A proof of the Extended Delta Conjecture
Jonah Blasiak, Mark Haiman, Jennifer Morse, Anna Pun, George H. Seelinger
aa r X i v : . [ m a t h . C O ] F e b A PROOF OF THE EXTENDED DELTA CONJECTURE
J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER
Abstract.
We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, acombinatorial formula for ∆ h l ∆ ′ e k e n , where ∆ ′ e k and ∆ h l are Macdonald eigenoperators and e n is an elementary symmetric function. We actually prove a stronger identity of infiniteseries of GL m characters expressed in terms of LLT series. This is achieved through newresults in the theory of the Schiffmann algebra and its action on the algebra of symmetricfunctions. Introduction
We prove the
Extended Delta Conjecture of Haglund, Remmel and Wilson [14] by adaptingmethods from our work in [2] on a generalized Shuffle Theorem and proving new results aboutthe action of the elliptic Hall algebra on symmetric functions. As in [2], we reformulate theconjecture as the polynomial truncation of an identity of infinite series of GL m characters,expressed in terms of LLT series. We then prove the stronger infinite series identity using aCauchy identity for non-symmetric Hall-Littlewood polynomials.The conjecture stemmed from studies of the diagonal coinvariant algebra DR n in two setsof n variables, whose character as a doubly graded S n module has remarkable links withboth classical combinatorial enumeration and the theory of Macdonald polynomials. It wasshown in [17] that this character is neatly given by the formula ∆ ′ e n − e n , where ∆ ′ f is acertain eigenoperator on Macdonald polynomials and e n is the n -th elementary symmetricfunction.The Shuffle Theorem , conjectured in [13] and proven by Carlsson and Mellit in [4], givesa combinatorial expression for ∆ ′ e n − e n in terms of Dyck paths—that is, lattice paths from(0 , n ) to ( n,
0) that lie weakly below the line segment connecting these two points.An expanded investigation led Haglund, Remmel and Wilson [14] to the
Delta Conjecture ,a combinatorial prediction for ∆ ′ e k e n , for all 0 ≤ k < n . This led to a flurry of activity(e.g. [6, 11, 14, 15, 16, 20, 21, 22, 25, 27]), including a conjecture by Zabrocki [26] that ∆ ′ e k e n captures the character of the super-diagonal coinvariant ring SDR n , a deformation of DR n involving the addition of a set of anti-commuting variables.The Delta Conjecture has been extended in two directions. One gives a Compositional generalization, just proved by D’Adderio and Mellit [7]. The other involves a second eigen-operator ∆ h l , where h l is the l -th homogeneous symmetric function. The Extended Delta
Mathematics Subject Classification.
Primary: 05E05; Secondary: 16T30.Authors were supported by NSF Grants DMS-1855784 (J. B.) and DMS-1855804 (J. M. and G. S.).
Conjecture [14, Conjecture 7.4] is, for l ≥ ≤ k ≤ n ,(1) ∆ h l ∆ ′ e k − e n = h z n − k i X λ ∈ D n + l X P ∈ L n + l,l ( λ ) q dinv( P ) t area( λ ) x wt + ( P ) Y r i ( λ )= r i − ( λ )+1 (cid:0) z t − r i ( λ ) (cid:1) , in which λ is a Dyck path and P is a certain type of labelling of λ (see § t = 0 specialization of theconjecture [5, 6]; Qiu and Wilson [20] reformulated the conjecture and established the q = 0specialization as well.Let us briefly outline the steps by which we prove (1).Feigin–Tsymbauliak [8] and Schiffmann–Vasserot [24] constructed an action of the ellipticHall algebra E of Burban and Schiffmann [3] on the algebra of symmetric functions. Theoperators ∆ f and ∆ ′ f are part of the E action. In Theorem 4.4.1, we use this to reformulatethe left hand side of (1) as the polynomial part of an explicit infinite series of virtual GL m characters with coefficients in Q ( q, t ). The proof of Theorem 4.4.1 requires new propertiesof distinguished elements of E introduced by Negut [19]. These properties are the commu-tator identity in Proposition 4.2.1 and the symmetry between the Negut elements and theirtransposes in Proposition 4.3.3.In Theorem 5.1.1, we also reformulate the right hand side of (1) as the polynomial part ofan infinite series, in this case expressed in terms of the LLT series introduced by Grojnowskiand Haiman in [12].With (1) now rewritten as an equality between the polynomial parts of two expressionsin (99), we ultimately arrive at Theorem 6.3.5—an identity of infinite series of GL m characterswhich implies the Extended Delta Conjecture by taking the polynomial part on each side.Although the Extended Delta Conjecture and the Compositional Delta Conjecture bothimply the Delta Conjecture, they generalize it in different directions and our methods arequite different from those of D’Adderio and Mellit. It would be interesting to know whethera common generalization is possible.2. The Extended Delta Conjecture
The Extended Delta Conjecture equates a “symmetric function side,” involving the actionof a Macdonald operator on an elementary symmetric function, with a “combinatorial side.”We begin by recalling the definitions of these two quantities.2.1.
Symmetric function side.
Integer partitions are written λ = ( λ ≥ · · · ≥ λ l ), some-times with trailing zeroes allowed. We set | λ | = λ + · · · + λ l and let ℓ ( λ ) be the numberof non-zero parts. We identify a partition λ with its French style Ferrers shape, the set oflattice squares (or boxes ) with northeast corner in the set(2) { ( i, j ) | ≤ j ≤ ℓ ( λ ) , ≤ i ≤ λ j } . The shape generator of λ is the polynomial(3) B λ ( q, t ) = X ( i,j ) ∈ λ q i − t j − . PROOF OF THE EXTENDED DELTA CONJECTURE 3
Let Λ = Λ k ( X ) be the algebra of symmetric functions in an infinite alphabet of vari-ables X = x , x , . . . , with coefficients in the field k = Q ( q, t ). We follow the notation ofMacdonald [18] for the graded bases of Λ. Basis elements are indexed by a partition λ and have homogeneous degree | λ | . Examples include the elementary symmetric functions e λ = e λ · · · e λ k , complete homogeneous symmetric functions h λ = h λ · · · h λ k , power-sums p λ = p λ · · · p λ k , monomial symmetric functions m λ , and Schur functions s λ .As is conventional, ω : Λ → Λ denotes the k -algebra involution defined by ωs λ = s λ ∗ , and h− , −i the symmetric bilinear inner product such that h s λ , s µ i = δ λ,µ .The basis of modified Macdonald polynomials, ˜ H µ ( X ; q, t ), is defined [9] from the integralform Macdonald polynomials J µ ( X ; q, t ) of [18] using the device of plethystic evaluation . Foran expression A in terms of indeterminates, such as a polynomial, rational function, or formalseries, p k [ A ] is defined to be the result of substituting a k for every indeterminate a occurringin A . We define f [ A ] for any f ∈ Λ by substituting p k [ A ] for p k in the expression for f as apolynomial in the power-sums p k , so that f f [ A ] is a homomorphism. The variables q, t from our ground field k count as indeterminates. The modified Macdonald polynomials aredefined by(4) ˜ H µ ( X ; q, t ) = t n ( µ ) J µ (cid:20) X − t − ; q, t − (cid:21) , where(5) n ( µ ) = X i ( i − µ i For any symmetric function f ∈ Λ, let f [ B ] denote the eigenoperator on the basis { ˜ H µ } ofΛ such that(6) f [ B ] ˜ H µ = f [ B µ ( q, t )] ˜ H µ . The left hand side of (1) is expressed in the notation of [14], where ∆ f = f [ B ] and ∆ ′ f = f [ B − symmetric function side of the Extended Delta Conjecture is(7) h l [ B ] e k − [ B − e n . The combinatorial side.
The right hand side of the Extended Delta Conjecture (1)is a combinatorial generating function that counts labelled lattice paths.
Definition 2.2.1. A Dyck path is a south-east lattice path lying weakly below the linesegment connecting the points (0 , N ) and ( N, D N . The staircase path δ is the Dyck path alternating between south and east steps.Each λ ∈ D N has area( λ ) = | δ/λ | defined to be the number of lattice squares lying above λ and below δ . Let r i ( λ ) be the area contribution from squares in the i -th row, numberedfrom north to south; in other words, r i is the distance from the i -th south step of λ to the J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER
Figure 1.
Partial labelling P ∈ L , ( λ ) of path λ below has area( λ ) = 10,dinv( P ) = 15, x wt + ( P ) = x x x x x x , and x wt( P ) = x x x x x x x .134 0 35 2 1 06 4 1234567891011 j c j i r i i -th south step of δ . Note that(8) r ( λ ) = 0 , r i ( λ ) ≤ r i − ( λ ) + 1 for i > , and N X i =1 r i ( λ ) = | δ/λ | . Definition 2.2.2. A labelling P = ( P , . . . , P N ) ∈ N N attaches a label in N = { , , . . . } toeach south step of λ ∈ D N so that the labels increase from north to south along vertical runsof south steps, as shown in Figure 1. The set of labellings is denoted by L N ( λ ), or simply L ( λ ). Given 0 ≤ l < N , a partial labelling of λ ∈ D N is a labelling where 0 occurs exactly l times and never on the line x = 0. We denote the set of these partial labellings by L N,l ( λ ).To each labelling P ∈ L ( λ ) is associated a statistic dinv( P ), defined to be the number ofpairs ( i < j ) such that either(9) ( r i ( λ ) = r j ( λ ) and P i < P j or r i ( λ ) = r j ( λ ) + 1 and P i > P j . The weight of a labelling P is defined so zero labels do not contribute, by(10) x wt + ( P ) = Y i ∈ [ N ]: P i =0 x P i . This is equivalent to letting x = 1 in x wt( P ) := Q i ∈ [ N ] x P i .The above defines the right hand side of (1), with h z n − k i denoting the coefficient of z n − k . Remark . In [14], a Dyck path is a north-east lattice path lying weakly above the linesegment connecting (0 ,
0) and (
N, N ), and labellings increase from south to north along
PROOF OF THE EXTENDED DELTA CONJECTURE 5 vertical runs. After reflecting the picture about a horizontal line, our conventions on paths,labellings, and the definition of dinv( P ) match those in [14]. Separately, [13] uses the sameconventions that we do for Dyck paths, but defines labellings to increase from south to north,and defines dinv( P ) with the inequalities in (9) reversed. However, since the sum(11) X P ∈ L ( λ ) q dinv( P ) x wt( P ) is a symmetric function [13], it is unchanged if we reverse the ordering on labels, after whichthe conventions in [13] agree with those used here.We prefer another slight modification based on the following lemma which was mentionedin [14] without details. Lemma 2.2.4.
For any Dyck path λ ∈ D N , we have (12) Y
0) to (0 , N ), as shownhere for the example in Figure 1.S S S E S EE S S EEE S E S E S S E S EETreating south and east steps as left and right parentheses, each south step pairs with aneast step to its right, and we have r i ( λ ) = c j ( λ ) if the i -th south step (numbered left to right)pairs with the j -th east step (numbered right to left). Furthermore, the leftmost member ofeach double south step pairs with the rightmost member of a double east step, as indicatedin the word displayed above.Since each index i − r i ( λ ) = r i − ( λ ) + 1 pairs with an index j − c j ( λ ) = c j − ( λ ) + 1, we have(13) Y
J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER
Setting N = n + l and m = k + l , the right hand side of (1), or the combinatorial sideof the Extended Delta Conjecture , is equal to(14) h z N − m i X λ ∈ D N P ∈ L N,l ( λ ) t | δ/λ | q dinv( P ) x wt + ( P ) Y
2. The Schiffmann algebra E is generatedby a central Laurent polynomial subalgebra F = k [ c ± , c ± ] and a family of subalgebrasΛ F ( X m,n ) isomorphic to the algebra of symmetric functions Λ F ( X ) over F , one for each pairof coprime integers ( m, n ). These are subject to defining relations spelled out below.For any algebra A containing a copy of Λ, there is an adjoint action of Λ on A arisingfrom the Hopf algebra structure of Λ. Using two formal alphabets X and Y to distinguishbetween the tensor factors in Λ ⊗ Λ ∼ = Λ( X )Λ( Y ), the coproduct and antipode for the Hopfalgebra structure are given by the plethystic substitutions(15) ∆ f = f [ X + Y ] , S ( f ) = f [ − X ] . The adjoint action of f ∈ Λ on ζ ∈ A is then given by(16) (Ad f ) ζ = X i f i ζ g i , where f [ X − Y ] = X i f i ( X ) g i ( Y ) , since the formula on the right is another way to write (1 ⊗ S )∆ f = P i f i ⊗ g i . More explicitly,we have(17) (Ad p n ) ζ = [ p n , ζ ] and (Ad h n ) ζ = X j + k = n ( − k h j ζ e k . PROOF OF THE EXTENDED DELTA CONJECTURE 7
The last formula can be expressed for all n at once as a generating function identity(18) (Ad Ω[ zX ]) ζ = Ω[ zX ] ζ Ω[ − zX ] , where(19) Ω( X ) = ∞ X n =0 h n ( X ) . We fix notation for the quantities(20) M = (1 − q )(1 − t ) , c M = (1 − ( q t ) − ) M, which play a role in the presentation of E and will be referred to again later.3.2.1. Basic structure and symmetries.
The algebra E is Z graded with the central subal-gebra F in degree (0 ,
0) and f ( X m,n ) in degree ( dm, dn ) for f ( X ) of degree d in Λ( X ).The universal central extension \ SL ( Z ) → SL ( Z ) acts on the set of tuples(21) { ( m, n, θ ) ∈ ( Z \ ) × R | θ is a value of arg( m + in ) } , lifting the SL ( Z ) action on pairs ( m, n ), with the central subgroup Z generated by the‘rotation by 2 π ’ map ( m, n, θ ) ( m, n, θ + 2 π ). The group \ SL ( Z ) acts on E by k -algebraautomorphisms, compatibly with the action of SL ( Z ) on the grading group Z . Beforegiving the defining relations of E , we specify how \ SL ( Z ) acts on the generators.For each pair of coprime integers ( m, n ), we introduce a family of alphabets X m,nθ , one foreach value θ of arg( m + in ), related by(22) X m,nθ +2 π = c m c n X m,nθ . We make the convention that X m,n without a subscript means X m,nθ with θ ∈ ( − π, π ]. Thesubalgebra Λ F ( X m,n ) = Λ F ( X m,nθ ) only depends on ( m, n ) and so does not depend on thechoice of branch for the angle θ . We will also refer below to the subalgebras Λ k ( X m,n ),which do depend on our choice of branch, unless we specialize the central parameters c , c to elements of k .The \ SL ( Z ) action is now given by ρ · f ( X m,nθ ) = f ( X m ′ ,n ′ θ ′ ) for f ( X ) ∈ Λ k ( X ) where ρ ∈ \ SL ( Z ) acts on the indexing data in (21) by ρ · ( m, n, θ ) = ( m ′ , n ′ , θ ′ ). Note that if m, n are coprime, then so are m ′ , n ′ . The action on F factors through the action of SL ( Z ) onthe group algebra k · Z ∼ = F .For instance, the ‘rotation by 2 π ’ element ρ ∈ \ SL ( Z ) fixes F , and has ρ · f ( X m,nθ ) = f ( X m,nθ +2 π ) = f [ c m c m X m,nθ ]. Thus ρ coincides with multiplication by c r c s in degree ( r, s ), andautomatically preserves all relations that respect the Z grading.We now turn to the defining relations of E . Apart from the relations implicit in F = k [ c ± , c ± ] being central and each Λ F ( X m,n ) being isomorphic to Λ F ( X ), these fall into threefamilies: Heisenberg relations, internal action relations and axis-crossing relations. J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER
Heisenberg relations.
Each pair of subalgebras Λ F ( X m,n ) and Λ F ( X − m, − n ) in degreesalong opposite rays in Z satisfy Heisenberg relations(23) [ p k ( X − m, − nθ ) , p l ( X m,nθ + π )] = δ k,l k p k [( c m c n − / c M ] , where c M is given by (20). As an exercise, the reader can check, using (22), that the relationsin (23) are consistent with swapping the roles of Λ F ( X m,n ) and Λ F ( X − m, − n ).3.2.3. Internal action relations.
The internal action relations describe the adjoint action ofeach Λ F ( X m,n ) on E . For simplicity, we write these relations, and also the axis-crossingrelations below, with Λ F ( X , ) distinguished. The full set of relations is understood to begiven by closing the stated relations under the \ SL ( Z ) action.Bearing in mind that X m,n means X m,nθ with θ ∈ ( − π, π ], the relations for the internalaction of Λ F ( X , ) are:(24) (Ad f ( X , )) p ( X m, ) = ( ωf )[ z ] (cid:12)(cid:12)(cid:12) z k p ( X m + k, )(Ad f ( X , )) p ( X m, − ) = ( ωf )[ − z ] (cid:12)(cid:12)(cid:12) z k p ( X m + k, − )3.2.4. Axis-crossing relations.
Again distinguishing Λ F ( X , ) and taking angles on thebranch θ ∈ ( − π, π ], the final set of relations is the closure under the \ SL ( Z ) action of(25) [ p ( X b, − ) , p ( X a, )] = − e a + b [ − c M X , ] c M for a + b > . More generally, rotating this relation by π determines [ p ( X b, − ) , p ( X a, )] for a + b < a + b = 0. Combining these gives(26) [ p ( X b, − ) , p ( X a, )] = − c M e a + b [ − c M X , ] a + b > − c − b c a + b = 0 − c − b c e − ( a + b ) [ − c M X − , ] a + b < . Further remarks.
Define upper and lower half subalgebras E ∗ ,> , E ∗ ,< ⊂ E to be gen-erated by the Λ F ( X m,n ) with n > n <
0, respectively. Using the \ SL ( Z ) image of therelations in (25), one can express any e k [ − c M X m,n ] for n > p ( X a, ). This shows that { p ( X a, ) | a ∈ Z } generates E ∗ ,> as an F -algebra. Similarly, { p ( X a, − ) | a ∈ Z } generates E ∗ ,< .The internal action relations give the adjoint action of Λ F ( X , ) on the space spanned by { p ( X a, ± ) | a ∈ Z } . Using the formula (Ad f )( ζ ζ ) = P ((Ad f (1) ) ζ )((Ad f (2) ) ζ ), where∆ f = P f (1) ⊗ f (2) in Sweedler notation, this determines the adjoint action of Λ F ( X , ) on E ∗ ,> and E ∗ ,< . The Heisenberg relations give the adjoint action of Λ F ( X , ) on Λ F ( X − , ),while Λ F ( X , ) acts trivially on itself, with (Ad f ) g = f [1] g .Together these determine the adjoint action of Λ F ( X , ) on the whole algebra E . Bysymmetry, the same holds for the adjoint action of any Λ F ( X m,n ). PROOF OF THE EXTENDED DELTA CONJECTURE 9
Anti-involution.
One can check from the defining relations above that E has a furthersymmetry given by an involutory anti-automorphism (product reversing automorphism)(27) Φ : E → E Φ g ( c , c ) = g ( c − , c − ) , Φ f ( X m,nθ ) = f ( X n,mπ/ − θ ) . Note that Φ is compatible with reflecting degrees in Z about the line x = y . Together with \ SL ( Z ) it generates a \ GL ( Z ) action on E for which ρ ∈ \ GL ( Z ) is an anti-automorphism if \ GL ( Z ) → GL ( Z ) det → {± } sends ρ to − Action of E on Λ . We write f • for the operator of multiplication by a function f tobetter distinguish between operator expressions such as ( ωf ) • and ω · f • . For f a symmetricfunction, f ⊥ denotes the h− , −i adjoint of f • .Here and again later on, we use an overbar to indicate inverting the variables in anyexpression; for example(28) M = (1 − q − )(1 − t − ) . We extend the notation in (6) accordingly, setting(29) f [ B ] ˜ H µ = f [ B µ ( q − , t − )] ˜ H µ . Proposition 3.3.1 ([2, Prop 3.3.1]) . There is an action of E on Λ characterized as follows.(i) The central parameters c , c act as scalars (30) c , c ( q t ) − . (ii) The subalgebras Λ k ( X ± , ) act as (31) f ( X , ) ( ωf )[ B − /M ] , f ( X − , ) ( ωf )[1 /M − B ] . (iii) The subalgebras Λ k ( X , ± ) act as (32) f ( X , ) f [ − X/M ] • , f ( X , − ) f ( X ) ⊥ . We will make particular use of operators representing the action on Λ of elements p ( X a, )and p ( X ,a ) in E . For the first we need the operator ∇ , defined in [1] as an eigenoperatoron the modified Macdonald basis by(33) ∇ ˜ H µ = t n ( µ ) q n ( µ ∗ ) ˜ H µ , where n ( µ ) is given by (5) and µ ∗ denotes the transpose partition.For the second, we introduce the doubly infinite generating series(34) D ( z ) = ω Ω[ z − X ] • ( ω Ω[ − zM X ]) ⊥ , where Ω( X ) is given by (19). Definition 3.3.2.
For a ∈ Z , we define operators on Λ = Λ k ( X ) E a = ∇ a e ( X ) • ∇ − a , (35) D a = h z − a i D ( z ) . (36)The operators D a are the same as in [2] and differ by a sign ( − a from those in [1, 10]. Proposition 3.3.3.
In the action of E on Λ given by Proposition 3.3.1:(i) the element p [ − M X ,a ] = − M p ( X ,a ) ∈ E acts as the operator D a ;(ii) the element p [ − M X a, ] = − M p ( X a, ) ∈ E acts as the operator E a .Proof. Part (i) is proven in [2, Prop 3.3.4].By (32), p [ − M X , ] acts on Λ as multiplication by p [ X ] = e ( X ). It was shown in[2, Proposition 3.4.1] that the action of E on Λ satisfies the symmetry ∇ f ( X m,n ) ∇ − = f ( X m + n,n ). More generally, this implies ∇ a f ( X m,n ) ∇ − a = f ( X m + an,n ) for every integer a .Hence, p [ − M X a, ] acts as ∇ a p [ − M X , ] ∇ − a = ∇ a e ( X ) • ∇ − a . (cid:3) l characters and the shuffle algebra. As usual, the weight lattice of GL l is Z l ,with Weyl group W = S l permuting the coordinates. A weight λ is dominant if λ ≥ · · · ≥ λ l .A polynomial weight is a dominant weight λ such that λ l ≥
0. In other words, polynomialweights of GL l are integer partitions of length at most l .The algebra of virtual GL l characters over k can be identified with the algebra of sym-metric Laurent polynomials k [ x ± , . . . , x ± l ] S l . If λ is a polynomial weight, the irreduciblecharacter χ λ is equal to the Schur function s λ ( x , . . . , x l ). Given a virtual GL l character f ( x ) = f ( x , . . . , x l ) = P λ c λ χ λ , the partial sum over polynomial weights λ is a symmetricpolynomial in l variables, which we denote by f ( x ) pol . We use the same notation for infiniteformal sums f ( x ) of irreducible GL l characters, in which case f ( x ) pol is a symmetric formalpower series.The Weyl symmetrization operator for GL l is(37) σ ( φ ( x , . . . , x l )) = X w ∈ S l w φ ( x ) Q i
0. We leave out thecentral subalgebra F , since the relations of E + (as we will see in a moment) do not dependon the central parameters.The image of E + under the anti-automorphism Φ in § E + generatedby the Λ k ( X m,n ) for n >
0. Note that our convention θ ∈ ( − π, π ] when the subscript isomitted yields Φ f ( X m,n ) = f ( X n,m ) for Λ k ( X m,n ) ⊂ E + , since the branch cut is in the thirdquadrant.Schiffmann and Vasserot [24] proved the following result. See [2, § Proposition 3.4.1 ([24, Theorem 10.1]) . There is an algebra isomorphism ψ : S → E + andan anti-isomorphism ψ op = Φ ◦ ψ : S → Φ E + , given on the generators by ψ ( z a ) = p [ − M X ,a ] and ψ op ( z a ) = p [ − M X a, ] . To be clear, on monomials in m variables, representing elements of tensor degree m in S ,the maps in Proposition 3.4.1 are given by ψ ( z a · · · z a m m ) = p [ − M X ,a ] · · · p [ − M X ,a m ](41) ψ op ( z a · · · z a m m ) = p [ − M X a m , ] · · · p [ − M X a , ](42)Later we will need the following formula for the action of ψ ( φ ( z )) on Λ( X ). Proposition 3.4.2 ([2, Proposition 3.5.2]) . Let φ ( z ) = φ ( z , . . . , z l ) be a Laurent polynomialrepresenting an element of tensor degree l in S , and let ζ = ψ ( φ ( z )) ∈ E + be its image under the map in (41) . With E acting on Λ as in Proposition 3.3.1, we have (43) ω ( ζ · x , . . . , x l ) = H q,t ( φ ( x )) pol . Schiffmann algebra reformulation of the symmetric function side
Distinguished elements D b and E a . Negut [19] defined a family of distinguishedelements D b ∈ E + , indexed by b ∈ Z l , which in the case l = 1 reduce to the elements inProposition 3.3.3(i). Here a remarkable symmetry between these elements and their images E a under the anti-involution Φ will play a crucial role. Specifically, we will derive twopreviously unknown identities—the commutator formula in Proposition 4.2.1 and symmetryidentity in Proposition 4.3.3—involving these elements. Definition 4.1.1 (see also [2, § . Given b = ( b , . . . , b l ) ∈ Z l , set(44) φ ( z ) = z b · · · z b l l Q l − i =1 (1 − q t z i /z i +1 ) . and let ν ( z ) = ν ( z , . . . , z l ) be a Laurent polynomial where H lq,t ( ν ( z )) = H lq,t ( φ ( z )); sucha ν ( z ) exists by [19, Proposition 6.1], and represents a well-defined element of the shufflealgebra S . The Negut element D b and the transposed Negut element E a , where a = ( b l , . . . , b )is the reversed sequence of indices, are defined by D b = D b ,...,b l = ψ ( ν ( z )) ∈ E + (45) E a = E b l ,...,b = Φ( D b ) = ψ op ( ν ( z )) ∈ Φ E + . (46)We should point out that, strictly speaking, the Negut elements in the case l = 1 are de-fined to be elements D a = p [ − M X ,a ] and E a = p [ − M X a, ] of E , while in Definition 3.3.2,we used the notation D a and E a for operators on Λ. However, by Proposition 3.3.3, theseNegut elements act as the operators with the same name, so no confusion should ensue.Later we will use the following product formulas, which are immediate from Defini-tion 4.1.1. D b ,...,b l D b l +1 ,...,b n = D b ,...,b n − q t D b ,...,b l +1 ,b l +1 − ,...,b n , (47) E a n ,...,a l +1 E a l ,...,a = E a n ,...,a − q t E a n ,...,a l +1 − ,a l +1 ,...,a . (48)As noted in § k ( X , ) on Φ E + .Using the anti-isomorphism between Φ E + and the shuffle algebra we can make this moreexplicit. Lemma 4.1.2.
Let φ ( z ) = φ ( z , . . . , z n ) be a Laurent polynomial representing an element oftensor degree n in S . Then (49) (Ad f ( X , )) ψ op ( φ ( z )) = ψ op (cid:0) ( ωf )( z , . . . , z n ) · φ ( z ) (cid:1) . As a particular consequence, we have (50) (Ad f ( X , )) E a n ,...,a = ψ op ( ωf )( z , . . . , z n ) · z a · · · z a n n Q n − i =1 (1 − q t z i /z i +1 ) ! . PROOF OF THE EXTENDED DELTA CONJECTURE 13
Proof.
This follows immediately from the rule in § f acting on a product. (cid:3) Commutator identity.
We now derive an identity giving a formula for the commu-tator of elements D a and D b , and a similar identity for E a and E b . The formulas involvesummations which are most conveniently expressed using the notation(51) b X i = a f i = P bi = a f i for a ≤ b + 1 − P a − i = b +1 f i for a ≥ b + 1 . As a mnemonic device, note that both cases can be interpreted as P ∞ i = a f i − P ∞ i = b +1 f i . Proposition 4.2.1.
For any a ∈ Z and b = ( b , . . . , b l ) ∈ Z l , we have [ D a , D b ,b ,...,b l ] = − M l X i =1 b i X k = a +1 D b ,...,b i − ,k,a + b i − k,b i +1 ,...,b l (52) [ E b l ,...,b ,b , E a ] = − M l X i =1 b i X k = a +1 E b l ,...,b i +1 ,a + b i − k,k,b i − ,...,b . (53)We will need the following lemma for the proof. The notation Ω( X ) is defined in (19).Since plethystic substitution into Ω( X ) is characterized by(54) Ω[ a + a + · · · − b − b − · · · ] = Q i (1 − b i ) Q i (1 − a i ) , we have(55) Ω[ M z ] = (1 − q z )(1 − t z )(1 − z )(1 − q t z ) and Ω[ − M z ] = (1 − z )(1 − q t z )(1 − q z )(1 − t z ) . Lemma 4.2.2.
For any f ( z ) = f ( z , . . . , z m ) antisymmetric in z i and z i +1 , we have (56) H mq,t (cid:0) Ω[ M z i /z i +1 ] f ( z ) (cid:1) = 0 . Proof.
The definition of H mq,t and (55) imply that H mq,t (cid:0) Ω[ M z i /z i +1 ] f ( z ) (cid:1) = X w ∈ S m w f ( z ) Y j = k − z j /z k Y j Identity (53) for [ E b l ,...,b , E a ] follows from (52) by applying theanti-homomorphism Φ, so we only prove (52), which can be written(58) D a D b − D b D a + M l X i =1 b i X k = a +1 D b ,...,b i − ,k,a + b i − k,b i +1 ,...,b l = 0 . Using Definition 4.1.1 and the isomorphism ψ : S → E + , we can prove (58) by showingthat a rational function representing the left hand side is in the kernel of the symmetrizationoperator H l +1 q,t . For this we can work directly with the rational functions φ ( z ) in (44); there isno need to replace them explicitly with Laurent polynomials having the same symmetrization.Let φ ( z ) be the function in (44) for D b , and set(59) φ (ˆ z i ) = φ ( z , . . . , z i − , z i +1 , . . . , z l +1 ) = z b · · · z b i − i − z b i i +1 · · · z b l l +1 (1 − q t z i − /z i +1 ) Q ≤ j ≤ lj = i,i − (1 − q t z j /z j +1 ) . To prove (58), we want to show(60) H l +1 q,t (cid:18) z a φ ( ˆ z ) − φ ( ˆ z l +1 ) z al +1 + M l P i =1 b i P k = a +1 z b · · · z b i − i − z ki z a + b i − ki +1 z b i +1 i +2 · · · z b l l +1 Q lj =1 (1 − q t z j /z j +1 ) (cid:19) = 0 . Since z ai φ (ˆ z i ) − φ (ˆ z i +1 ) z ai +1 is antisymmetric in z i and z i +1 , Lemma 4.2.2 implies(61) l X i =1 H l +1 q,t (cid:18) Ω[ M z i /z i +1 ]( z ai φ (ˆ z i ) − φ (ˆ z i +1 ) z ai +1 ) (cid:19) = 0The first formula in (55) is algebraically the same asΩ[ M z ] = 1 − M (1 − z − )(1 − q t z ) . After substituting this into (61), the linearity of H l +1 q,t gives(62) H l +1 q,t (cid:18) l X i =1 (cid:16) z ai φ ( ˆ z i ) − φ ( ˆ z i +1 ) z ai +1 − M z ai φ ( ˆ z i ) − φ ( ˆ z i +1 ) z ai +1 (1 − z i +1 /z i )(1 − q t z i /z i +1 ) (cid:17)(cid:19) = 0 . The terms z ai φ ( ˆ z i ) − φ ( ˆ z i +1 ) z ai +1 telescope, reducing this to(63) H l +1 q,t (cid:18) z a φ ( ˆ z ) − φ ( ˆ z l +1 ) z al +1 − M l X i =1 z ai φ ( ˆ z i ) − φ ( ˆ z i +1 ) z ai +1 (1 − z i +1 /z i )(1 − q t z i /z i +1 ) (cid:19) = 0 . PROOF OF THE EXTENDED DELTA CONJECTURE 15 If we use the convention z = 0 and z l +2 = ∞ , collecting terms in z ai φ ( ˆ z i ) and some furtheralgebra manipulations give l X i =1 z ai φ ( ˆ z i ) − φ ( ˆ z i +1 ) z ai +1 (1 − z i +1 z i )(1 − q t z i z i +1 ) = l +1 X i =1 " − z i +1 z i )(1 − q t z i z i +1 ) − − z i z i − )(1 − q t z i − z i ) z ai φ ( ˆ z i )= l +1 X i =1 z ai φ ( ˆ z i )(1 − q t z i − z i +1 )(1 − q t z i − z i )(1 − q t z i z i +1 ) (cid:16) − z i +1 z i − − z i z i − (cid:17) = l +1 X i =1 z ai φ ( ˆ z i )(1 − q t z i − z i +1 )(1 − q t z i − z i )(1 − q t z i z i +1 ) − z ai +1 φ ( ˆ z i +1 )(1 − q t z i z i +2 )(1 − q t z i z i +1 )(1 − q t z i +1 z i +2 )1 − z i +1 z i . Expanding the definition (59) of φ ( ˆ z i ) for each i yields z ai φ ( ˆ z i )(1 − q t z i − /z i +1 )(1 − q t z i − /z i )(1 − q t z i /z i +1 ) = z b · · · z b i − i − z ai z b i i +1 · · · z b l l +1 Q lj =1 (1 − q t z j /z j +1 ) , so that l X i =1 z ai φ ( ˆ z i ) − φ ( ˆ z i +1 ) z ai +1 (1 − z i +1 /z i )(1 − q t z i /z i +1 ) = P li =1 z b · · · z b i − i − · z ai z b i i +1 − z b i i z ai +1 − z i +1 /z i · z b i +1 i +2 · · · z b l l +1 Q lj =1 (1 − q t z j /z j +1 )= − P li =1 z b · · · z b i − i − · (cid:0) b i P k = a +1 z ki z a + b i − ki +1 (cid:1) · z b i +1 i +2 · · · z b l l +1 Q lj =1 (1 − q t z j /z j +1 )Identity (60) follows by substituting this back into (63). (cid:3) Symmetry identity for D b and E a . Next we will prove an identity between certaininstances of the Negut elements D b ∈ E + and transposed Negut elements E a ∈ Φ E + . Beforestating the identity we need to describe how the indices a and b will correspond. Definition 4.3.1. A south-east lattice path γ from (0 , n ) to ( m, m, n , is admissible if it starts with a south step and ends with an east step; that is, γ has astep from (0 , n ) to (0 , n − 1) and one from ( m − , 0) to ( m, b ( γ ) = ( b , . . . , b m )by taking b i = (vertical run of γ at x = i − 1) for i = 1 , . . . , m and a ( γ ) = ( a n , . . . , a ) with a j = (horizontal run of γ at y = j − 1) for j = 1 , . . . , n . Set D γ = D b ( γ ) and E γ = E a ( γ ) .Note that if γ ∗ is the transpose of an admissible path γ with b ( γ ) = ( b , . . . , b m ) and a ( γ ) = ( a n , . . . , a ), as above, then a ( γ ∗ ) = ( b m , . . . , b ) and b ( γ ∗ ) = ( a , . . . , a n ), and E γ = Φ( D γ ∗ ). Example . Paths γ and γ ∗ below are both admissible. γ is from (0 , 8) to (4 , 0) with b ( γ ) = (2 , , , 2) and a ( γ ) = (0 , , , , , , , γ ∗ is from (0 , 4) to (8 , 0) and has a ( γ ∗ ) = (2 , , , 2) and b ( γ ∗ ) = (1 , , , , , , , γ γ ∗ Proposition 4.3.3. For every admissible path γ we have D γ = E γ .Proof. Let γ be an admissible path γ from (0 , n ) to ( m, m, n are positive integers.We first establish the case when n = 1. In this case, E γ = E m = p [ − M X m, ] and D γ = D m − . If m = 1, these are E = D = p [ − M X , ]. In general, (24) im-plies E m = p [ − M X m, ] = (Ad p ( X , )) m − p [ − M X , ] = (Ad p ( X , )) m − D , while(17) and the commutator identity (52) imply (Ad p ( X , )) D k = [ p ( X , ) , D k ] = − (1 /M )[ D , D k ] = D k +1 , and therefore (Ad p ( X , )) m − D = D m − .Using the involution Φ, we can deduce the m = 1 case from the n = 1 case:(64) D γ = D n = Φ E n = Φ D , n − = E n − , = E γ . For m, n > 1, we proceed by induction, assuming that the result holds for paths from(0 , n ′ ) to ( m ′ , 0) when m ′ ≤ m and n ′ ≤ n and ( m ′ , n ′ ) = ( m, n ).For a given m, n , there are finitely many admissible paths γ , and thus a finite dimensionalspace V of linear combinations P γ c γ D γ involving these paths. Let V ′ ⊆ V denote thesubspace consisting of linear combinations which form the left hand side of a valid instanceof the identity(65) X γ c γ D γ = X γ c γ E γ . Note that V ′ = V if and only if D γ = E γ for all the paths γ in question.We will use the induction hypothesis to construct enough instances of (65) to reduce each D γ modulo V ′ to a scalar multiple of D γ , where γ is the path with a south run from (0 , n )to (0 , 0) followed by an east run to ( m, D γ , showing that V ′ = V .Suppose now that γ = γ . Then γ contains an east step from ( m − , n ) to ( m , n )and a south step from ( m , n ) to ( m , n − 1) for some m + m = m and n + n = n .In particular, γ = ν · η for shorter admissible paths ν and η , where ν · η is defined to bethe lattice path obtained by placing ν and η end to end; thus ν · η traces a copy of ν from(0 , n + n ) to ( m , n ) and then traces a copy of η from ( m , n ) to ( m + m , PROOF OF THE EXTENDED DELTA CONJECTURE 17 Define γ ′ = ν · ′ η to be the admissible path obtained from ν · η by replacing the east-southcorner at ( m , n ) with a south-east corner at ( m − , n − γ ′ contains a south step from( m − , n ) to ( m − , n − 1) and an east step from ( m − , n − 1) to ( m , n − D ν D η = D ν · η − q t D ν · ′ η and E ν E η = E ν · η − q t E ν · ′ η . By induction, D ν = E ν and D η = E η , so (66) implies D γ − q t D γ ′ = E γ − q t E γ ′ . In otherwords, in terms of the space V ′ defined above, we have D γ ≡ q t D γ ′ (mod V ′ ). Using thisrepeatedly, we obtain D γ ≡ ( q t ) h ( γ ) D γ (mod V ′ ) for every path γ , where h ( γ ) is the areaenclosed by the path γ and the x and y axes.To complete the proof it suffices to establish one more identity of the form (65), for whichthe congruences D γ ≡ ( q t ) h ( γ ) D γ (mod V ′ ) reduce the left hand side to a non-zero scalarmultiple of D γ .We can assume by induction that D n, m − = E n − ,m − , since this case has the same n anda smaller m . Taking the commutator with p ( X , ) on both sides gives(67) − M [ D , D n, m − ] = [ p ( X , ) , D n, m − ] = (Ad p ( X , )) E n − ,m − . Using (52) on the left hand side and (50) on the right hand side gives(68) n − X k =0 D ( n − k,k, m − ) = n − X k =0 E (0 n − ,m − ε n − k . Now, for 1 ≤ k ≤ n − 1, we have D ( n − k,k, m − ) = D γ and E (0 n − ,m − ε n − k = E γ for anadmissible path with h ( γ ) = k , as displayed below. n − kkm − This shows that (68) is an instance of (65). The previous congruences reduce the left handside of (68) to (1 + q t + · · · + ( q t ) n − ) D γ . Since the coefficient is non-zero, we have nowestablished a set of instances of (65) whose left hand sides span V . (cid:3) Corollary 4.3.4. For any indices a , . . . , a l , we have (69) E a l ,...,a ,a · E a l ,...,a , · . Proof. To rephrase, we are to show that E a l ,...,a ,a · a . The symmetry f ( X m,n ) f ( X m + rn,n ) of E + sends E a l ,...,a to E a l + r,...,a + r . By [2, Proposition 3.4.1], theaction of E on Λ satisfies ∇ r f ( X m,n ) ∇ − r = f ( X m + rn,n ), and since ∇ (1) = 1, this gives ∇ r E a l ,...,a ,a · E a l + r,...,a + r,a + r · 1. Hence, we can reduce to the case that a i > i .By [2, Lemma 3.6.3], we have that D b ,...,b n , ,..., · b i ≥ i and b > 0, this and Proposition 4.3.3 imply that E a l ,...,a · a , provided that a i ≥ i and a > 0. However, wealready saw that this suffices. (cid:3) Shuffling the symmetric function side of the Extended Delta Conjecture. Wecan now give the promised reformulation of (7). Theorem 4.4.1. For ≤ l < m ≤ N , we have (70) (cid:0) ω ( h l [ B ] e m − l − [ B − e N − l ) (cid:1) ( x , . . . , x m ) = H mq,t ( φ ( x )) pol , where (71) φ ( x ) = x · · · x m Q i (1 − q t x i /x i +1 ) h N − m ( x , . . . , x m ) e l ( x , . . . , x m ) , and e l ( x , . . . , x m ) = e l ( x − , . . . , x − m ) by our convention on the use of the overbar.Proof. For any symmetric function f and any ζ ∈ E , set g ( X ) = ( ωf )[ X + 1 /M ]; then (31)gives an identity in Λ(72) f [ B ] ζ · g ( X , ) ζ · X ((Ad g (1) ( X , )) ζ ) g (2) ( X , ) · , where g [ X + Y ] = P g (1) ( X ) g (2) ( Y ) in Sweedler notation and we used the general formula g ζ = P ((Ad g (1) ) ζ ) g (2) . Since g [ X + Y ] = ( ωf )[ X + Y + 1 /M ], and h [ B ] · h [0] · h ( X ), the right hand side of (72) is equal to(73) X ((Ad ( ωf ) (1) ( X , )) ζ ) ( ωf ) (2) [ X , + 1 /M ] · X ((Ad ( ωf ) (1) ( X , )) ζ ) ( ωf ) (2) [0] · ωf )( X , )) ζ ) · . Let n = N − l . Taking ζ = E a n ,...,a and using (50), this gives(74) f [ B ] E a n ,...,a · f ( z n , . . . , z ) | z r n n · · · z r E a n + r n ,...,a + r ,a + r · . By Corollary 4.3.4, the right hand side is a function of f ( z n , . . . , z , z r does not depend on the exponent r . Expressing f ( z n , . . . , z , 1) as f [ z n + · · · + z + 1] and then substituting f [ X − 1] for f ( X ) yields(75) f [ B − E a n ,...,a · f [ z n + · · · + z ] | z r n n · · · z r E a n + r n ,...,a + r ,a · . PROOF OF THE EXTENDED DELTA CONJECTURE 19 By [19, Proposition 6.7] (see also [2, Proposition 3.6.1]), E n = e n [ − M X , ]. Using (75),we therefore obtain(76) e k − [ B − e n = e k − [ z n + · · · + z ] (cid:12)(cid:12)(cid:12) z r n n · · · z r E a n + r n ,...,a + r ,a · X | I | = k − E ε I , · X | I | = k − E ε I , · , where the sum is over subsets I ⊂ [ n − 1] and ε I = P i ∈ I ε i . The terms in the last sum arejust E a ( ν ) · ν from (0 , n ) to ( k, 0) with single east steps on any k − y = j for j ∈ [ n − y = 0. Denote the set of these admissible pathsby P k,n . For instance, with n = 8 and k = 4, the path γ in Example 4.3.2 corresponds to E γ = E , , , , , , , .By (74), applying h l [ B ] to (76) gives(77) h l [ B ] e k − [ B − e n = X ν ∈P k,n X r ∈ N n | r | = l E r + a ( ν ) · . This last expression is the sum of E γ · γ from (0 , n ) to ( k + l, k − j ∈ [ n − 1] for which γ has at least one east step onthe line y = j . We can consider these indices as distinguishing k − γ . However, we can also distinguish these corners by their x coordinates, that is, by a set of k − i ∈ [ k + l − 1] for which γ has at least one south step on the line x = i . Setting m = k + l and using Proposition 4.3.3, this yields the identity(78) h l [ B ] e m − l − [ B − e n = X s ∈ N m : | s | = n − kI ⊂ [2 ,m ] , | I | = l D s +(1 m ) − ε I · . Now, since(79) X s ∈ N m : | s | = n − kI ⊂ [2 ,m ] , | I | = l x s +(1 m ) − ε I = x x · · · x m h n − k ( x , . . . , x m ) e l ( x , . . . , x m ) , the definition of D b and Proposition 3.4.2 imply that(80) ω (cid:18) X s ∈ N m : | s | = n − kI ⊂ [2 ,m ] , | I | = l D s +(1 m ) − ε I · (cid:19) ( x , . . . , x m ) = H mq,t ( φ ( x )) pol with φ ( x ) given by (71). (cid:3) Remark . For any b ∈ Z m , [2, Corollary 3.7.2] gives that the Schur expansion of ω ( D b · s λ ( X ) with ℓ ( λ ) ≤ m . Hence, although Theorem 4.4.1 is a statement in m variables, it determines ω ( h l [ B ] e m − l − [ B − e N − l ) by (78). Reformulation of the combinatorial side z N − m ; the natural resultinvolves a q -weighted tableau generating function N β/α rather than partially labelled paths.For now, we work only with the tableau description of N β/α , but in § N β/α is a truncation of LLT series introduced by Grojnowski and Haiman in [12].The q -weight in our reformulation involves two auxiliary statistics: for η, τ ∈ N m , define(81) d ( η, τ ) = X ≤ j For ≤ l < m ≤ N , we have (83) h z N − m i X λ ∈ D N P ∈ L N,l ( λ ) t | δ/λ | Y
0) + ε J , and N β/α is given by Definition 5.2.1. Definition of N β/α . For α, β ∈ Z l such that α j ≤ β j for all j , define β/α to be thetuple of single row skew shapes ( β j ) / ( α j ) such that the x coordinates of the right edges ofboxes a in the j -th row are the integers α j + 1 , . . . , β j . The boxes just outside the j -th row,adjacent to the left and right ends of the row, then have x coordinates α j and β j + 1. Weconsider these two boxes to be adjacent to the ends of an empty row, with α j = β j , as well.Given a tuple of skew row shapes β/α , three boxes ( u, v, w ) form a w -triple when box v is in row r of β/α , boxes u and w are in or adjacent to a row j with j > r , and the x -coordinates i u , i v , i w of these boxes satisfy i u = i v and i w = i v + 1. These triples are aspecial case of σ -triples defined for any σ ∈ S l in [2]. We denote the number of w -triples in β/α by h w ( β/α ). The reader can verify that(84) h w ( β/α ) = X r For β = (12211123233), α = (11000121220), there are h w ( β/α ) = 29 w -triples in β/α . The row strict tableau S of shape β/α has h w ( S ) = 15 increasing w -triples, x wt + ( S ) = x x x x x x , and x wt( S ) = x x x x x x x .A w -triple ( u, v, w ) is an increasing w -triple in S if S ( u ) < S ( v ) < S ( w ), with theconvention that S ( u ) = −∞ if u is adjacent to the left end of a row of β/α , and S ( w ) = ∞ if w is adjacent to the right end of a row. Let h w ( S ) be the number of increasing w -triplesin S .For S ∈ RST( β/α, N ), define(85) x wt + ( S ) = Y u ∈ β/α, S ( u ) =0 x S ( u ) and x wt( S ) = Y u ∈ β/α x S ( u ) . Definition 5.2.1. For α, β ∈ N m , define(86) N β/α = N β/α ( X ; q ) = X S ∈ RST( β/α, Z + ) q h w ( S ) x wt( S ) . Note that if α j > β j for any j then N β/α = 0 by our convention that RST( β/α, A ) = ∅ . Remark . It is shown in [2, Proposition 4.5.2] and its proof that, for α, β ∈ N m , N β/α isa symmetric function whose Schur expansion involves only s λ where ℓ ( λ ) ≤ m .5.3. Transforming the combinatorial side. To prove (83), we first associate each Dyckpath with a tuple of row shapes recording vertical runs. Definition 5.3.1. The LLT data associated to a path λ ∈ D N is β = (1 , c ( λ ) + 1 , . . . , c N ( λ ) + 1) and α = ( c ( λ ) , . . . , c N ( λ ) , , where c i ( λ ) counts lattice squares between λ and the line segment connecting (0 , N ) to ( N, i , numbered from right to left, as in Lemma 2.2.4.Figure 2 shows the LLT data β, α associated to the path λ in Figure 1. Note that β i (resp. α i ) is the furthest (resp. closest) distance from the diagonal to the path λ on the line x = N − i , so that β i − α i is the number of south steps of λ on that line.This association allows us to relate q -weighted sums over partial labellings to the N β/α . Lemma 5.3.2. For λ ∈ D N and its associated LLT data α, β , we have (87) X P ∈ L N,l ( λ ) q dinv( P ) x wt + ( P ) = X I ⊂ [ N − | I | = l q h I ( α ) N β/ ( α + ε I ) ( X ; q ) . Proof. There is a natural weight-preserving bijection mapping P ∈ L N ( λ ) to S ∈ RST( β/α, N ), where the labels of column x = i of P , read north to south, are placedinto row N − i of β/α , west to east. See Figures 1 and 2. Moreover, dinv( P ) = h w ( S ).To see this, let ˆ P be the same labelling as P but with the ordering on letters taken to be0 > > · · · . It is proven in [2, Proposition 6.1.1] that dinv ( ˆ P ) = h w ( S ), where dinv ( ˆ P )was introduced in [13] and matches dinv( P ) as discussed in Remark 2.2.3. The bijectionrestricts to a bijection from L N,l ( λ ) to the subset of tableaux S ∈ RST( β/α, N ) with exactly l N . This gives(88) X P ∈ L N,l ( λ ) q dinv( P ) x wt + ( P ) = X I ⊂ [ N − | I | = l X S ∈ RST( β/α, N )0 in rows i ∈ I q h w ( S ) x wt + ( S ) . The claim then follows from Definition 5.2.1 and the following Lemma. (cid:3) Lemma 5.3.3. For α, β ∈ N N and S ∈ RST( β/α, N ) , let I ⊂ [ N ] be the rows of S containinga zero and let T be the tableau in RST( β/ ( α + ε I ) , Z + ) obtained by deleting all zeros from S . Then (89) h w ( T ) = h w ( S ) − h I ( α ) , where h I ( α ) is defined in (82) .Proof. Consider an increasing w -triple ( u, v, w ) of S ; the entries satisfy S ( u ) < S ( v ) < S ( w ), v lies in some row r , and both u and w lie in a row j > r . When r I , either j I so that( u, v, w ) is an increasing w -triple of T with the same entries as S , or j ∈ I and S ( u ) = 0changes to T ( u ) = −∞ where still ( u, v, w ) is an increasing w -triple of T . However, if r ∈ I , S ( v ) = 0 changes to T ( v ) = −∞ and thus ( u, v, w ) is not an increasing w -triple of T . Notethe increasing condition implies that this happens only when j I and α r = α j − S ( u ) < < S ( w ). Thus (89) follows. (cid:3) Definition 5.3.4. Given a = ( a , . . . , a m − ) ∈ N m − and τ = ( τ , . . . , τ m ) ∈ N m , we definetwo sequences β a τ and α a τ of length | τ | + m as follows.The sequence β a τ is the concatenation of sequences (1 , , . . . , τ i + 1) and ( a i − + 1 , a i − +2 , . . . , a i − + τ i + 1) for i = 2 , . . . , m . The sequence α a τ is the same as β a τ except in the PROOF OF THE EXTENDED DELTA CONJECTURE 23 positions corresponding to the ends of the concatenated subsequences. In these positions,we change the entries τ + 1 , a + τ + 1 , . . . , a m − + τ m + 1 in β a τ to a , a , . . . , a m − , α a τ is the same as the sequence obtained by subtracting 1 from all entries of β a τ and shifting one place to the left, deleting the first entry and adding a zero at the end. Example . For a = (130012) and τ = (2311022),(90) (0 , a ) + (1 m ) + τ = ( 3 5 5 2 1 4 5) β a τ = (1 2 3 2 3 4 5 4 5 1 2 1 2 3 4 3 4 5) α a τ = (1 2 1 2 3 4 3 4 0 1 0 1 2 3 2 3 4 0)( a , 0) = ( 1 3 0 0 1 2 0) . The wider spaces show the division into blocks of size τ i + 1. The last entry of α a τ in eachblock is a i , and the next block in α a τ and β a τ starts with a i + 1. Lemma 5.3.6. For ≤ l < m ≤ N , (91) h z N − m i X λ ∈ D N P ∈ L N,l ( λ ) t | δ/λ | Y
Use Lemma 5.3.2 to rewrite the left hand side of (91) as(92) h z N − m i X λ ∈ D N t | δ/λ | Y
0) are the LLT data for λ .Note that a tuple c = ( c , c , . . . , c N ) ∈ N N is the sequence of column heights c i ( λ ) of a path λ ∈ D N if and only if c s ≤ c s − + 1 for all s > c = 0; in this case, | δ/λ | = | c | . Replace D N in (92) by these tuples, and expand the product to obtain(93) h z N − m i X A ⊂ [ N ] \{ } X c i = c i − +1 ∀ i ∈ Ac i ≤ c i − +1 ∀ i A t | c |− P i ∈ A c i z | A | X I ⊂ [ N − | I | = l q h I ( α ) N β/ ( α + ε I ) = X { }⊂ J ⊂ [ N ] | J | = m X c j = c j − +1 ∀ j / ∈ J t P j ∈ J c j X I ⊂ [ N − | I | = l q h I ( α ) N β/ ( α + ε I ) , where the equality comes from re-indexing with J = [ N ] \ A and noting that we can dropthe condition c j ≤ c j − + 1 ∀ j ∈ J because N β/ ( α + ε I ) = 0 if any ( α + ε I ) j ≥ α j > β j .If we replace the sum over J by a sum over { τ ∈ N m : | τ | = N − m } using J = { , τ +2 , τ + τ + 3 , . . . , τ + · · · + τ m − + m } , then, for fixed J (or fixed τ ), the sum over c can bereplaced by a sum over(94) c = (0 , , , . . . , τ , a , a + 1 , . . . , a + τ , a , . . . , a m − + τ m ) ((0 , a ) + (1 m ) + τ ) / ( a , rr − ... j β a τ /α a τ r ↑ y ( r − ↑ ... j ↑ τ r Figure 3. Comparing the tuples of rows β a τ /α a τ and ((0 , a )+(1 m )+ τ ) / ( a , a ∈ N m − and τ ∈ N m . Here a j = 2, a r − = 0 , a r = 3, and τ r = 5.for a ranging over N m − . Note that P j ∈ J c j = | a | . With this encoding of c , we have β/α = β a τ /α a τ in the notation of Definition 5.3.4, and (93) becomes the right hand side of(91). (cid:3) We make a final adjustment to the right hand side of (91). This sum runs over tuples β a τ / ( α a τ + ε I ) with | τ | necessarily empty rows which can be removed at the cost of a q factor.We introduce some notation depending on a given a ∈ N m − , τ = ( τ , . . . , τ m ) ∈ N m , and theassociated β a τ /α a τ from Definition 5.3.4. First we set j ↑ = j + P x ≤ j τ x for j ∈ [ m ], so theentry of β a τ in position j ↑ is a j − + τ j + 1, or τ + 1 if j = 1, and the entry of α a τ in the sameposition is a j , or 0 if j = m . For a subset J ⊆ [ m ], we set J ↑ = { j ↑ : j ∈ J } . In positions i [ m ] ↑ , the sequences β a τ and α a τ agree, so row i is empty in β a τ /α a τ . The tuple of rowshapes obtained by deleting these empty rows from β a τ /α a τ is ((0 , a ) + (1 m ) + τ ) / ( a , j ∈ [ m ] corresponds to row j ↑ of β a τ /α a τ ; note that rows ( j − ↑ and j ↑ areseparated by τ j empty rows. See Figure 3. Lemma 5.3.7. For J ⊂ [ m ] , a ∈ N m − and τ ∈ N m , let I = J ↑ . Then (95) N β a τ / ( α a τ + ε I ) = q d ((0 , a ) ,τ ) − h ′ J ( a ,τ ) N ((0 , a )+(1 m )+ τ ) / (( a , ε J )) , where h ′ J ( a , τ ) = |{ ( j < r ) : j ∈ J, r ∈ [ m ] , a j ∈ [ a r − , a r − + τ r − }| with a = 0 , and d ((0 , a ) , τ ) is defined by (81) .Proof. Set a = 0. We can assume a j + ( ε J ) j ≤ a j − + τ j + 1 for all j ∈ [ m ] since otherwiseboth sides of (95) vanish by Definition 5.2.1. Hence, each side is a q -generating function forrow strict tableaux on tuples of single row skew shapes; rows of β a τ / ( α a τ + ε I ) on the lefthand side differ from the right hand side only by the removal of empty rows r [ m ] ↑ . Thus,the two sides agree up to a factor q d , where d counts w -triples of β a τ / ( α a τ + ε I ) involvingone of these empty rows. PROOF OF THE EXTENDED DELTA CONJECTURE 25 To evaluate d , consider such an empty row ( b ) / ( b ), coming from b ∈ { a r − +1 , . . . , a r − + τ r } for some r ∈ [ m ]. The adjacent boxes on the left and right of this empty row form a w -triple, increasing in every tableau, with one box in each non-empty lower row j ↑ , of the form( a j − + τ j + 1) / ( a j + ( ε J ) j ), such that b ∈ [ a j + ( ε J ) j + 1 , a j − + τ j + 1]. Hence, d = X ≤ j Consider a summand t | a | q h I ( α a τ ) N β a τ / ( α a τ + ε I ) on the right hand sideof identity (91) for I ⊂ [ N − a ∈ N m − , τ ∈ N m ,. It vanishes unless I = J ↑ for some J ⊂ [ m − 1] since N β/ ( α + ε I ) = 0 when ( α + ε I ) i > β i for some index i . For I = J ↑ , we canreplace the summand with t | a | q d ((0 , a ) ,τ )+ h I ( α a τ ) − h ′ J ( a ,τ ) N ((0 , a )+(1 m )+ τ ) / (( a , ε J ) , by Lemma 5.3.7.It now suffices to prove that for α = α a τ ,(97) h I ( α ) = h ′ J ( a , τ ) + h J ( a ) . We recall N = m ↑ and note that [ N ] \ I = N \ [ m ] ↑ ⊔ [ m ] ↑ \ I = N \ [ m ] ↑ ⊔ ([ m ] \ J ) ↑ . Hence, h I ( α ) = |{ ( x < y ) : x ∈ I, y ∈ [ N ] \ I, α y = α x + 1 }| = | S | + | S | for S = { ( x < y ) : x ∈ J ↑ , y ∈ [ N ] \ [ m ] ↑ , α y = α x + 1 } S = { ( x < y ) : x ∈ J ↑ , y ∈ ([ m ] \ J ) ↑ , α y = α x + 1 } . Since α m ↑ = 0 implies ( x, m ↑ ) S for any x < [ m ] ↑ , we use that a u = α u ↑ for every u ∈ [ m − 1] to see that(98) h J ( a ) = (cid:12)(cid:12) S (cid:12)(cid:12) = (cid:12)(cid:12) { ( j < r ) : j ∈ J, r ∈ [ m − \ J, a r = a j + 1 } (cid:12)(cid:12) . Furthermore, { ( j < r ) : j ∈ J, r ∈ [ m ] , a r − +1 ≤ a j +1 ≤ a r − + τ r } and S are equinumerous,as we can see by letting a pair ( j < r ) in the first set correspond to the pair ( j ↑ < y ) in S ,where y is the unique row index in the range ( r − ↑ < y < r ↑ such that α y = α j ↑ +1 = a j +1,as illustrated in Figure 3. (cid:3) Stable unstraightened extended delta theorem Extended Delta Conjecture is equivalent to (99) H mq Q i +1 0) + ε J , and (cid:0) ωN β/α (cid:1) ( x , . . . , x m ; q ) is ωN β/α ( X ; q )evaluated in m variables.Although this is an identity in only m variables, it does amount to the Extended DeltaConjecture by Remarks 4.4.2 and 5.2.2: both ω ( h l [ B ] e m − l − [ B − e N − l ) and ωN β/α ( X ; q )for the α, β arising in (99) are linear combinations of Schur functions s λ with ℓ ( λ ) ≤ m .We will show in Proposition 6.2.2 that the functions ωN β/α on the right hand side of (99)are the polynomial parts of ‘LLT series’ introduced in [12], making each side of (99) thepolynomial part of an infinite series of GL m characters. We then prove (99) as a consequenceof a stronger identity between these infinite series.Hereafter, we use x to abbreviate the alphabet x , . . . , x m .6.2. LLT series. We will work with the (twisted) non-symmetric Hall-Littlewood polyno-mials as in [2]. For a GL m weight λ ∈ Z m and σ ∈ S m , the twisted non-symmetric Hall-Littlewood polynomial E σλ ( x ; q ) is an element of Z [ q ± ][ x ± , . . . , x ± m ] defined using an actionof the Hecke algebra on this ring; we refer the reader to [2, § F σλ ( x ; q ) = E σw − λ ( x ; q ) , recalling that f ( x , . . . , x m ; q ) = f ( x − , . . . , x − m ; q − ).For any weights α, β ∈ Z m and a permutation σ ∈ S m , the LLT series L σβ/α ( x ; q ) = L σβ/α ( x , . . . , x m ; q ) is defined in [2, § h χ λ iL σ − β/α ( x ; q − ) = h E σβ i χ λ · E σα . Alternatively, [2, Proposition 4.4.2] gives the following expression in terms of the Hall-Littlewood symmetrization operator in (38):(102) L σβ/α ( x ; q ) = H lq ( w ( F σ − β ( x ; q ) E σ − α ( x ; q ))) , PROOF OF THE EXTENDED DELTA CONJECTURE 27 where w denotes the permutation of maximum length here and after. We will only needthe LLT series for σ = w and σ = id , although most of what follows can be generalized toany σ .In addition to the above formulas, we have the following combinatorial expressions for thepolynomial truncations of LLT series as tableau generating functions with q weights thatcount triples. As usual, a semistandard tableau on a tuple of skew row shapes ν = β/α is amap T : ν → [ m ] which is weakly increasing on rows. Let SSYT( ν ) denote the set of these,and define x wt( T ) = Q b ∈ ν x T ( b ) . Proposition 6.2.1 ([2, Remark 4.5.5 and Corollary 4.5.7]) . If α i ≤ β i for all i , then (103) L w β/α ( x ; q ) pol = X T ∈ SSYT( β/α ) q h ′ w ( T ) x wt( T ) , where h ′ w ( T ) is the number of w -triples ( u, v, w ) of β/α such that T ( u ) ≤ T ( v ) ≤ T ( w ) . Proposition 6.2.2 ([2, Proposition 4.5.2]) . For any α, β ∈ Z m , (104) L w β/α ( x ; q ) pol = (cid:0) ωN β/α (cid:1) ( x ; q ) . Extended Delta Theorem. We now give several lemmas on non-symmetric Hall-Littlewood polynomials, then conclude by using the Cauchy formula for these polynomialsto prove Theorem 6.3.5, below, yielding the stronger series identity that implies (99). Lemma 6.3.1. For a ∈ N m − and w ∈ S m and ˜ w ∈ S m − the permutations of maximumlength, we have E w ( a , ( x , . . . , x m ; q ) = E ˜ w a ( x , . . . , x m − ; q )(105) F w (0 , a ) ( x , . . . , x m ; q ) = F ˜ w a ( x , . . . , x m ; q ) . (106) Proof. The factorizations E w ( a , ( x , . . . , x m ; q ) = E ˜ w a ( x , . . . , x m − ; q ) E id (0) ( x m ; q ) and E id (0 , − a ) ( x , . . . , x m ; q ) = E id (0) ( x ; q ) E id − a ( x , . . . , x m ; q ) are given by [2, Lemma 4.3.4]. Theclaim then follows from the definition F σ a = E w σ − a and noting that E id (0) ( x m ; q ) = 1 = F id ( x ; q ). (cid:3) For any µ ∈ R l , define Inv( µ ) = { ( i < j ) | µ i > µ j } . When σ ∈ S l and ǫ > µ + ǫσ ) is the set of pairs i < j such that either µ i > µ j , or µ i = µ j and σ i > σ j . Lemma 6.3.2. For l ≤ m , a ∈ Z m , and σ ∈ { id, w } ⊂ S m , we have (107) e l ( x ) E σ − a ( x ; q ) = X I ⊂ [ m ]: | I | = l q ι I ( a ,σ ) E σ − a + ε I ( x ; q ) , where ι I ( a , σ ) = | Inv( a + ε I + ǫσ ) \ Inv( a + ǫσ ) | = ( |{ ( i < j ) | a i = a j , i ∈ I, j I }| for σ = id |{ ( i < j ) | a j = a i + 1 , i ∈ I, j / ∈ I }| for σ = w . Proof. In [2, Lemma 4.5.1], it is shown that for all σ ∈ S m ,(108) e l ( x ) E σ − a ( x ; q ) = X I ⊂ [ m ]: | I | = l q − ι I ( a ,σ ) E σ − a + ε I ( x ; q ) . Inverting all variables and specializing σ = w or σ = id yields the result here. (cid:3) A version of the next lemma holds in arbitrary Lie type, but for simplicity we only deriveit for symmetric polynomials. Lemma 6.3.3. For any a , α ∈ N m and any symmetric polynomial f ( x ) ∈ Z [ q ± ][ x , . . . , x m ] S m , (109) h E w w a ( x ; q − ) i f ( x ) E w w α ( x ; q − ) = h E idα ( x ; q ) i f ( x ) E id a ( x ; q ) . Proof. We can reduce to the case f = e λ by linearity. Taking f = e λ · · · e λ k on the left and e λ k · · · e λ on the right, we can further reduce to the case f ( x ) = e l ( x ),(110) h E w w a ( x ; q − ) i e l ( x ) E w w α ( x ; q − ) = h E idα ( x ; q ) i e l ( x ) E id a ( x ; q ) . Given α, a ∈ N m , we prove (110) as follows. On the one hand, inverting q in (108) gives(111) e l ( x ) E w w α ( x ; q − ) = X J ⊂ [ m ]: | J | = l q ι J ( w α,w ) E w w α + ε J ( x ; q − ) . Thus the left hand side of (110) is 0 unless w a − w α = ε J for some J ⊂ [ m ] with | J | = l ,in which case it is q to the power ι J ( w α, w ) = | Inv( w α + ε J + ǫ w ) \ Inv( w α + ǫ w ) | = | Inv( α + w ε J + ǫ id ) c \ Inv( α + ǫ id ) c | .On the other hand, observe that e m − l ( x , . . . , x m ) = x · · · x m e l ( x , . . . , x m ) and x · · · x m E id a + ε I ( x ; q ) = E id a − (1 m )+ ε I ( x ; q ). Replacing l = m − l in (107) thus gives(112) e l ( x ) E id a ( x ; q ) = X I ⊂ [ m ]: | I | = m − l q ι I ( a ,id ) E id a − (1 m )+ ε I ( x ; q ) . The statistic ι I ( a , id ) = | Inv( a − (1 m ) + ε I + ǫ id ) \ Inv( a + ǫ id ) | = | Inv( a + ǫ id ) c \ Inv( a − (1 m ) + ε I + ǫ id ) c | agrees with ι J ( w α, w ) when a = α + w ε J and J = [ m ] \ I . The desired(110) then follows. (cid:3) Lemma 6.3.4. For w the maximal permutation in S m and η ∈ N m , we have (113) h l ( x ) F w η ( x ; q ) = X τ ∈ N m | τ | = l q d ( η,τ ) F w η + τ ( x ; q ) , recalling from (81) that d ( η, τ ) = P j By specializing all but one variable in (103) to zero, Proposition 6.2.1 implies that thecoefficient of h l in L w w ( a /α ) ( x ; q ) pol is q h ′ w ( T ) for T the semistandard tableau of shape w ( a /α )filled with a single letter; h ′ w ( T ) is the number of w -triples of w ( a /α ) as defined in (84).When α = − η − τ and a = − η , this number is d ( η, τ ) as claimed. (cid:3) Theorem 6.3.5. For ≤ l < m ≤ N and w ∈ S m the maximum length permutation, wehave Q i +1 Our starting point is the Cauchy formula [2, Theorem 5.1.1] for the twisted non-symmetric Hall-Littlewood polynomials associated to any ˜ σ ∈ S m − :(115) Q i It suffices to prove the reformulation in (99); thisfollows by applying H mq and (102) to the identity of Theorem 6.3.5, taking the polynomialpart, and using Proposition 6.2.2. 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