion tothe case of two sets of variables, Proposition 1 holds in the more general context. It iseasy to show that D n is a subspace of the space of diagonal harmonics. It is in fact thewhole space for one (classical) and two (see [12]) sets of variables, and it seems to be soin general. For more on the case of 3 or more sets of variables, see [2, 6]. The related spaces L ∂ [ f ( x )∆ n ( x )] (with no application of the E -operators) are studied in [4]. The spaces D n ; f are graded by the (vector) degree deg( f ) := (deg x ( f ) , deg y ( f ) , deg z ( f ) , . . . ) , (3)and the action of S n respects this grading, thus D n ; f is a graded S n -module. Recall that,for a (vector-degree) graded S n -module V = M d ∈ N r V d , the coefficients of the Frobenius characteristic (here denoted by V ( w ; q )), in the Schurbasis of the S µ ( w ), correspond to (graded) multiplicities of irreducibles in the S n -module V . This is to say that V ( w ; q ) = X d =( d ,d ,...,d r ) q d q d · · · q d r r V d ( w ) , with V d ( w ) = X µ ⊢ n a µ ; d S µ ( w ) , where a µ ; q is the multiplicity of the irreducible accounted for S µ ( w ) in the homogeneouscomponent V d of degree d . Observe that the fact that V is graded over N r is encodedby the number of variables considered in q .When f ( x ) is a Schur function s µ ( x ), we write D n ; µ for D n ; f . To clarify any possibleresulting ambiguity, let us agree that for f ( x ) = s ( x ) = 1, we denote by D n the resultingspace. Our first objective here is to relate the Frobenius D n ; k ( w ; q ) to conjectures of [5]concerning an operator ∇ on symmetric functions. This operator is defined below interms of Macdonald operators. On the way, we also deduce new identities concerningthis operator ∇ and some of its generalizations.2. Macdonald polynomials and the ∇ operator Macdonald polynomials.
To go on with our story, we need to recall some basic factsabout Macdonald polynomials and operators for which they are common eigenfunctions.As usual (see [16]), we denote by λ (cid:22) µ the dominance order on partitions. This isthe (partial) order characterized by the fact that λ + λ + · · · + λ k ≤ µ + µ + · · · + µ k , for all k ≥ , setting µ i = 0 or λ i = 0, whenever i is larger than the number of parts of the underlyingpartition. Also as usual, µ ′ stands for the conjugate partition of µ .Recall that the integral form Macdonald polynomials H µ ( w ; q, t ), with µ a partitionof n (we write µ ⊢ n ), expand in the Schur function basis as H µ ( w ; q, t ) = X λ ⊢ n K λ,µ ( q, t ) S λ ( w ) . (4) RELIMINARY VERSION 3
The coefficients K λ,µ ( q, t ) are known as the q, t - Kostka polynomials . They have beenshown, in [12], to have positive integer coefficients. They form a linear basis of the ringΛ, of symmetric functions in the variables w , and are characterized by the equations(i) h S λ ( w ) , H µ [(1 − q ) w ; q, t ] i = 0 , if λ µ, (ii) h S λ ( w ) , H µ [(1 − t ) w ; q, t ] i = 0 , if λ µ ′ , and(iii) h S n ( w ) , H µ ( w ; q, t ) i = 1 , (5)involving the usual “Hall” scalar product on symmetric functions (for which the Schurfunctions are orthonormal). Stated otherwise, (iii) says that H µ ( w ; q, t ) is normalized sothat K n,µ ( q, t ) = 1, for all µ . Since the one-part partition ( n ) is largest in dominanceorder, the equations in (5) imply that H n ( w ; q, t ) = H n ( w ; q ) = e n (cid:20) w − q (cid:21) n Y k =1 (1 − q k ) , (6)using plethystic substitution notation (see [1] and section 4). Observe that we may dropthe parameter t , since it plays no role here. This is a special case of a Hall-Littlewoodpolynomial (see [16]), so that H n ( w ; q ) has coefficients K λ,n ( q ) = X λ ( τ )= µ q coch( τ ) , (7)where the sum is over the set of standard tableaux τ , of shape λ ( τ ) = µ , with coch( τ )standing for the cocharge statistic (see [16]) of τ . For example, H ( w ; q ) = S ( w ) ,H ( w ; q ) = S ( w ) + q S ( w ) ,H ( w ; q ) = S ( w ) + (cid:0) q + q (cid:1) S ( w ) + q S ( w ) ,H ( w ; q ) = S ( w ) + (cid:0) q + q + q (cid:1) S ( w ) + (cid:0) q + q (cid:1) S ( w )+ (cid:0) q + q + q (cid:1) S ( w ) + q S ( w ) . It has been shown in [11] that H µ ( w ; t, q ) = H µ ( w ; q, t ) , where H µ := L ∂ [∆ µ ] stands for the Garsia-Haiman module , obtained as the linearspan of all partial derivatives of the determinant∆ µ := det (cid:0) x ai y bi (cid:1) ≤ i ≤ n ( a,b ) ∈ µ , with ( a, b ) ∈ µ meaning that 0 ≤ b ≤ ℓ ( µ ) −
1, and 0 ≤ a ≤ µ i − S n -submodules (see [3]): H µ ↓ := \ ν → µ H ν , F. BERGERON where ν → µ means that the partition ν precedes µ in the Young lattice. Using aset of heuristics discussed in [3], we can calculate H µ ↓ ( w ; q, t ) in terms of Macdonaldpolynomials. The ∇ operator. Since the H µ ( w ; q, t ) form a basis of Λ, we can define a linear operator ∇ (read “nabla”) on degree n symmetric functions by imposing that ∇ ( H µ ( w ; q, t )) = H µ ( w ; q, t ) Y ( a,b ) ∈ µ q a t b . Thus the H µ ( w ; q, t ) are eigenfunctions of ∇ with simple, explicit eigenvalues expressedin term of the partition statistic n ( µ ) := X i ( i − µ i . Indeed, the above definition is equivalent to ∇ ( H µ ( w ; q, t )) := q n ( µ ′ ) t n ( µ ) H µ ( w ; q, t ). Re-call that the main result in this context is that D n ( w ; q, t ) = ∇ ( e n ( w )) , (8)where D n is being consider over two sets of n variables, as expressed by the use of thetwo parameters q and t (rather than q and q ).As discussed in [5], explicit calculations reveal that the effect of ∇ on Schur functions isstriking. To expresse this, let us consider the matrix ∇ ( n ) := ( ∇ λµ ) λ,µ ⊢ n , with ∇ λµ ( q, t ) := h∇ ( S λ ( w )) , S µ ( w ) i . Here, we order partitions in decreasing lexicographic order, i.e.: 3, 21, 111 for n = 3.The coefficients of these matrices are symmetric polynomials in the variables q and t . Weexpand them as Schur polynomials (denoted with a lowercase “s” to further distinguishthem from the S µ ( w )), omitting the variables q and t . For n = 2, we get ∇ (2) = (cid:18) − s s (cid:19) , and for n = 3, ∇ (3) = s s − s − s s + s s + s . Inspection of these matrices, together with some theoretical considerations, leads to
Conjecture 1 (see [5]) . For all λ and µ , ( − m ( λ ) ∇ λµ ( q, t ) has positive integer coefficients when expanded in terms of Schur polynomials, with m ( λ ) := (cid:18) k (cid:19) + X λ ′ i < ( i − ( i − − λ ′ i ) , (9) k being the number of parts of λ . RELIMINARY VERSION 5
Similar conjectures have also been formulated in [5] for iterates of ∇ , and for more generaloperators (see Section 4). Conjecture 1 has been shown to hold in the special case t = 1by Lenart in [14]. Among other results along these lines, let us also mention the work ofCan and Loehr in [7].3. Representation theoretic conjecture for ∇ ( − S , n − )Going back to the spaces introduced in section 2, we have the following. Proposition 1.
The space D n ; n is isomorphic to the restriction to S n of the S n +1 -module D n +1 . Proof.
Let us denote by x ′ the set of n + 1 variables x , . . . , x n , x n +1 , obtained by addingone extra variable to x . We check that D n ; n = L ∂,E [ e n ( x )∆ n ( x )] is isomorphic to therestriction to S n of D n +1 = L ∂,E [∆ n +1 ( x ′ )] as follows. Considering its expansion as apolynomial in the variable x n +1 , it is easily seen that we may write ∆ n +1 ( x ′ ) in the form∆ n +1 ( x ′ ) = e n ( x )∆ n ( x ) + x n +1 g ( x ′ ) , (10)for some polynomial g ( x ′ ). Since ( ∂x + . . . + ∂x n +1 )∆ n +1 ( x ′ ) = 0, it is also easy to checkthat we may construct a basis of D n +1 by application on ∆ n +1 ( x ′ ) of some operatorsbuilt using only the E -operators and partial derivatives involving variables x k , for which1 ≤ k ≤ n . It follows that application of these same operators on the “leading” part e n ( x )∆ n ( x ) of (10), we get a basis of D n ; n . Clearly this is all compatible with the actionof S n on the first n variables in x ′ . (cid:3) By general principles (see [1]), we conclude that we have the Frobenius equality D n ; n ( w ; q, t ) = ∂p D n +1 ( w ; q, t ) . (11)Using opeartors calculus outlined in section 4, it follows that Proposition 2.
In the case of two sets of variables, we have D n ; n ( w ; q, t ) = n X k =1 [ k + 1] q,t ∇ ( e k ( w ) e n − k ( w )) , (12) where [ a ] q,t := q a − t a q − t = q a − + q a − t + . . . + qt a − + t a − . Based on experiments and observations, we conjecture that more generally
Conjecture 2.
For all k between and n − , D n ; k ( w ; q, t ) = k X j =0 [ k − j + 1] q,t ∇ ( e j ( w ) e n − j ( w )) . (13) F. BERGERON
Observe that, for k = 1, (13) takes the form D n ;1 ( w ; q, t ) = ∇ ( e j ( w ) e n − j ( w )) + ( q + t ) ∇ ( e n ( w )) . In view of the identity S n − ( w ) = e ( w ) e n − ( w ) − e n ( w ), we calculate directly thatConjecture 2 implies that ∇ ( − S n − ( w )) = D n ;1 ( w ; q, t ) − ( q + t + 1) D n ;0 ( w ; q, t ) . (14)Now, it is easy to check that D n ⊕ e ( x ) D n ⊕ e ( y ) D n ⊆ D n ;1 . (15)This leads us to consider the orthogonal complement O Y (in the space D n ;1 ) of thesubspace corresponding to the left-hand-side of (15). The case k = 1 of Conjecture 2 isthus seen to be equivalent to Conjecture 3. O Y ( w ; q, t ) = ∇ ( − S n − ( w )) . Similar descriptions can be obtained for other values of k . A recent conjecture of N. Loehrand G. Warrington (see [15]) describes in a combinatorial manner, for all partition λ , theexpansion of ∇ ( S λ ( w )) in terms of monomial symmetric functions. However, this givesonly a purely enumerative description of the coefficients of the expansion of ∇ ( S λ ( w ))in terms of the LLT symmetric polynomials (see [13]).4. Operators
We have already used plethystic substitution to describe H n ( w ; q ) in (6). Recall that thisoperation turns symmetric functions into operators on the ring Λ, in the following man-ner. We first expand every symmetric function in terms of power sum p λ = p λ · · · p λ k ,and then apply the following rules: • p k [ c ] = c if c is a constant; • p k [ x ] = x k if x is a variable; • p k [ p j ] = p k · j ; • F [ f + g ] = F [ f ] + F [ g ], and ( F + G )[ f ] = F [ f ] + G [ f ]; • F [ f · g ] = F [ f ] · F [ g ], and ( F · G )[ f ] = F [ f ] · G [ f ].For the purpose of such calculations, w is identified with w + w + 2 + . . . = p ( w ). Asin [5], we can now introduce the operators D m (on symmetric functions f ( w )) definedby the generating function identity ∞ X m = −∞ D m ( f ( w )) ξ m := f [ w + α/ξ ] Ω ′ ( w ; − ξ ) , (16) RELIMINARY VERSION 7 with α = α ( q, t ) := (1 − q )(1 − t ), andΩ ′ ( w ; ξ ) := X k ≥ e k ( w ) ξ k = n Y i =1 (1 + r i ξ ) . Now, recall from [5] that we have p ⊥ ∇ = α − ∇ D − . (17)For a symmetric function f , the operator f ⊥ is dual to multiplication by f , for the Hallscalar product. We apply both sides of this operator identity to e n +1 ( w ), and calculate(directly using (16)) that ∂p ∇ ( e n +1 ( w )) = ∇ ( α − D − ( e n +1 ( w )))= ∇ n X k =0 [ k + 1] q,t e k ( w ) e n − k ( w ) ! = n X k =0 [ k + 1] q,t ∇ ( e k ( w ) e n − k ( w )) . Using (8), we conclude that (12) holds, since D n ; n ( w ; q, t ) = ∂p ∇ ( e n +1 ( w ))= n X k =0 [ k + 1] q,t ∇ ( e k ( w ) e n − k ( w )) . This proves Proposition 2.Let us recall some more operators identities of [5] and [8]. Following [9, (3.24)] definethe symmetric functions ε n,j ( w ; q ): ε n,j ( w ; q ) = n X k =0 F k ( w ; q ) Z k (cid:12)(cid:12)(cid:12) t j , with n X k =0 F k ( w ; q ) z k = e n (cid:20) w (1 − z )1 − q (cid:21) , and Z k := k X i =0 ( − i q ( i +12 ) − k i ( q ; q ) i (cid:20) ki (cid:21) t i . One can show (see [9, Exercise 3.33]) that( − q ) n − ε n, ( w ; q ) = S n ( w ) . (18)It is also shown in [10, Formula (221)] that q ( n ) ε n,n − ( w ; q ) = − [ n − q H ( n, ↓ ( w ) . (19) Observe that we have had to change the notation used in [9] from E n,j to ε n,j ( w ; q ). Otherwise, wewould have had too many objects denoted by “ E ”. Although without the use H µ ↓ which is tied with conjectures in [3]. F. BERGERON
One can further check (see [9]) that e n ( w ) = n X j =1 ε n,j ( w ; q ) . (20)For any symmetric function f , as in [5] we now consider the operators ∇ f having theMacdonald polynomials H µ ( w ; q, t ) as eigenfunctions with eigenvalue f [ B µ ], writing B µ for the polynomial P ( a,b ) ∈ µ q a t b . Implicitly, a partition is here identified with the set ofcells of its Ferrers diagram. In formula, the above definition can be stated as ∇ f ( H µ ( w ; q, t )) := f [ B µ ] H µ ( w ; q, t ) . (21)Observe that we clearly have the operator identities ∇ f + g = ∇ f + ∇ g and ∇ f g = ∇ f ∇ g .Moreover all these operators commute among themselves since they share a commonbasis of eigenfunctions. It has been shown in [12] that, for all partitions µ and α , onehas the property h∇ S µ ∇ ( e n ( w )) , S α ( w ) i ∈ N [ q, t ] . (22)In light of (20) and computer calculations, the following refinement of Formula (22) wasalso conjectured to hold (see [9, Conj. 4.12]): h∇ S µ ∇ ( ε n,k ( w ; q )) , S α ( w ) i ∈ N [ q, t ] , for all 1 ≤ k ≤ n. (23)On the other hand, it was conjectured in [5] that Conjecture 4 (BGHT 1999) . h∇ S µ ( e n ( w )) , S α ( w ) i ∈ N [ q, t ] , (24) for all µ and α . For an analogous statement for the functions ε n,k ( w ; q ), see (39).We denote by ∇ k the operator obtained by choosing f = e k (the elementary symmetricfunction). In particular, when restricted to the subspace of degree n homogeneous sym-metric functions, the operator ∇ n is simply the usual ∇ . In other terms, the usual ∇ is ∇ = X k ∇ k π k , with π k denoting the projection of the ring of symmetric functions on its homogeneouscomponent of degree k . For the case f = P k =0 e k u k , we denote by Ψ the operator ∆ f .In other words, we have Ψ( H µ ) = Y ( a,b ) ∈ µ (1 − q a t b u ) H µ . (25)When restricted to degree n symmetric functions, the operator Ψ clearly expands asΨ = Id + u ∇ + u ∇ + . . . + u n − ∇ n − + u n ∇ n . (26) RELIMINARY VERSION 9
For the sake of clarity, we now denote by ρ the operator p ⊥ . Let us also write θ forthe operator α − D − , so that formulas appearing in formulas (I.12) of [5] may now berewritten as (a) D = Id − α ∇ , (b) α D k +1 = D k ι − ι D k , (c) D = −∇ ι ∇ − , (d) θ = ∇ − ρ ∇ , (e) Ψ − e Ψ = e + u θ, (f) e ∇ k = ∇ k e + ∇ k − θ. We are now ready to derive more identities in the line of these.
Proposition 3.
We have the linear operators identities: Ψ − ρ Ψ = ρ + u θ, (27) ρ ∇ k = ∇ k ρ + ∇ k − θ. (28) Proof.
The first identity is easy to “check” by direct application on the basis of Mac-donald polynomials. Indeed, recalling the dual Pieri formula of Macdonald ρ H µ = X ν → µ c µ,ν H ν , with ν → µ indicating that ν is covered by µ in the Young lattice of partitions, and the c µ,ν = c µ,ν ( q, t ) being explicit rational fractions in q and t (see [8]). Applying the lefthand side of (27) to H µ we getΨ − ρ Ψ( H µ ) = X ν → µ c µ,ν (1 − q a ν t bν u ) H ν , where ( a ν , b ν ) stands for the coordinates of the corner cell by which ν differs from µ .The right hand side of this last equality can clearly be written in the formΨ − ρ Ψ( H µ ) = ρ H µ + u ∇ − ρ ∇ ( H ν ) , and we conclude using the operator equality θ = ∇ − ρ ∇ (see [5, Formulas I.12]). Com-paring coefficients of u k in the expansion in (26), it follows from (27) that (28) holds. (cid:3) Iterative applications of (28) gives that ρ n ∇ m = n X j =0 X α ∈A j,n − j ∇ m − j α, (29)with A j,k denoting the set of (cid:0) j + kk (cid:1) operators that can be obtained by composing j copiesof ρ and k copies of θ in all possible order. Applying both sides of (29) to e n ( w ), we findthat ρ n ∇ n − ( e n ( w )) = n − X k =0 θ k ρ θ n − − k e n ( w ) , (30)since all other terms of the resulting sum vanish, and ∇ = Id. It happens (as we will seebelow) that ∇ n − ( e n ( w )) corresponds to the bigraded Frobenius characteristic of some S n -module. Formula (30) can be used to get an explicit formula for the Hilbert series ofthis module. Other exciting consequences of (27) and (28) are that ρ Ψ( p n ( w )) = ( − n − u [ n ] t [ n ] q Ψ( e n ( w )) , (31) ∇ n − ( p n ( w )) = [ n ] t [ n ] q q n − t n − ∇ ( h n ( w )) . (32)It is also interesting that we have Proposition 4. ∇ ( p n ( w )) = ( − n − [ n ] t [ n ] q e n ( w ) , (33) ∇ ( h n ( w )) = n X k =1 ( − q t ) n − k S k, n − k ( w ) , (34) ∇ ( e n ( w )) = n X k =1 [ k ] q,t e n − k ( w ) e k ( w ) . (35) Proof.
These are all derived using the fact that ∇ = 1(1 − t )(1 − q ) (Id − D ) , (36)which is directly derived from formula (1.11) of [8]. The first one is easiest since we have(Id − D )( p n ( w )) = p n ( w ) − p n (cid:20) w + (1 − t )(1 − q ) ξ (cid:21) Ω ′ ( w ; − ξ ) (cid:12)(cid:12) ξ = − (1 − t n )(1 − q n ) 1 ξ n Ω ′ ( w ; − ξ ) (cid:12)(cid:12) ξ = ( − n − (1 − t n )(1 − q n ) e n ( w ) . For (34), we use h n ( w + s ) = n X k =0 h k ( w ) h n − k ( s )to calculate that h n (cid:20) w + (1 − t )(1 − q ) ξ (cid:21) = n X k =0 ξ k h n − k ( w ) h k ((1 − t )(1 − q ))= h n ( w ) + n X k =1 ξ k h n − k ( w ) (1 − t )(1 − q )(1 − q k t k )(1 − q t ) . RELIMINARY VERSION 11
Thus we have ∇ ( h n ( w )) = 1(1 − t )(1 − q ) (Id − D )( h n ( w ))= − n X k =1 h n − k ( w ) (1 − q k t k )(1 − q t ) 1 ξ k Ω ′ ( w ; − ξ ) (cid:12)(cid:12) ξ = − n X k =1 ( − k (1 − q k t k )(1 − q t ) h n − k ( w ) e k ( w )= n X k =1 ( − q t ) n − k S k, n − k ( w ) . To get (35), we calculate (now using e n ( w + s ) = P nk =0 e k ( w ) e n − k ( s )) that(Id − D )( e n ( w )) = e n ( w ) − e n (cid:20) w + (1 − t )(1 − q ) ξ (cid:21) Ω ′ ( w ; − ξ ) (cid:12)(cid:12) ξ = − n X k =1 ξ k e k ((1 − t )(1 − q )) e n − k ( w )Ω ′ ( w ; − ξ ) (cid:12)(cid:12) ξ = (1 − t )(1 − q ) n X k =1 [ k ] q,t e k ( w ) e n − k ( w ) . (cid:3) We can also see that ∇ n − ( e n ( w )) = ( − n − q n − t n − ∇∇ ( h n ( w )) (37)= ( − k − n X k =1 ( − q t ) − k ∇ ( S k, n − k ( w )) , (38)with the second equality obtained using (34). Observe that, because of (18), this is alsoequal to ∇ ∇ ( ε n, ( w ; q ) /t n − ) so that we have a link here with (23).For the symmetric functions ε n,j ( w ; q ), it has been shown (see [9, (3.34)]) that q ( n ) ε n,n ( w ; q ) = H n ( w ; q, t ) . As already discussed (see (7)), it is well-known that H n ( w ; q, t ) expands as H n ( w ; q, t ) = X µ ⊢ n K µ, ( n ) ( q ) S µ ( w ) , with the K µ, ( n ) ( q ) polynomials in only the parameter q and having nonnegative integercoefficients. More generally, Garsia and Haglund have conjectured that (see [9, Conj.3.8]) for all j . Conjecture (Garsia-Haiman 2001) . ∇ ( ε n,j ( w ; q )) = X µ ⊢ n c ( n,j ) µ ( q, t ) S µ ( w ) , (39) with the coefficients c ( n,j ) µ ( q, t ) polynomials in q and t with positive integer coefficients. Moreover, a combinatorial formula is proposed for (39) (see [9, (6.61)]). In view of For-mula (18), both (39) and conjecture 1 can be applied here. Sure enough they agree. For-mula (8) implies that we can give a representation theoretic meaning to ∇ ( ε n,n − ( w ; q )).Indeed, we first show that ∇ ( ε n,n − ( w ; q )) = coefficient of t in ∇ ( e n ( w )) , (40)and then use Formula (19) to calculate ∇ ( ε n,n − ( w ; q )) using results of [3]. Furtherinterest in this arises from the known combinatorial formula [9, Prop. 3.9.1] for the lefthand side of this equality. References [1]
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