A Shuffle Theorem for Paths Under Any Line
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 SHUFFLE THEOREM FOR PATHS UNDER ANY LINE
J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER
Abstract.
We generalize the shuffle theorem and its ( km, kn ) version, as conjecturedby Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit,respectively. In our version the ( km, kn ) Dyck paths on the combinatorial side are replacedby lattice paths lying under a line segment whose x and y intercepts need not be integers, andthe algebraic side is given either by a Schiffmann algebra operator formula or an equivalentexplicit raising operator formula.We derive our combinatorial identity as the polynomial truncation of an identity of infiniteseries of GL l characters, expressed in terms of infinite series versions of LLT polynomials.The series identity in question follows from a Cauchy identity for non-symmetric Hall-Littlewood polynomials. Introduction shuffle theorem , conjectured by Haglund et al. [15] and proven by Carlsson andMellit [6], is a combinatorial identity that expresses the symmetric polynomial ∇ e k as a sumover LLT polynomials indexed by Dyck paths—that is, lattice paths from (0 , k ) to ( k,
0) thatlie weakly below the line segment connecting these two points. Here e k is the k -th elementarysymmetric function, and ∇ is an operator on symmetric functions with coefficients in Q ( q, t )that arises in the theory of Macdonald polynomials [3]. The polynomial ∇ e k is significantbecause it gives the doubly graded character of the ring of diagonal coinvariants for thesymmetric group S k [18, Proposition 3.5].More generally, Haglund et al. conjectured an identity expressing ∇ m e k as a sum overLLT polynomials indexed by lattice paths below the line segment from (0 , k ) to ( km, e k [ − M X m,n ] · , kn ) to ( km, km, kn ) with m, n coprime. Here we have written e k [ − M X m,n ] for the operatoron symmetric functions given by a certain element of the elliptic Hall algebra E of Burbanand Schiffmann [5], such that for n = 1 the symmetric polynomial e k [ − M X m,n ] · ∇ m e k . This notation is explained in § 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.). arbitrary points (0 , s ) and ( r,
0) on the positive real axes, and reducing to the shuffle theoremof Bergeron et al. and Mellit when ( r, s ) = ( km, kn ) are integers.Our generalized shuffle theorem (Theorem 5.5.1) is an identity(1) D b · X λ t a ( λ ) q dinv p ( λ ) ω ( G ν ( λ ) ( X ; q − )) , whose ingredients we now summarize briefly, deferring full details to later parts of the paper.The sum on the right hand side of (1) is over lattice paths λ from (0 , ⌊ s ⌋ ) to ( ⌊ r ⌋ ,
0) thatlie below the line segment from (0 , s ) to ( r, a ( λ ) is the number of latticesquares enclosed between the path λ and the highest such path δ . We set p = s/r anddefine dinv p ( λ ) to be the number of ‘ p -balanced’ hooks in the (French style) Young diagramenclosed by λ and the x and y axes. A hook is p -balanced if a line of slope − p passes throughthe segments at the top of its leg and the end of its arm (Definition 5.4.1 and Figure 3).In the remaining factor ω ( G ν ( λ ) ( X ; q − )), the function G ν ( X ; q ) is an ‘attacking inversions’LLT polynomial (Definition 4.1.1), and ω ( h µ ) = e µ is the standard involution on symmetricfunctions. The index ν ( λ ) is a tuple of one-row skew Young diagrams of the same lengthsas runs of contiguous south steps in λ , arranged so that the reading order on boxes of ν ( λ )corresponds to the ordering on south steps in λ given by decreasing values of y + px at theupper endpoint of each step.The operator D b = D b ,...,b l on the left hand side of (1) is one of a family of special elementsdefined by Negut [28] in the Schiffmann algebra E . Letting δ again denote the highest pathunder the given line segment, the index b is defined by taking b i to be the number of southsteps in δ on the line x = i − s = kn and r slightly largerthan km , so p = n/m − ǫ for a small ǫ >
0. The segment from (0 , s ) to ( r,
0) has the samelattice paths below it as the segment from (0 , kn ) to ( km, p ( λ ) reduces to theversion of dinv( λ ) in the original conjectures. The element D b associated to the highest path δ below the segment from (0 , kn ) to ( km,
0) is equal to e k [ − M X m,n ]. Hence, (1) reduces tothe ( km, kn ) shuffle theorem.1.2.
Preview of the proof.
We prove our generalized shuffle theorem by a remarkablysimple method, which we now outline to help orient the reader in following the details.In § ω and evaluating in l = ⌊ r ⌋ +1variables x , . . . , x l , becomes the polynomial part(2) ω ( D b · x , . . . , x l ) = H b ( x ) pol of an explicit infinite series of GL l characters(3) H b ( x ) = X w ∈ S l w x b · · · x b l l Q i +1 When ν is a tuple of one-row skew shapes ( β i ) / ( α i ), the LLT polynomial G ν ( x ; q − ) is equal,up to a factor of the form q d , to the polynomial part of an infinite GL l character series(4) q d G ν ( x ; q − ) = L β/α ( x ; q ) pol defined by Grojnowski and the second author [13]. In Theorem 5.3.1 we establish an identityof infinite series(5) H b ( x ) = X a ,...,a l − ≥ t | a | L σ (( b l ,...,b )+(0; a )) / ( a ;0) ( x ; q ) , where L σβ/α ( x ; q ) is a ‘twisted’ variant of L β/α ( x ; q ) (see § ω omitted.In fact, (4) holds when α i ≤ β i for all i , and otherwise L β/α ( x ; q ) pol = 0. When we takethe polynomial part in (5), this leaves a non-vanishing term t | a | q d G ν ( λ ) ( x ; q − ) for each path λ under the given line segment, with a giving the number of lattice squares in each columnbetween λ and the highest path δ , so t | a | = t a ( λ ) . The factor q d from (4) turns out to beprecisely q dinv p ( λ ) , yielding (1).Finally, the infinite series identity (5) from Theorem 5.3.1 is essentially a corollary toa Cauchy formula for non-symmetric Hall-Littlewood polynomials, Theorem 5.1.1. ThisCauchy formula is quite general and can be applied in other situations, some of which wewill take up elsewhere.1.3. Further remarks. (i) The conjectures in [4, 15] and proofs in [6, 27] use a versionof dinv( λ ) that coincides with dinv p ( λ ) for p = n/m − ǫ . Alternatively, one can tilt thesegment from (0 , kn ) to ( km, 0) in the other direction, taking r = km and s slightly largerthan kn , to get a version of the original conjectures with a variant of dinv( λ ) that coincideswith dinv p ( λ ) for p = n/m + ǫ . Our theorem implies this version as well.(ii) Haglund, Zabrocki and the third author [16] formulated a ‘compositional’ generaliza-tion of the original shuffle conjecture, in which the sum over Dyck paths is decomposed intopartial sums over paths touching the diagonal at specified points. Bergeron et al. also gavea compositional form of their ( km, kn ) shuffle conjecture in [4]. The proofs by Carlsson andMellit [6] and Mellit [27] include the compositional forms of the conjectures, and indeed thisseems to be essential to their methods.Our results here do not cover the compositional shuffle conjectures. For the generalizationto paths under any line, it is not yet clear whether something like a compositional extensionis possible or what form it might take.(iii) By [15, Proposition 5.3.1], the LLT polynomials G ν ( λ ) ( x ; q ) in (1) are q Schur positive,meaning that their coefficients in the basis of Schur functions belong to N [ q ]. The right handside of (1) is therefore q, t Schur positive. In the cases corresponding to the ( km, kn ) shuffletheorem for k = 1, this can also be seen from the representation theoretic interpretation ofthe right hand side given by Hikita [19]. J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER Identity (1) therefore implies that the expression D b · q, t Schurpositive. In the cases where the left hand side coincides with ∇ m e k , this can be explainedusing the representation theoretic interpretations of ∇ e k in [18] and ∇ m e k in [15, Proposition6.1.1]. In § D b · q, t Schur positive.(iv) The algebraic left hand side of (1) is manifestly symmetric in q and t . Hence, thecombinatorial right hand side is also symmetric in q and t . No direct combinatorial proof ofthis symmetry is currently known.2. Symmetric functions and GL l characters This section serves to fix notation and terminology for standard notions concerning sym-metric functions and characters of the general linear groups GL l .2.1. Symmetric functions. Integer partitions are written λ = ( λ ≥ · · · ≥ λ l ), possiblywith trailing zeroes. We set | λ | = λ + · · · + λ l and let ℓ ( λ ) be the number of non-zero parts.The partitions of a given integer n are partially ordered by(6) λ ≤ µ if λ + · · · + λ k ≤ µ + · · · + µ k for all k, where the sums include trailing zeroes for k > ℓ ( λ ) or k > ℓ ( µ ).The (French style) Young diagram of a partition λ is the set of lattice points { ( i, j ) | ≤ j ≤ ℓ ( λ ) , ≤ i ≤ λ j } . We often identify λ and its diagram with the set of lattice squares,or boxes , with northeast corner at a point ( i, j ) ∈ λ . We write λ ∗ for the transpose of λ . The diagram generator of λ is the polynomial(7) B λ ( q, t ) = X ( i,j ) ∈ λ q i − t j − . We will occasionally refer to the quantity(8) n ( λ ) = X i ( i − λ i . Let Λ = Λ k ( X ) be the algebra of symmetric functions in an infinite alphabet of variables X = x , x , . . . , with coefficients in the field k = Q ( q, t ). We follow Macdonald’s notation [25]for various graded bases of Λ, such as 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 . The involutory k -algebra auto-morphism ω : Λ → Λ mentioned in the introduction may be defined by any of the formulas(9) ω e k = h k , ω h k = e k , ω p k = ( − k − p k , ω s λ = s λ ∗ . We also need the symmetric bilinear inner product h− , −i on Λ defined by any of(10) h s λ , s µ i = δ λ,µ , h h λ , m µ i = δ λ,µ , h p λ , p µ i = z λ δ λ,µ , where z λ = Q i r i ! i r i if λ = (1 r , r , . . . ).We write f • for the operator of multiplication by a function f . Otherwise, the custom ofwriting f for both the operator and the function would make it hard to distinguish between SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 5 operator expressions such as ( ωf ) • and ω · f • . When f is a symmetric function, we write f ⊥ for the h− , −i adjoint of f • .2.2. We briefly recall the device of plethystic evaluation . If A is an expression in terms ofindeterminates, such as a polynomial, rational function, or formal series, we define p k [ A ] tobe the result of substituting a k for every indeterminate a occurring in A . We define f [ A ]for any f ∈ Λ by substituting p k [ A ] for p k in the expression for f as a polynomial in thepower-sums p k , so that f f [ A ] is a homomorphism.The variables q, t from our ground field k count as indeterminates.As a simple example, the plethystic evaluation f [ x + · · · + x l ] is just the ordinary evaluation f ( x , . . . , x l ), since p k [ x + · · · + x l ] = x k + · · · + x kl . This also works in infinitely many variables.When X = x , x , . . . is the name of an infinite alphabet of variables, we use f ( X ), withround brackets, as an abbreviation for f ( x , x , . . . ) ∈ Λ( X ). In this situation we also makethe convention that when X appears inside plethystic brackets, it means X = x + x + · · · . With this convention, f [ X ] is another way of writing f ( X ).As a second example and caution to the reader, the formula in (9) for ω p k implies theidentity ωf ( X ) = f [ − zX ] | z = − . Note that f [ − zX ] | z = − does not reduce to f ( X ), as it mightat first appear, since specializing the indeterminate z to a number does not commute withplethystic evaluation.Plethystic evaluation of a symmetric infinite series is allowed if the result converges as aformal series. The series(11) Ω = 1 + X k> h k = exp X k> p k k , or Ω[ a + a + · · · − b − b − · · · ] = Q i (1 − b i ) Q i (1 − a i )is particularly useful. The classical Cauchy identity can be written using this notation as(12) Ω[ XY ] = X λ s λ [ X ] s λ [ Y ] . Taking the inner product with f ( X ) in (12) yields f [ A ] = h Ω[ AX ] , f ( X ) i , which implies(13) h Ω[ AX ]Ω[ BX ] , f ( X ) i = f [ A + B ] = h Ω[ BX ] , f [ X + A ] i . As B is arbitrary, Ω[ BX ] is in effect a general symmetric function, so (13) implies(14) Ω[ AX ] ⊥ f ( X ) = f [ X + A ] . Note that although Ω[ AX ] ⊥ = P k h k [ AX ] ⊥ is an infinite series, it converges formally as anoperator applied to any f ∈ Λ( X ), since h k [ AX ] ⊥ has degree − k , and so kills f for k ≫ p k , we have(15) p ⊥ k = k ∂∂p k . In fact, taking A = z and f = p k in (14) shows that exp( P ( p ⊥ k z k ) /k ) is the operator thatsubstitutes p k + z k for p k in any polynomial f ( p , p , . . . ). This operator can also be writtenexp( P z k ∂∂p k ), giving (15). J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER Another consequence of (14) is the operator identity(16) Ω[ AX ] ⊥ Ω[ BX ] • = Ω[ AB ] Ω[ BX ] • Ω[ AX ] ⊥ with notation Ω[ BX ] • as in § l characters. The weight lattice of GL l is X = Z l , with Weyl group W = S l permuting the coordinates. Letting ε , . . . , ε l be unit vectors, the positive roots are ε i − ε j for i < j , with simple roots α i = ε i − ε i +1 for i = 1 , . . . , l − 1. The standard pairingon Z l in which the ε i are orthonormal identifies the dual lattice X ∗ with X . Under thisidentification, the coroots coincide with the roots, and the simple coroots α ∨ i with the simpleroots. A weight λ ∈ Z l is dominant if λ ≥ · · · ≥ λ l ; a weight is regular (has trivial stabilizerin S l ) if λ , . . . , λ l are distinct.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 ( k X ) W 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 , we denote the partial sum over polynomial weights λ by f ( x ) pol . Thus, f ( x ) pol is a symmetric polynomial in l variables. We also use this notation forinfinite formal sums f ( x ) of irreducible GL l characters, in which case f ( x ) pol is a symmetricformal power series.The Weyl symmetrization operator for GL l is(17) σ f ( x , . . . , x l ) = X w ∈ S l w f ( x ) Q i SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 7 Lemma 2.3.1. For any GL l weights λ, µ and Laurent polynomial φ ( x ) = φ ( x , . . . , x l ) , wehave χ λ Y i Given a Laurent polynomial φ ( x , . . . , x l ) over afield k containing Q ( q ), we define(23) H q ( φ ( x )) = σ (cid:16) φ ( x ) Q i J. BLASIAK, M. HAIMAN, J. MORSE, A. PUN, AND G. H. SEELINGER on the algebra of symmetric functions Λ, constructed by Feigin and Tsymbauliak [7] andSchiffmann and Vasserot [30].From a certain point of view, this material is unnecessary: two of the three quantitiesequated by (1) and (2) are defined without reference to the Schiffmann algebra, and wecould take “Shuffle Theorem” to mean the identity between these two, namely(24) H b ( x ) pol = X λ t a ( λ ) q dinv p ( λ ) G ν ( λ ) ( x , . . . , x l ; q − ) , with H b ( x ) as in (3). This identity still has the virtue of equating the combinatorial righthand side, involving Dyck paths and LLT polynomials, with a simple algebraic left hand sidethat is manifestly symmetric in q and t . Furthermore, our proof of (1) in Theorem 5.5.1proceeds by combining (2) with a proof of (24) (via Theorem 5.3.1) that makes no use ofthe Schiffmann algebra.What we need the Schiffmann algebra for is to provide the link between our shuffle theoremand the classical and ( km, kn ) versions. Indeed, the very definition of the algebraic side in the( km, kn ) shuffle theorem is e k [ − M X m,n ] · e k [ − M X m,n ] ∈ E , whilethe classical shuffle theorems refer implicitly to the same operators through the identity ∇ m e k = e k [ − M X m, ] · ∇ m e k and( H b ) pol to the action of the elements e k [ − M X m,n ] and D b on Λ. For ease of reference, wehave collected most of the statements that will be used elsewhere in the paper in § §§ The algebra E . Let k = Q ( q, t ). The Schiffmann algebra E is generated by subalgebrasΛ( X m,n ) isomorphic to the algebra Λ k of symmetric functions, one for each pair of coprimeintegers ( m, n ), and a central Laurent polynomial subalgebra k [ c ± , c ± ], subject to somedefining relations which we will not list in full here, but only invoke a few of them as needed.For purposes of comparison with [5, 29, 30], our notation (on the left hand side of eachformula) is related as follows to that in [5, Definition 6.4] (on the right hand side). Notethat our indices ( m, n ) ∈ Z correspond to transposed indices ( n, m ) in [5].(25) q = σ − , t = ¯ σ − ,c m c n = κ − n,m ,ωp k ( X m,n ) = κ ε n,m kn,km u kn,km ,e k [ − c M X m,n ] = κ ε n,m kn,km θ kn,km , where ε n,m , which is equal to (1 − ǫ n,m ) / ε n,m = ( n < m, n ) = ( − , . SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 9 The expression e k [ − c M X m,n ] in (25) uses plethystic substitution ( § c M = (1 − qt ) M where M = (1 − q )(1 − t ) . The quantity M will be referred to repeatedly.3.3. Action of E on Λ . A natural action of E by operators on Λ( X ) has been constructedin [7, 30]. Actually these references give several different normalizations of essentially thesame action. The action we use is a slight variation of the action in [30, Theorem 3.1].To write it down we need to recall some notions from the theory of Macdonald polynomials.Let ˜ H µ ( X ; q, t ) denote the modified Macdonald polynomials [10], which can be defined interms of the integral form Macdonald polynomials J µ ( X ; q, t ) of [25] by(28) ˜ H µ ( X ; q, t ) = t n ( µ ) J µ [ X − t − ; q, t − ] , with n ( µ ) as in (8). For any symmetric function f ∈ Λ, let f [ B ], f [ B ] denote the eigenop-erators on the basis { ˜ H µ } of Λ such that(29) f [ B ] ˜ H µ = f [ B µ ( q, t )] ˜ H µ , f [ B ] ˜ H µ = f [ B µ ( q, t )] ˜ H µ with B µ ( q, t ) as in (7) and B µ ( q, t ) = B µ ( q − , t − ). More generally we use the overbar tosignify inverting the variables in any expression, for example(30) M = (1 − q − )(1 − t − ) . The next proposition essentially restates the contents of [30, Theorem 3.1 and Proposition4.10] in our notation. To be more precise, since these two theorems refer to different actionsof E on Λ, one must first use the plethystic transformation in [30, § u ± ,l then yields the following. Proposition 3.3.1. There is an action of E on Λ characterized as follows.(i) The central parameters c , c act as scalars (31) c , c ( q t ) − . (ii) The subalgebras Λ( X ± , ) act as (32) f ( X , ) ( ωf )[ B − /M ] , f ( X − , ) ( ωf )[1 /M − B ] . (iii) The subalgebras Λ( X , ± ) act as (33) f ( X , ) f [ − X/M ] • , f ( X , − ) f ( X ) ⊥ , using the notation in § Remark . The subalgebras Λ( X ± ( m,n ) ) ⊆ E satisfy Heisenberg relations that depend onthe central element c m c n . If c m c n = 1, the Heisenberg relation degenerates and Λ( X ± ( m,n ) )commute. In particular, the value c X ± , ) commute, consistent with(32). The value c / ( qt ) makes Λ( X , ± ) satisfy Heisenberg relations consistent with(33).We will show in Proposition 3.3.4, below, that the elements p [ − M X ,a ] ∈ E act on Λ asoperators D a given by the coefficients D a = h z − a i D ( z ) of a generating series(34) D ( z ) = X a ∈ Z D a z − a defined by either of the equivalent formulas(35) D ( − z ) = Ω[ − z − X ] • Ω[ zM X ] ⊥ or D ( z ) = ( ω Ω[ z − X ]) • ( ω Ω[ − zM X ]) ⊥ , using the operator notation from § D a differ by a sign ( − a from thosestudied in [3, 11], and by a plethystic transformation from operators previously introducedby Jing [21]. Lemma 3.3.3. We have the identities (36) [ ( ωp k [ − X/M ]) • , D a ] = − D a + k , [ ( ωp k ( X )) ⊥ , D a ] = D a − k . Proof. We start with the second identity, which is equivalent to(37) [ ( ωp k ( X )) ⊥ , D ( z ) ] = z − k D ( z ) . Since all operators of the form f ( X ) ⊥ commute with each other, (37) follows from thedefinition of D ( z ) and(38) [ ( ωp k ( X )) ⊥ , ( ω Ω[ z − X ]) • ] = z − k ( ω Ω[ z − X ]) • . To verify the latter identity, note first that (15) and Ω[ z − X ] = exp P k> p k z − k /k imply(39) [ p k ( X ) ⊥ , Ω[ z − X ] • ] = z − k Ω[ z − X ] • . Conjugating both sides by ω and using ( ωf ) • = ω · f • · ω and ( ωf ) ⊥ = ω · f ⊥ · ω gives (38).For the first identity in (36), consider the modified inner product(40) h f, g i ′ = h f [ − M X ] , g i = h f, g [ − M X ] i . The second equality here, which shows that h− , −i ′ is symmetric, follows from orthogonalityof the power-sums p λ . For any f , the operators f ⊥ and f [ − X/M ] • are adjoint with respectto h− , −i ′ . Using this and the definition of D ( z ), we see that D ( z − ) is the h− , −i ′ adjointof D ( z ), hence D − a is adjoint to D a . Taking adjoints on both sides of the second identity in(36) now implies the first. (cid:3) Proposition 3.3.4. In the action of E on Λ given by Proposition 3.3.1, the element p [ − M X ,a ] = − M p ( X ,a ) ∈ E acts as the operator D a defined by (34) . SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 11 Proof. It is known [17, Proposition 2.4] that D ˜ H µ = (1 − M B µ ) ˜ H µ . From Proposition 3.3.1(ii), we see that p [ − M X , ] acts by the same operator, giving the case a = 0.Among the defining relations of E are(41) [ ωp k ( X , ) , p ( X ,a ) ] = − p ( X ,a + k ) , [ ωp k ( X , − ) , p ( X ,a ) ] = p ( X ,a − k ) . Multiplying these by − M and using Proposition 3.3.1 (iii) to compare with (36) reduces thegeneral result to the case a = 0. (cid:3) The operator ∇ . As in [3], we define an eigenoperator ∇ on the Macdonald basis by(42) ∇ ˜ H µ = t n ( µ ) q n ( µ ∗ ) ˜ H µ , with n ( µ ) as in (8). Since t n ( µ ) q n ( µ ∗ ) = e n [ B µ ( q, t )] for n = | µ | , we see that ∇ coincides indegree n with e n [ B ]. Although the operators e n [ B ] belong to E acting on Λ, the operator ∇ does not. Its role, rather, is to internalize a symmetry of this action. Lemma 3.4.1. Conjugation by the operator ∇ provides a symmetry of the action of E on Λ , namely (43) ∇ f ( X m,n ) ∇ − = f ( X m + n,n ) . Proof. For m = ± n = 0, this says that ∇ commutes with the other Macdonald eigenop-erators, which is clear.It is known from [5] that the group of k -algebra automorphisms of E includes one whichacts on the subalgebras Λ( X m,n ) by f ( X m,n ) f ( X m + n,n ), and on the central subalgebra k [ c ± , c ± ] by an automorphism which fixes the central character in Proposition 3.3.1 (i).The Λ( X m,n ) for n > E generated by the elements p ( X a, ). To prove (43) for n > 0, it therefore suffices to verify the operator identity ∇ p ( X a, ) ∇ − = p ( X a +1 , ).In E there are relations(44) [ ωp k ( X , ) , p ( X a, ) ] = p ( X a + k, ) , [ c − k ωp k ( X − , ) , p ( X a, ) ] = − p ( X a − k, ) . Since ∇ commutes with the action of Λ( X ± , ), these relations reduce the problem to thecase a = 0, that is, to the identity ∇ p ( X , ) ∇ − = p ( X , ). By Propositions 3.3.1 and3.3.4, this is equivalent to the operator identity ∇ · p ( X ) • · ∇ − = D , which is [3, I.12 (iii)].We leave the similar argument for the case n < (cid:3) Shuffle algebra. The operators of interest to us belong to the ‘right half-plane’ subal-gebra E + ⊆ E generated by the Λ( X m,n ) for m > 0, or equivalently by the elements p ( X ,a ).The subalgebra E + acts on Λ as the algebra generated by the operators D a . It was shownin [30] that E + is isomorphic to the shuffle algebra constructed in [7] and studied further in[28], whose definition we now recall.We fix the rational function(45) Γ( x/y ) = 1 − q t x/y (1 − y/x )(1 − q x/y )(1 − t x/y ) , and define, for each l , a q, t analog of the symmetrization operator H q in (23) by(46) H q,t ( φ ( x , . . . , x l )) = X w ∈ S l φ ( x ) · Y i Given a Laurent polynomial φ = φ ( x , . . . , x l ) , let ζ = ψ ( φ ) ∈ E + be itsimage under the isomorphism in Proposition 3.5.1. Then with the action of E on Λ given byProposition 3.3.1, we have (48) ω ( ζ · x , . . . , x l ) = H q,t ( φ ) pol . Moreover, the Schur function expansion of the symmetric function ω ( ζ · X ) contains onlyterms s λ with ℓ ( λ ) ≤ l , so (48) determines ζ · . SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 13 Proof. It suffices to consider the case when φ ( x ) = x a = x a · · · x a l l and thus (by Proposition3.3.4) ζ acts on Λ as the operator D a · · · D a l . To find ζ · 1, we use (16) to compute(49) D ( z ) · · · D ( z l ) = (cid:0)Y i Given a rational function φ ( x , . . . , x l ), it may happen thatwe have an identity of rational functions H q,t ( φ ) = H q,t ( η ) for some Laurent polynomial η ( x ). In this case, H q,t ( φ ) is the representative of the image of η in S , or of ψ ( η ) ∈ E + ,even though φ ( x ) is not necessarily a Laurent polynomial. For the shuffle algebra S underconsideration here, Negut [28, Proposition 6.1] showed that this happens for(51) φ ( x ) = x b · · · x b l l Q l − i =1 (1 − q t x i /x i +1 ) . Accordingly, there are distinguished elements(52) D b ,...,b l = ψ ( η ) ∈ E + , where ψ : S → E + is the isomorphism in Proposition 3.5.1 and η ( x ) is any Laurent polynomialsuch that H q,t ( φ ) = H q,t ( η ) for the function φ in (51).Negut identified certain of the elements D b ,...,b l as (in our notation) ribbon skew Schurfunctions s R [ − M X m,n ]. The following result is a special case. Proposition 3.6.1 ([28, Proposition 6.7]) . Let m, n, k be positive integers with m, n coprime.For i = 1 , . . . , km , let b i = ⌈ in/m ⌉ − ⌈ ( i − n/m ⌉ be the number of south steps at x = i − in the highest south-east lattice path weakly below the line from (0 , kn ) to ( km, . Then (53) e k [ − M X m,n ] = D b ,...,b km Lemma 3.6.2. For any indices b , . . . , b l we have (54) D b ,...,b l , · D b ,...,b l · . Proof. Using Proposition 3.5.2, each side of (54) is characterized by its evaluation ω ( D b ,...,b l · x , . . . , x l ) = H ( l ) q,t (cid:16) x b · · · x b l l Q l − i =1 (1 − q t x i /x i +1 ) (cid:17) pol (55) ω ( D b ,...,b l , · x , . . . , x l +1 ) = H ( l +1) q,t (cid:16) x b · · · x b l l (1 − q t x l /x l +1 ) Q l − i =1 (1 − q t x i /x i +1 ) (cid:17) pol . (56)Terms with a negative exponent of x l +1 inside the parenthesis in (56) contribute zero afterwe apply H q,t ( − ) pol . We can therefore drop all but the constant term of the geometric seriesfactor 1 / (1 − q t x l /x l +1 ), since the other factors are independent of x l +1 . This shows thatthe right hand side of (56) is the same as in (55), except that it has H ( l +1) q,t in place of H ( l ) q,t .It follows that ω ( D b ,...,b l , · 1) is a linear combination of Schur functions s λ ( X ) with ℓ ( λ ) ≤ l ,and that ω ( D b ,...,b l , · 1) and ω ( D b ,...,b l · 1) evaluate to the same symmetric function in l variables. Hence, they are identical. (cid:3) Summary. Most of what we use from this section in other parts of the paper can besummarized as follows. Definition 3.7.1. Given b = ( b , . . . , b l ) ∈ Z l , the infinite series of GL l characters H b ( x ) = H b ,...,b l ( x , . . . , x l ) is defined by(57) H b ( x ) = H q,t (cid:16) x b Q l − i =1 (1 − q t x i /x i +1 ) (cid:17) = H q (cid:16) x b Q i +1 For the Negut element D b ∈ E acting on Λ , the symmetric function ω ( D b · evaluated in l variables is given by (58) ω ( D b · x , . . . , x l ) = H b ( x ) pol . Moreover, all terms s λ in the Schur expansion of ω ( D b · X ) have ℓ ( λ ) ≤ l , so ω ( D b · is determined by its evaluation in l variables. In the special cases where the index b is the sequence of south runs in the highest ( km, kn )Dyck path, D b · Corollary 3.7.3. For i = 1 , . . . , l = km + 1 , let b i be the number of south steps at x = i − in the highest south-east lattice path weakly below the line from (0 , kn ) to ( km, , including b l = 0 . Then the Negut element D b ,...,b l and the operator e k [ − M X m,n ] agree when applied to ∈ Λ , that is, we have (59) D b ,...,b l · e k [ − M X m,n ] · . SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 15 Proof. Immediate from Proposition 3.6.1 and Lemma 3.6.2. (cid:3) Corollary 3.7.4 (also proven in [4]) . In the case n = 1 of (59) , we further have (60) ∇ m e k ( X ) = e k [ − M X m, ] · . Proof. By Proposition 3.3.1 (iii), e k [ − M X , ] · e k ( X ). Since ∇ (1) = 1, the result nowfollows from Lemma 3.4.1. (cid:3) Remark . Equations (46), (54), (57), (58), (59), and (60) imply the raising operatorformula ( ω ∇ m e k +1 )( x , . . . , x l ) = σ x x m +1 x m +1 · · · x km +1 Q i +1 In this section we review the definition of the combinatorial LLT polynomials G ν ( X ; q ),using the attacking inversions formulation from [15], which is better suited to our purposesthan the original ribbon tableau formulation in [23].We also define and prove some results on the infinite LLT series L β/α ( x ; q ) introducedin [13]. Since [13] is unpublished, due for revision, and doesn’t cover the ‘twisted’ variants L σβ/α ( x ; q ), we give here a self-contained treatment of the material we need.4.1. Combinatorial LLT polynomials. The content of a box a = ( i, j ) in row j , column i of any skew Young diagram is c ( a ) = i − j .Let ν = ( ν (1) , . . . , ν ( k ) ) be a tuple of skew Young diagrams. When referring to boxes of ν , we identify ν with the disjoint union of the ν ( i ) . Fix ǫ > kǫ < adjusted content of a box a ∈ ν ( i ) is ˜ c ( a ) = c ( a ) + iǫ . A reading order is any totalordering of the boxes a ∈ ν on which ˜ c ( a ) is increasing. In other words, the reading orderis lexicographic, first by content, then by the index i for which a ∈ ν ( i ) , with boxes of thesame content in the same component ν ( i ) ordered arbitrarily.Boxes a, b ∈ ν attack each other if 0 < | ˜ c ( a ) − ˜ c ( b ) | < 1. If a ∈ ν ( i ) precedes b ∈ ν ( j ) in the reading order, the attacking condition means that either c ( a ) = c ( b ) and i < j , or c ( b ) = c ( a ) + 1 and i > j . We also say that a, b form an attacking pair in ν .By a semistandard Young tableau on the tuple ν we mean a map T : ν → Z + whichrestricts to a semistandard tableau on each component ν ( i ) . We write SSYT( ν ) for the setof these. The weight of T ∈ SSYT( ν ) is x T = Q a ∈ ν x T ( a ) . An attacking inversion in T isan attacking pair a, b such that T ( a ) > T ( b ), where a precedes b in the reading order. Wedefine inv( T ) to be the number of attacking inversions in T . Definition 4.1.1. The combinatorial LLT polynomial indexed by a tuple of skew Youngdiagrams ν is the generating function(62) G ν ( X ; q ) = X T ∈ SSYT( ν ) q inv( T ) x T . In [15] it was shown that G ν ( X ; q − ) coincides up to a factor q e with a ribbon tableauLLT polynomial as defined in [23], and is therefore a symmetric function. A direct and moreelementary proof that G ν ( X ; q ) is symmetric was given in [14].It is useful in working with the LLT polynomials G ν ( X ; q ) to consider a more generalcombinatorial formalism, as in [14, § A = A + ` A − be a ‘signed’ alphabet with a positive letter v ∈ A + and a negative letter v ∈ A − for each v ∈ Z + , and an arbitrary totalordering on A .A super tableau on a tuple of skew shapes ν is a map T : ν → A , weakly increasing alongrows and columns, with positive letters increasing strictly on columns and negative lettersincreasing strictly on rows. A usual semistandard tableau is thus the same thing as a supertableau with all entries positive. Let SSYT ± ( ν ) denote the set of super tableaux on ν .An attacking inversion in a super tableau is an attacking pair a, b , with a preceding b inthe reading order, such that either T ( a ) > T ( b ) in the ordering on A , or T ( a ) = T ( b ) = v with v negative. As before, inv( T ) denotes the number of attacking inversions. Lemma 4.1.2 ([14, (81–82) and Proposition 4.2]) . We have the identity (63) ω Y G ν [ X + Y ; q ] = X T ∈ SSYT ± ( ν ) q inv( T ) x T + y T − , where the weight is given by (64) x T + y T − = Y a ∈ ν ( x T ( a ) , T ( a ) ∈ A + ,y T ( a ) , T ( a ) ∈ A − . This holds for any choice of the ordering on the signed alphabet A . Corollary 4.1.3. We have (65) ω G ν ( X ; q ) = X T ∈ SSYT − ( ν ) q inv( T ) x T , where the sum is over super tableaux T with all entries negative, and we abbreviate x T − to x T in this case. Proposition 4.1.4. Given a tuple of skew Young diagrams ν = ( ν (1) , . . . , ν ( k ) ) , let ν R =(( ν (1) ) R , . . . , ( ν ( k ) ) R ) , where ( ν ( i ) ) R is the ◦ rotation of the transpose ( ν ( i ) ) ∗ , positioned sothat each box in ν R has the same content as the corresponding box in ν . Then (66) ω G ν ( X ; q ) = q I ( ν ) G ν R ( X ; q − ) , where I ( ν ) is the total number of attacking pairs in ν . SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 17 Proof. Use Corollary 4.1.3 on the left hand side, ordering the negative letters as 1 > > · · · .Given a negative tableau T on ν , let T R be the tableau on ν R obtained by reflecting thetableau along with ν and changing negative letters v to positive letters v . Then T R is anordinary semistandard tableau, and T T R is a weight preserving bijection from negativetableaux on ν to SSYT( ν R ). An attacking pair in ν is an inversion in T if and only if thecorresponding attacking pair in ν R is a non-inversion in T R , hence inv( T R ) = I ( ν ) − inv( T ).This implies (66). (cid:3) Example . Consider a tuple ν = ( ν (1) , . . . , ν ( k ) ) in which each skew shape is a column sothat ν R is a tuple of rows. The super tableau of shape ν R with all entries a positive letter 1has no inversions, whereas the distinguished tableau T of shape ν with all entries a negativeletter 1 has(67) inv( T ) = I ( ν ) , where I ( ν ) is the total number of attacking pairs in ν . Lemma 4.1.6. The LLT polynomial G ν ( X ; q ) is a linear combination of Schur functions s λ ( X ) such that ℓ ( λ ) is bounded by the total number of rows in the diagram ν .Proof. Let r be the total number of rows in ν . It is equivalent to show that ω G ν ( X ; q ) is alinear combination of monomial symmetric functions m λ ( X ) such that λ ≤ r . By Propo-sition 4.1.4, ω G ν ( X ; q ) has a monomial term q I ( ν ) − inv( T ) x T for each semistandard tableau T ∈ SSYT( ν R ) on the tuple of reflected shapes ν R . Since a letter can appear at most oncein each column of T , the exponents of x T are bounded by r . (cid:3) Reminder on Hecke algebras. We recall, in the case of GL l , the Hecke algebraaction on the group algebra of the weight lattice, as in Lusztig [24] or Macdonald [26] anddue originally to Bernstein and Zelevinsky.For GL l , we identify the group algebra k X of the weight lattice X = Z l with the Laurentpolynomial algebra k [ x ± , . . . , x ± l ]. Here k is any ground field containing Q ( q ).The Demazure-Lusztig operators(68) T i = qs i + (1 − q ) 11 − x i +1 /x i ( s i − i = 1 , . . . , l − S l on k [ x ± , . . . , x ± l ]. Wehave normalized the generators so that the quadratic relations are ( T i − q )( T i + 1) = 0. Theelements T w , defined by T w = T i T i · · · T i m for any reduced expression w = s i s i · · · s i m ,form a k -basis of the Hecke algebra, as w ranges over S l .Let R + be the set of positive roots and Q + = N R + the cone they generate in the rootlattice Q . For dominant weights we define λ ≤ µ if µ − λ ∈ Q + . For polynomial weights ofGL l , this coincides with the standard partial ordering (6) on partitions.For any weight λ , let λ + denote the dominant weight in the orbit S l · λ .Let conv( S l · λ ) be the convex hull of the orbit S l · λ in the coset λ + Q of the root lattice,i.e., the set of weights that occur with non-zero multiplicity in the irreducible character χ λ + .Note that conv( S l · λ ) ⊆ conv( S l · µ ) if and only if λ + ≤ µ + . Each orbit S l · λ + has a partial ordering induced by the Bruhat ordering on S l . Moreexplicitly, this ordering is the transitive closure of the relation s i λ > λ if h α ∨ i , λ i > 0. Weextend this to a partial ordering on all of X = Z l by defining λ ≤ µ if λ + < µ + , or if λ + = µ + and λ ≤ µ in the Bruhat order on S l · λ + .Suppose h α ∨ i , λ i ≥ 0. If h α ∨ i , λ i = 0, that is, if λ = s i λ , then(69) T i x λ = q x λ . Otherwise, if h α ∨ i , λ i > 0, then(70) T i x λ ≡ q x s i λ + ( q − x λ ,T i x s i λ ≡ x λ modulo the space spanned by monomials x µ for µ strictly between λ and s i λ on the rootstring λ + Z α i . Note that µ < λ for these weights µ , since they lie on orbits strictly insideconv( S l · λ ). Furthermore, the set of all monomials µ ≤ s i λ is s i -invariant and has convexintersection with every root string ν + Z α i , hence the space k · { x µ | µ ≤ s i λ } is closed under T i . It follows that if h α ∨ i , λ i ≥ 0, then T i applied to any Laurent polynomial of the form x λ + P µ<λ c µ x µ yields a result of the form(71) T i (cid:0) x λ + X µ<λ c µ x µ (cid:1) = q x s i λ + X µ For each GL l weight λ ∈ Z l , wedefine the non-symmetric Hall-Littlewood polynomial (72) E λ ( x ; q ) = E λ ( x , · · · x l ; q ) = q − ℓ ( w ) T w x λ + , where w ∈ S l is such that λ = w ( λ + ). If λ has non-trivial stabilizer then w is not unique,but it follows from (69)–(71) that E λ ( x ; q ) is independent of the choice of w and has themonic and triangular form(73) E λ ( x ; q ) = x λ + X µ<λ c µ x µ . See Figure 1 for examples.For context, we remark that several distinct notions of ‘non-symmetric Hall-Littlewoodpolynomial’ can be found the literature. Our E λ (and F λ , below) coincide with special-izations of non-symmetric Macdonald polynomials considered by Ion in [20, Theorem 4.8].The twisted variants E σλ below are specializations of the ‘permuted basement’ non-symmetricMacdonald polynomials studied (for GL l ) by Alexandersson [1] and Alexandersson and Sawh-ney [2].For any µ ∈ R l we define Inv( µ ) = { ( i < j ) | µ i > µ j } . In the case of a permutation,Inv( σ ) is then the usual inversion set of σ = ( σ (1) , . . . , σ ( l )) ∈ S l .Taking ρ as in § ǫ > λ + ǫρ ) denotes the set of pairs i < j such that λ i ≥ λ j . SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 19 E = 1 F = 1 E = x F = y E = (1 − q ) x + x F = y E = (1 − q ) x + (1 − q ) x + x F = y E = x x F = y y E = (1 − q ) x x + x x F = y y E = (1 − q ) x x + (1 − q ) x x + x x F = y y E = x F = y + (1 − q ) y y + (1 − q ) y y E = (1 − q ) x + (1 − q ) x x + x F = y + (1 − q ) y y E = (1 − q ) (cid:0) x + (1 − q ) x x + x x + x + x x (cid:1) + x F = y Figure 1. Non-symmetric Hall-Littlewood polynomials E σ a ( x , x , x ; q − )and F σ a ( y , y , y ; q ) for l = 3, σ = 1, and | a | ≤ σ ∈ S l , we define twisted non-symmetric Hall-Littlewood polynomials E σλ ( x ; q ) = q | Inv( σ − ) ∩ Inv( λ + ǫρ ) | T − σ − E σ − ( λ ) ( x ; q )(74) F σλ ( x ; q ) = E σw − λ ( x ; q ) = E σw − λ ( x − , . . . , x − l ; q − ) , (75)where w ∈ S l is the longest element, given by w ( i ) = l + 1 − i . The normalization in (74)implies the recurrence(76) E σλ = ( q − I ( λ i ≤ λ i +1 ) T i E s i σs i λ , s i σ > σq I ( λ i ≥ λ i +1 ) T − i E s i σs i λ , s i σ < σ, where I ( P ) = 1 if P is true, I ( P ) = 0 if P is false. Together with the initial condition E σλ = x λ for all σ if λ = λ + , this determines E σλ for all σ and λ . Corollary 4.3.1. E σλ has the monic form in (73) for all σ .Proof. This follows from (71) and (76). (cid:3) Proposition 4.3.2. For every σ ∈ S l , the E σλ ( x ; q ) and F σλ ( x ; q ) are dual bases of k [ x ± , . . . , x ± l ] with respect to the inner product defined by (77) h f, g i q = h x i f g Y i Lemma 4.3.3. The Demazure-Lusztig operators T i in (68) are self-adjoint with respect to h− , −i q .Proof. It’s the same to show that T i − q is self-adjoint. A bit of algebra gives(78) T i − q = q − q − x i /x i +1 − x i /x i +1 ( s i − , and therefore(79) h ( T i − q ) f, g i q = q h x i ( s i ( f ) g − f g ) Y − x j /x k − q − x j /x k , where the product is over j < k with ( j, k ) = ( i, i + 1). We want to show that this issymmetric in f and g , i.e., that the right hand side is unchanged if we replace s i ( f ) g with f s i ( g ). Let ∆ denote the product factor in (79), and note that ∆ is symmetric in x i and x i +1 . The constant term h x i ϕ ( x ) of any ϕ ( x , . . . , x l ) is equal to h x i s i ( ϕ ( x )). In particular, h x i s i ( f ) g ∆ = h x i f s i ( g ) ∆, which implies the desired result. (cid:3) Proof of Proposition 4.3.2. The desired identity is just a tidy notation for h E σλ , E σw − µ i q = δ λµ .By (76), for every i , we have either h E σλ , E σw − µ i q = q e h T i E s i σs i λ , T − i E s i σw − s i µ i q or h E σλ , E σw − µ i q = q e h T − i E s i σs i λ , T i E s i σw − s i µ i q , depending on whether s i σ > σ or s i σ < σ , for some exponent e .Moreover, if λ = µ , then q e = 1.Since T i is self-adjoint, we get h E σλ , E σw − µ i q = q e h E s i σs i λ , E s i σw − s i µ i q in either case. Repeatingthis gives an identity(80) h E σλ , E σw − µ i q = q e h E vσvλ , E vσw − vµ i q for all λ, µ ∈ Z l and all σ, v ∈ S l , again with q e = 1 if λ = µ .Choose v ∈ S l such that µ − = v ( µ ) is antidominant. Then (80) gives(81) h E σλ , E σw − µ i q = q e h E vσvλ , x − ( µ − ) i q = q e h x µ − i ∆ E vσvλ , where ∆ is the product factor in (77). Let supp( f ) denote the set of weights ν for which x ν occurs with non-zero coefficient in f . Since supp(∆) = Q + , and supp( E vσvλ ) ⊆ conv( S l · λ ),it follows from (81) that if h E σλ , E σw − µ i q = 0, then ( µ − − Q + ) ∩ conv( S l · λ ) = ∅ and therefore µ − − λ − ∈ Q + . Since w ( Q + ) = − Q + , this is equivalent to λ + ≥ µ + .By symmetry, exchanging λ with − µ and σ with σw , if h E σλ , E σw − µ i q = 0 then we also have( − λ ) − − ( − µ ) − ∈ Q + , hence λ + − µ + ∈ − Q + , that is, λ + ≤ µ + . Hence, h E σλ , E σw − µ i q = 0implies λ + = µ + , so λ and µ belong to the same S l orbit. This reduces the problem to thecase that S l · λ = S l · µ .In this case, ( µ − − Q + ) ∩ conv( S l · λ ) = { µ − } . Furthermore, if λ = µ , then vλ = µ − , andCorollary 4.3.1 implies that ( µ − − Q + ) ∩ supp( E vσvλ ) = ∅ , hence h E σλ , E σw − µ i q = 0.If λ = µ , then the right hand side of (81) reduces to h x µ − i ∆ E vσµ − . Since supp(∆) = Q + and supp( E vσµ − ) ⊂ µ − + Q + , only the constant term of ∆ and the x µ − term of E vσµ − contributeto the coefficient of x µ − in ∆ E vσµ − , and we have h x µ − i E vσµ − = 1 by Corollary 4.3.1. Hence, h E σλ , E σw − λ i q = 1. (cid:3) SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 21 Lemma 4.3.4. Given λ ∈ Z l , suppose there is an index k such that λ i ≥ λ j for all i ≤ k and j > k . Given σ ∈ S l , let σ ∈ S k and σ ∈ S l − k be the permutations such that σ (1) , . . . , σ ( k ) are in the same relative order as σ (1) , . . . , σ ( k ) , and σ (1) , . . . , σ ( l − k ) arein the same relative order as σ ( k + 1) , . . . , σ ( l ) . Then (82) E σ − λ ( x , . . . , x l ; q ) = E σ − ( λ ,...,λ k ) ( x , . . . , x k ; q ) E σ − ( λ k +1 ,...,λ l ) ( x k +1 , . . . , x l ; q ) . Proof. If λ is dominant, the result is trivial. Otherwise, the recurrence (76) determines E σ − λ by induction on | Inv( − λ ) | . The condition on λ implies that we only need to use (76) for i = k , that is, for s i in the Young subgroup S k × S l − k ⊂ S l . For i = k , the right hand sideof (82) satisfies the same recurrence. (cid:3) LLT series.Definition 4.4.1. Given GL l weights α, β ∈ Z l and a permutation σ ∈ S l , the LLT se-ries L σβ/α ( x , . . . , x l ; q ) is the infinite formal sum of irreducible GL l characters in which thecoefficient of χ λ is defined by(83) h χ λ i L σ − β/α ( x ; q − ) = h E σβ i χ λ · E σα , where E σλ ( x ; q ) are the twisted non-symmetric Hall-Littlewood polynomials from § E σλ ( x ; q ) are polynomials in q − , so the convention ofinverting q in (83) makes the coefficients of L σβ/α ( x ; q ) polynomials in q . Inverting σ as wellleads to a more natural statement and proof in Corollary 4.5.7, below. Proposition 4.4.2. We have the formula (84) L σβ/α ( x ; q ) = H q ( w ( F σ − β ( x ; q ) E σ − α ( x ; q ))) , where H q is the Hall-Littlewood symmetrization operator in (23) and w ( i ) = l + 1 − i is thelongest element in S l .Proof. By Proposition 4.3.2, the coefficient h E σ − β ( x ; q − ) i χ λ · E σ − α ( x ; q − ) of χ λ in L σβ/α isgiven by the constant term(85) h x i χ λ F σ − β ( x − ; q ) E σ − α ( x ; q − ) Y i We now work out a tableau formalism which relates thepolynomial part L σβ/α ( x ; q ) pol to a combinatorial LLT polynomial G ν ( x ; q ). Lemma 4.5.1. For all σ ∈ S l , λ ∈ Z l and k , the product of the elementary symmetricfunction e k ( x ) and the non-symmetric Hall-Littlewood polynomial E σ − λ ( x ; q ) is given by (88) e k ( x ) E σ − λ ( x ; q ) = X | I | = k q − h I E σ − λ + ε I ( x ; q ) , where I ⊆ { , . . . , l } has k elements, ε I = P i ∈ I ε i is the indicator vector of I , and (89) h I = | Inv( λ + ε I + ǫσ ) \ Inv( λ + ǫσ ) | . Equivalently, h I is the number of pairs i < j such that i ∈ I , j I , and we have λ j = λ i if σ ( i ) < σ ( j ) , or λ j = λ i + 1 if σ ( i ) > σ ( j ) .Proof. First consider the case σ = 1. Being symmetric, e k ( x ) commutes with T w , giving(90) e k E λ = q − ℓ ( w ) T w e k x λ + = q − ℓ ( w ) X | J | = k T w x λ + + ε J , where λ = w ( λ + ), as in (72). To fix the choice, we take w maximal in its coset w · Stab( λ + ).In each term of the sum in (90), the weight µ = λ + + ε J can fail to be dominant at worstby having some entries µ j = µ i + 1 for indices i < j such that ( λ + ) i = ( λ + ) j , i J and j ∈ J . Let v be the minimal permutation such that µ + = v ( µ ), that is, the permutationthat moves indices j ∈ J to the left within each block of constant entries in λ + . The formula T i x ai x a +1 i +1 = x a +1 i x ai +1 is immediate from the definition of T i , and implies that T v x µ = x µ + .By the maximality of w , since v ∈ Stab( λ + ), there is a reduced factorization w = uv , hence T w = T u T v . Then(91) T w x λ + + ε J = T u x µ + = q ℓ ( u ) E λ + w ( ε J ) , since λ + w ( ε J ) = w ( µ ) = u ( µ + ).Now, ℓ ( v ) is equal to the number of pairs i < j such that µ i < µ j , that is, such that( λ + ) i = ( λ + ) j , i J and j ∈ J . By maximality, the permutation w carries these to the pairs j ′ = w ( i ), i ′ = w ( j ) such that i ′ < j ′ , λ i ′ = λ j ′ , i ′ ∈ I and j ′ I , where I = w ( J ). For σ = 1,the definition of h I is the number of such pairs i ′ , j ′ , so we have ℓ ( u ) − ℓ ( w ) = − ℓ ( v ) = − h I .Hence, the term for J in (90) is q − ℓ ( w ) T w x λ + + ε J = q − h I E λ + ε I . As J ranges over subsets ofsize k , so does I = w ( J ), giving (88) in this case. SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 23 −∞ ∞−∞ ∞−∞ ∞ −∞ −∞ 42 4 85 σ -triples for σ = 1 S ∞ ∞ −∞ ∞ σ -triples for σ = (1 , , (3 , , ∞ Figure 2. A negative tableau S on β/α = (6 , , / (2 , , 2) and the σ -triplesin β/α , for two choices of σ , shown with their entries from S . Triples inboldface are increasing in S .Substituting σ ( λ ) for λ and σ ( I ) for I in the σ = 1 case, and acting on both sides with T − σ , yields(92) q −| Inv( σ ) ∩ Inv( λ + ǫρ ) | e k E σ − λ = X | I | = k q −| Inv( σ ( λ + ε I )) \ Inv( σ ( λ )) | q −| Inv( σ ) ∩ Inv( λ + ε I + ǫρ ) | E σ − λ + ε I . Combining powers of q gives the desired identity (88) if we verify that(93) | Inv( σ ) ∩ Inv( λ + ε I + ǫρ ) | − | Inv( σ ) ∩ Inv( λ + ǫρ ) | = | Inv( λ + ε I + ǫσ ) \ Inv( λ + ǫσ ) | − | Inv( σ ( λ + ε I )) \ Inv( σ ( λ )) | . On the left hand side, cancelling the contribution from the intersection of the two sets leaves(94) | Inv( σ ) ∩ (Inv( λ + ε I + ǫρ ) \ Inv( λ + ǫρ )) | − | Inv( σ ) ∩ (Inv( λ + ǫρ ) \ Inv( λ + ε I + ǫρ )) | . The first term in (94) counts pairs i < j such that i ∈ I , j I , σ ( i ) > σ ( j ), and λ j = λ i + 1.The second term counts pairs i > j such that i ∈ I , j I , σ ( i ) < σ ( j ), and λ j = λ i . Thefirst term on the right hand side of (93) counts pairs i < j such that i ∈ I , j I , and λ j = λ i if σ ( i ) < σ ( j ), or λ j = λ i + 1 if σ ( i ) > σ ( j ). The second term on the right hand side of (93)counts the set of pairs whose images under σ − are pairs i, j (in either order) such that i ∈ I , j I , σ ( i ) < σ ( j ) and λ i = λ j . The cases in the first term with σ ( i ) < σ ( j ) cancel those inthe second term with i < j . The remaining cases in the first term, with σ ( i ) > σ ( j ), matchthe first term in (94), while the remaining cases in the second term, with i > j , match thesecond term in (94). This proves (93) and completes the proof of the lemma. (cid:3) Given α, β ∈ Z l such that α j ≤ β j for all j , we let β/α denote the tuple of single row skewYoung diagrams ( β j ) / ( α j ) such that the x coordinates of the right edges of boxes a in the j -th row are the integers α j + 1 , . . . , β j . The boxes just outside the j -th row, adjacent tothe left and right ends of the row, then have x coordinates α j and β j + 1. In the case of an empty row with α j = β j , we still consider these two boxes to be adjacent to the ends of therow.For each box a , let i ( a ) denote the x coordinate of its right edge and j ( a ) the index of therow containing it.For σ ∈ S l , we define σ ( β/α ) = σ ( β ) /σ ( α ). If a is a box in row j ( a ) of β/α , then σ ( a )denotes the corresponding box with coordinate i ( σ ( a )) = i ( a ) in row σ ( j ( a )) of σ ( β/α ). Theadjusted content of σ ( a ), as defined in § c ( σ ( a )) = i ( a ) + ǫσ ( j ( a )). Hence, thereading order on σ ( β/α ) corresponds via σ to the ordering of boxes in β/α by increasingvalues of i ( a ) + ǫσ ( j ( a )).We define a σ -triple in β/α to consist of any three boxes ( a, b, c ) arranged as follows:boxes a and c are in or adjacent to the same row j ( a ) = j ( c ), and are consecutive, thatis, i ( c ) = i ( a ) + 1, while box b is in a row j ( b ) < j ( a ), and we have a < b < c in theordering corresponding to the reading order on σ ( β/α ). More explicitly, this means that if σ ( j ( b )) < σ ( j ( a )), then i ( b ) = i ( c ), while if σ ( j ( b )) > σ ( j ( a )), then i ( b ) = i ( a ). The box b isrequired to be a box of β/α , but box a is allowed to be outside and adjacent to the left endof a row, while c is similarly allowed to be outside and adjacent to the right end of a row.An example of a tuple β/α , with all its σ -triples for two different choices of σ , is shownin Figure 2.A negative tableau on β/α is a map S : β/α → Z + strictly increasing on each row. In theterminology of § S is a super tableau on β/α with entries in Z + , considered as a negativealphabet ordered by 1 < < · · · . We say that a σ -triple ( a, b, c ) in β/α is increasing in S if S ( a ) < S ( b ) < S ( c ), with the convention that S ( a ) = −∞ if a is just outside the left end ofa row, and S ( c ) = ∞ if c is just outside right end of a row. Along with the σ -triples in β/α ,Figure 2 also displays which triples are increasing in a sample tableau S . Proposition 4.5.2. Given α, β ∈ Z l such that α i ≤ β i for all i , and σ ∈ S l , let (95) N σβ/α ( X ; q ) = X S ∈ SSYT − ( β/α ) q h σ ( S ) x S be the generating function for negative tableaux S on the tuple of single-row skew diagrams ( β i ) / ( α i ) , weighted by q h σ ( S ) , where h σ ( S ) is the number of increasing σ -triples in S . Then N σβ/α ( X ; q ) is a symmetric function, and ωN σβ/α ( X ; q ) evaluates in l variables to (96) ( ωN σβ/α )( x , . . . , x l ; q ) = L σβ/α ( x , . . . , x l ; q ) pol . If we do not have α i ≤ β i for all i , then L σβ/α ( x ; q ) pol = 0 .Proof. Let L σ ( X ; q ) be the unique symmetric function such that (i) L σβ/α ( X ; q ) is a linearcombination of Schur functions s λ with ℓ ( λ ) ≤ l , and (ii) in l variables, it evaluates to(97) L σβ/α ( x , . . . , x l ; q ) = L σβ/α ( x , . . . , x l ; q ) pol . What we need to prove is that ω L σβ/α ( X ; q ) = N σβ/α ( X ; q ). SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 25 The definition of L σβ/α ( x ; q ) implies that L σβ/α ( X ; q ) satisfies(98) h s λ ( X ) , L σβ/α ( X ; q ) i = h E σ − β ( x ; q − ) i s λ ( x , . . . , x l ) E σ − α ( x ; q − )for every partition λ , including when ℓ ( λ ) > l , since then both sides are zero. By linearity,we can replace s λ by any symmetric function f , giving(99) h f ( X ) , L σβ/α ( X ; q ) i = h E σ − β ( x ; q − ) i f ( x ) E σ − α ( x ; q − ) . The coefficient of m µ ( X ) in ω L σβ/α ( X ; q ) is given by taking f = e µ .To show that ω L σβ/α ( X ; q ) is given by the tableau generating function in (95), we useLemma 4.5.1 to express(100) h E σ − β ( x ; q − ) i e µ ( x ) E σ − α ( x ; q − )as a sum of powers of q indexed by negative tableaux. In particular, this coefficient willvanish unless we have α i ≤ β i for all i , giving the last conclusion in the proposition.Multiplying by e µ through e µ n successively and keeping track of one chosen term in eachproduct gives a sequence of terms E σ − α (0) , E σ − α (1) , . . . , E σ − α ( n ) , in which α (0) = α and α ( m +1) = α ( m ) + ε I for a set of indices I of size µ m , for each m . Each sequence with α ( n ) = β contributes to (100).If we record these data in the form of a tableau S : β/α → Z + with S ( a ) = m for a ∈ ( α ( m ) /α ( m − ), S satisfies the condition that it is a negative tableau of weight x S = x µ .The contribution to (100) from the corresponding sequence of terms is the product of the q h I with h I as in (89) for k = µ m , λ = α ( m − , and I the set of indices j such that S ( a ) = m for some box a in row j .We now express the h I corresponding to ( α ( m ) /α ( m − ) = S − ( { m } ) as an attribute of S .For h I to count a pair i < j , we must have i ∈ I , which means that S ( b ) = m for a box b inrow i , and j I , and one of the following two situations.If σ ( i ) < σ ( j ), we must have α ( m − j = α ( m − i . Since there is no m in row j of S , thismeans that the boxes a and c in row j with coordinates i ( a ) = i ( b ) − i ( c ) = i ( b ) have S ( a ) < S ( b ) < S ( c ), with the same convention as above that S ( a ) = −∞ if a is to the leftof a row of β/α , and S ( c ) = ∞ if c is to the right of a row.If σ ( i ) > σ ( j ), we must have α ( m − j = α ( m − i + 1. This means that the boxes a and c inrow j with coordinates i ( a ) = i ( b ), i ( c ) = i ( b ) + 1 have S ( a ) < S ( b ) < S ( c ), with the sameconvention as before.These two cases establish that h I is equal to the number of increasing σ -triples in S forwhich S ( b ) = m . Summing them up gives the total number of increasing σ -triples, implyingthat the coefficient in (100) is the sum of q h σ ( S ) over negative tableaux S of weight x S = x µ on β/α , where h σ ( S ) is the number of increasing σ -triples in S . (cid:3) Lemma 4.5.3. Given σ ∈ S l and α, β ∈ Z l with α i ≤ β i for all i , for T ∈ SSYT − ( σ ( β/α )) , (101) h σ ( β/α ) − inv( T ) = h σ ( S ) , where S = σ − ( T ) = T ◦ σ . Proof. Recall from § T ( a ) ≥ T ( b ), where a, b is an attacking pair with a preceding b in the reading order.One can verify from the definition of σ -triple that the attacking pairs in σ ( β/α ), orderedby the reading order, are precisely the pairs σ ( a, b ) or σ ( b, c ) for ( a, b, c ) a σ -triple such thatthe relevant boxes are in β/α . Moreover, every attacking pair occurs in this manner exactlyonce.If all three boxes of a σ -triple ( a, b, c ) are in β/α , and T is a negative tableau on σ ( β/α ),then since T ( σ ( a )) < T ( σ ( c )), at most one of the attacking pairs σ ( a, b ), σ ( b, c ) can be anattacking inversion in T . The condition that neither pair is an attacking inversion is that S ( a ) < S ( b ) < S ( c ) in the negative tableau S = σ − ( T ) = T ◦ σ on β/α . This also holds fortriples not contained in β/α with our convention that S ( a ) = −∞ or S ( c ) = ∞ for a or c outside the tuple β/α . Hence, the result follows. (cid:3) Example . Let S be as in Figure 2 and σ = (1 , , (3 , , T = S ◦ σ − = , so T is a negative tableau on σ ( β/α ) = (3 , , / (1 , , T by reading orderon σ ( β/α ) gives 134285967. The pairs ( T ( a ) , T ( b )) for attacking inversions ( a, b ) in T are(3 , , , , T ) = 4. From the bottom row of Figure 2, we have h σ ( β/α ) = 5, h σ ( S ) = 1, so h σ ( β/α ) − inv( T ) = h σ ( S ) is indeed satisfied.Now let σ = 1 and T = S ◦ σ − = S . Reading T by reading order on β/α gives 123458697.The pairs ( T ( a ) , T ( b )) for the attacking inversions are (8 , 6) and (9 , T ) = 2. FromFigure 2, we see h σ ( β/α ) = 7, h σ ( S ) = 5, so again h σ ( β/α ) − inv( T ) = h σ ( S ) holds. Remark . If we define an increasing σ -triple for T ∈ SSYT( σ ( β/α )) to be any σ -triple( a, b, c ) satisfying T ( a ) ≤ T ( b ) ≤ T ( c ), then a similar argument gives the relation(102) h σ ( β/α ) − inv( T ) = h σ ( S )for ordinary semi-standard tableaux as well, where again S = σ − ( T ) = T ◦ σ . Remark . Given a tuple of rows β i /α i , if we shift the i -th row to the right by ǫσ ( i ) fora small ǫ > 0, then h σ ( β/α ) is equal to the number of alignments between a box boundaryin row j and the interior of a box in row i for i < j , where a boundary is the edge commonto two adjacent boxes which are either in or adjacent to the row. An empty row has oneboundary.The relation between G and L can now be made precise by applying Proposition 4.5.2 andLemma 4.5.3 to the expression for q h σ ( β/α ) G σ ( β/α ) ( X ; q − ) given in Corollary 4.1.3. Corollary 4.5.7. Given α, β ∈ Z l and σ ∈ S l , (103) L σβ/α ( x ; q ) pol = ( q h σ ( β/α ) G σ ( β/α ) ( x ; q − ) if α i ≤ β i for all i otherwise , SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 27 where h σ ( β/α ) is the number of σ -triples in β/α , and the right hand side is evaluated in l variables x , . . . , x l . The generalized shuffle theorem Cauchy identity. In this section we derive our main results, Theorems 5.3.1 and 5.5.1.The key point is the following delightful ‘Cauchy identity’ for non-symmetric Hall-Littlewoodpolynomials. Theorem 5.1.1. For any permutation σ ∈ S l , the twisted non-symmetric Hall-Littlewoodpolynomials E σλ ( x ; q ) and F σλ ( x ; q ) in (74, 75) satisfy the identity (104) Q i 0, the polynomials E a ( x , . . . , x l ; q − ) and F a ( y l , . . . , y ; q ) have co-efficients in Z [ q ] and specialize at q = 0 to the key polynomial K a ( x , . . . , x l ) and Demazureatom b K w ( a ) ( y , . . . , y l ), respectively. Our Cauchy formula (104) specializes to Lascoux’s non-symmetric Cauchy formula [8, 22] upon setting q = 0, t = 1 and reversing the y variables.5.2. Winding permutations. We will apply Theorem 5.1.1 in cases for which the twistingpermutation has a special form, allowing the Hall-Littlewood polynomial F σ a ( y ; q ) to bewritten another way. Definition 5.2.1. Let [ x ] = x − ⌊ x ⌋ denote the fractional part of a real number x . Let c , . . . , c l be the sequence of fractional parts c i = [ a i + b ] of an arithmetic progression, where a is assumed irrational, so the c i are distinct. Let σ ∈ S l be the permutation such that σ ( c , . . . , c l ) is increasing, i.e., σ (1) , . . . , σ ( l ) are in the same relative order as c , . . . , c l .A permutation σ of this form is a winding permutation . The descent indicator of σ is thevector ( η , . . . , η l − ) defined by(107) η i = ( σ ( i ) > σ ( i + 1) , σ ( i ) < σ ( i + 1) . The head and tail of the winding permutation σ are the respective permutations τ, θ ∈ S l − such that τ (1) , . . . , τ ( l − 1) are in the same relative order as σ (1) , . . . , σ ( l − θ (1) , . . . , θ ( l − 1) are in the same relative order as σ (2) , . . . , σ ( l ).Adding an integer to a in the above definition doesn’t change the c i , so we can assumethat 0 < a < 1. In that case the descent indicator of σ is characterized by(108) η i = 1 ⇔ c i > c i +1 ⇔ c i +1 = c i + a − ,η i = 0 ⇔ c i < c i +1 ⇔ c i +1 = c i + a. Proposition 5.2.2. Let σ ∈ S l be a winding permutation, with descent indicator η , andhead and tail τ, θ ∈ S l − . For every λ ∈ Z l − we have identities E θ − λ ( x ; q ) = x η E τ − λ − η ( x ; q ) , (109) F θ − λ ( x ; q ) = x η F τ − λ − η ( x ; q )(110) of Laurent polynomials in x , . . . , x l − . The proof uses the following lemma. Lemma 5.2.3. With τ, θ and η as in Proposition 5.2.2, and for every w ∈ S l − , there is anidentity of operators on k [ x ± , . . . , x ± l − ](111) T − τw T τ x − η T − θ T θw = q e x − w − ( η ) , for some exponent e depending on w . SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 29 Proof. We prove (111) by induction on the length of w . The base case w = 1 is trivial.Suppose now that w = vs i is a reduced factorization. We can write the left hand side of(111) as(112) T ε i T − τv T τ x − η T − θ T θv T ε i , where(113) ε = ( +1 if τ vs i < τ v − τ vs i > τ v ε = ( +1 if θvs i > θv − θvs i < θv . Assuming by induction that (111) holds for v , and substituting this into (112), we are leftto show that(114) T ε i x − v − ( η ) T ε i = q e x − s i v − ( η ) for some exponent e . We now consider the possible values for h α ∨ i , − v − ( η ) i , which is equalto η k − η j , where v ( i ) = j , v ( i + 1) = k .Case 1: If η j = η k , we see from (108) that c j +1 − c j = c k +1 − c k , hence c j +1 < c k +1 ⇔ c j < c k .This implies that σ ( j + 1) < σ ( k + 1) ⇔ σ ( j ) < σ ( k ), and therefore that τ v ( i ) < τ v ( i + 1) ⇔ θv ( i ) < θv ( i + 1). Hence, in this case we have ε = − ε .Case 2: If η j = 1 and η k = 0, then from (108) we get c j +1 = c j + a − c k +1 = c k + a .Then c k +1 − c j +1 = c k − c j + 1. Since c k +1 − c j +1 and c k − c j both have absolute value lessthan 1, this implies c k < c j and c k +1 > c j +1 . It follows in the same way as in Case 1 that τ v ( i ) > τ v ( i + 1) and θv ( i ) < θv ( i + 1). Hence, in this case we have ε = ε = 1.Case 3: If η j = 0 and η k = 1 we reason as in Case 2, but with j and k exchanged, toconclude that in this case we have ε = ε = − T − i x µ T i = T i x µ T − i = x µ = x s i µ if h α ∨ i , µ i = 0 T i x µ T i = q x s i µ if h α ∨ i , µ i = − T − i x µ T − i = q − x s i µ if h α ∨ i , µ i = 1 (cid:3) Proof of Proposition 5.2.2. Let w ∈ S l and w ′ ∈ S l − be the longest permutations. Then w ′ θ , w ′ τ are the head and tail of the winding permutation w σ , and the descent indicatorof w σ is η ′ i = 1 − η i . Using these facts and the definition (75) of F πλ ( x ; q ), one can checkthat (109) implies (110).To prove (109), we begin by observing that for any given λ there exists w ∈ S l such thatboth w ( λ ) and w ( λ − η ) are dominant. To see this, first choose any v such that v ( λ ) = λ + is dominant. Since η is { , } -valued, the weight µ = v ( λ − η ) = λ + − v ( η ) has the propertythat for all i < j , if µ i < µ j then ( λ + ) i = ( λ + ) j . Hence, there is a permutation u thatfixes λ + and sorts µ into weakly decreasing order, so u ( µ ) = uv ( λ − η ) is dominant. Since uv ( λ ) = λ + is also dominant, w = uv works. Now, Lemma 5.2.3 implies(116) T − τw T τ x − η T − θ T θw ( x w − ( λ ) ) ∼ x − w − ( η ) x w − ( λ ) , where ∼ signifies that the expressions are equal up to a q power factor. Equivalently,(117) T − θ T θw x w − ( λ ) ∼ x η T − τ T τw x w − ( λ − η ) . Writing out the definitions of E θ − λ and E τ − λ − η , while ignoring q power factors, and using thefact that λ + = w − ( λ ) and ( λ − η ) + = w − ( λ − η ) for this w , (117) implies that (109) holdsup to a scalar factor q e . But we know that the x λ term on each side has coefficient 1, so(109) holds exactly. (cid:3) Stable shuffle theorem. We now prove an identity of formal power series with co-efficients in GL l characters, that is, symmetric Laurent polynomials in variables x , . . . , x l .When truncated to the polynomial part, this identity will reduce to our shuffle theorem forpaths under a line (Theorem 5.5.1). Theorem 5.3.1. Let p, s be real numbers with p positive and irrational. For i = 1 , . . . , l , let b i = ⌊ s − p ( i − ⌋ − ⌊ s − pi ⌋ . Let c i = [ s − p ( i − , and let σ ∈ S l be the permutation such that σ (1) , . . . , σ ( l ) are inthe same relative order as c l , . . . , c , i.e., σ ( c l , . . . , c ) is increasing. For any non-negativeintegers u, v we have the identity of formal power series in t (118) H b + u,b ,...,b l − ,b l − v = X a ,...,a l − ≥ t | a | L σ (( b l ,...,b )+( − v,a l − ,...,a )) / ( a l − ,...,a , − u ) ( x ; q ) , where H b is given by Definition 3.7.1.Remark . If δ is the highest south-east lattice path weakly below the line y + px = s ,starting at (0 , ⌊ s ⌋ ) and extending forever (not stopping at the x axis), then b i is the numberof south steps in δ along the line x = i − 1, and c i is the gap along x = i − δ beneath it. Proof of Theorem 5.3.1. We will prove that for b , σ as in the hypothesis of the theorem, wehave the stronger ‘unstraightened’ identity(119) x u x − vl x b Q i +1 In particular, b l ≤ b l − + 1, hence b l − v ≤ b l − + a l − + 1, and if equality holds, then b l − = ⌊ p ⌋ , so σ (1) < σ (2). Using Lemma 4.3.4, and recalling that the definition (75) of F σλ ( x ; q ) is E σw − λ ( x ; q ), we have E σ − ( a l − ,...,a , − u ) ( x ; q ) = x − ul E τ − ( a l − ,...,a ) ( x , . . . , x l − ; q )(121) F σ − ( b l ,...,b )+( − v,a l − ,...,a ) ( x ; q ) = x b l − v F θ − ( b l − ,...,b )+( a l − ,...,a ) ( x , . . . , x l ; q ) , (122)where τ , θ are the head and tail of σ , as in Proposition 5.2.2. Note that σ is a windingpermutation. From (120) we also see that ( b l − , . . . , b ) = η + ⌊ p ⌋ · (1 , . . . , η is thedescent indicator of σ . Adding a constant vector k · (1 , . . . , 1) to the index λ multiplies any F πλ ( x ; q ) by ( Q i x i ) k . Using this and Proposition 5.2.2, we can replace (122) with(123) F σ − ( b l ,...,b )+( − v,a l − ,...,a ) ( x ; q ) = x b l − v x b l − · · · x b l F τ − ( a l − ,...,a ) ( x , . . . , x l ; q )Using the Cauchy formula (104) from Theorem 5.1.1 in l − τ − , and substituting x − i for x i , we obtain(124) Q i Our next goal is to deduce the combinatorial versionof our shuffle theorem—that is, the identity (1) previewed in the introduction and restated as(133), below—from Theorem 5.3.1. To do this we first need to define the data that will serveto attach LLT polynomials to lattice paths, and relate these to the combinatorial statisticdinv p ( λ ).We will be concerned with lattice paths λ lying weakly below the line segment(125) y + p x = s ( p = s/r )between arbitrary points (0 , s ) and ( r, 0) on the positive y and x axes.We always assume that the slope − p of the line is irrational. Clearly it is possible toperturb any line slightly so as make its slope irrational, without changing the set of latticepoints, and therefore also the set of lattice paths, that lie below the line. All dependence on p in the combinatorial constructions to follow comes from comparisons between p and variousrational numbers. By taking p to be irrational, we avoid the need to resolve ambiguities thatwould result from equality occurring in the comparisons. Definition 5.4.1. Let λ be a south-east lattice path in the first quadrant with endpointson the axes. Let Y be the Young diagram enclosed by the positive axes and λ . The arm and leg of a box y ∈ Y are, as usual, the number of boxes in Y strictly east of y and strictlynorth of y , respectively. Given a positive irrational number p , we define dinv p ( λ ) to be the Figure 3. number of boxes in Y whose arm a and leg l satisfy(126) la + 1 < p < l + 1 a , where we interpret ( l + 1) /a as + ∞ if a = 0.Geometrically, condition (126) means that some line of slope − p crosses both the east stepin λ at the top of the leg and the south step at the end of the arm, as shown in Figure 3.Since p is irrational, such a line can always be assumed to pass through the interiors of thetwo steps.To each lattice path weakly below the line (125) we now attach a tuple of one-row skewshapes β/α and a permutation σ . The index ν ( λ ) of the LLT polynomial in (1) and (133)will be defined in terms of these data. Definition 5.4.2. Let λ be a south-east lattice path from (0 , ⌊ s ⌋ ) to ( l − , 0) which is weaklybelow the line y + p x = s in (125), where l − ≤ r and p is irrational. For i = 1 , . . . , l , let(127) d i = ⌊ s − p ( i − ⌋ be the y coordinate of the highest lattice point weakly below the given line at x = i − 1. Let(128) α = ( α l , . . . , α ) , β = ( β l , . . . , β )be the vectors of integers 0 ≤ α i ≤ β i , written in reverse order, such that the south steps in λ on the line x = i − y = d i − α i to y = d i − β i . Let(129) c i = s − p ( i − − d i = [ s − p ( i − x = i − 1. Let σ ∈ S l be the permutation with σ (1) , . . . , σ ( l ) in the same relative order as c l , . . . , c , i.e., such that σ ( c l , . . . , c ) is increasing. The vectors α and β and the permutation σ are the LLT data associated with λ and the given line. Example . The first diagram in Figure 4 shows a line y + px = s with p ≈ . s ≈ . λ below it from (0 , ⌊ s ⌋ ) = (0 , 9) to ( l − , 0) = (6 , 0) with l = 7.In this example, the y coordinates of the highest lattice points below the line at x = 0 , . . . , d , . . . , d ) = (9 , , , , , , λ go from y coordinates SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 33 ( β ) / ( α ) = (1) / (1)( β ) / ( α ) = (2) / (1)( β ) / ( α ) = (2) / (2)( β ) / ( α ) = (4) / (2)( β ) / ( α ) = (3) / (0)( β ) / ( α ) = (1) / (1)( β ) / ( α ) = (3) / (0) Figure 4. (i) A path λ under y + px = s with p ≈ . s ≈ . l = 7. (ii)Transformed path λ ′ under y = s , with gaps c i marked. (iii) Bottom to top:tuple of rows ( β , . . . , β ) / ( α , . . . , α ) offset by ( c , . . . , c ).(9 , , , , , , 0) to (6 , , , , , , d i and listing them in reverseorder gives(130) α = (1 , , , , , , , β = (1 , , , , , , . The gaps, in reverse order, are ( c , . . . , c ) ≈ ( . , . , . , . , . , . , . σ = (1 , , , , , , . Proposition 5.4.4. Given the line (125) and a lattice path λ weakly below it satisfying theconditions in Definition 5.4.2, let α, β, σ be the associated LLT data. Then (132) dinv p ( λ ) = h σ ( β/α ) , where dinv p ( λ ) is given by Definition 5.4.1 and h σ ( β/α ) is as in Corollary 4.5.7.Proof. Let λ ′ be the image of λ under the transformation in the plane that sends ( x, y )to ( x, y + px ). Then λ ′ is a path composed of unit south steps and sloped steps (1 , p )(transforms of east steps), which starts at (0 , ⌊ s ⌋ ) and stays weakly below the horizontal line y = s (transform of the line y + p x = s ).The south steps in λ ′ on the line x = i − y = s − ( c i + α i ) to y = s − ( c i + β i ). Thismeans that if we offset the i -th row ( β l +1 − i ) / ( α l +1 − i ) in the tuple of one-row skew diagrams β/α by c l +1 − i , then the x coordinate on each box of β/α covers the same unit interval asdoes the distance below the line y = s on the south step in λ ′ corresponding to that box.See Figure 4 for an example.Since 0 < c i < c l , . . . , c are in the same relative order as σ (1) , . . . , σ ( l ),the description of h σ ( β/α ) in Remark 4.5.6 still applies if we offset row i by c l +1 − i insteadof ǫσ ( i ). Mapping this onto λ ′ , we see that h σ ( β/α ) is the number of horizontal alignmentsbetween any endpoint of a step in λ ′ and the interior of a south step occurring later in λ ′ . Toput this another way, for each south step S in λ ′ , let B S denote the interior of the horizontal S S S S Figure 5. band of height 1 to the left of S in the plane. Then h σ ( β/α ) is the number of pairs consistingof a south step S and a point P ∈ B S which is an endpoint of a step in λ ′ .For comparison, dinv p ( λ ) is the number of pairs consisting of a south step S in λ ′ and asloped step which meets B S . To complete the proof it suffices to show that each band B S contains the same number of step endpoints P as the number of sloped steps that meet B S .In fact, we make the following stronger claim: within each band B S , step endpoints alternatefrom left to right with fragments of sloped steps, starting with a step endpoint and endingwith a sloped step fragment. To see this, consider a connected component C of λ ′ ∩ B S . Each component C either enters B S from above along a south step or from below along a sloped step, and exits B S either atthe top along a sloped step or at the bottom along a south step, except in two degeneratesituations. One of these occurs if C contains the starting point (0 , ⌊ s ⌋ ) of λ ′ . In this case weregard C as entering B S from above. The other is if C contains a sloped step that adjoins S . Then we regard C as exiting B S at the top.Each component C thus belongs to one of four cases shown in Figure 5. Note that since B S has height 1, it cannot contain a full south step of λ ′ . In Figure 5 we have chosen p < B S might contain full sloped steps of λ ′ . If p > B S can only meet sloped steps in proper fragments.On each component C , step endpoints clearly alternate with sloped step fragments, start-ing with a step endpoint if C enters from above, or with a sloped step fragment if C entersfrom below, and ending with a step endpoint if C exits at the bottom, or with a sloped stepfragment if C exits at the top. Since the distance from the line y = s to the starting point of λ ′ is less than 1, the leftmost component C of λ ′ ∩ B S always enters B S from the top. Eachsubsequent component from left to right must enter B S from the same side (top or bottom)that the previous component exited. This implies the claim stated above. (cid:3) Shuffle theorem for paths under a line. We now prove the identity previewed as(1) in the introduction. Theorem 5.5.1. Let r, s > be positive real numbers with p = s/r irrational. We have theidentity (133) D b ,...,b l · X λ t a ( λ ) q dinv p ( λ ) ω ( G ν ( λ ) ( X ; q − )) , where the sum is over lattice paths λ from (0 , ⌊ s ⌋ ) to ( ⌊ r ⌋ , lying weakly below the line (125) through (0 , r ) and ( s, , and the other pieces of (133) are defined as follows. SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 35 The integer a ( λ ) is the number of lattice squares enclosed between λ and δ , where δ is thehighest path from (0 , ⌊ s ⌋ ) to ( ⌊ r ⌋ , weakly below the given line. The index b i is the numberof south steps in δ along the line x = i − , for i = 1 , . . . , l , where l = ⌊ r ⌋ + 1 . The integer dinv p ( λ ) is given by Definition 5.4.1.The LLT polynomial G ν ( λ ) ( X ; q ) is indexed by a tuple of one-row skew shapes ν ( λ ) = σ ( β/α ) = ( β σ − (1) , . . . , β σ − ( l ) ) / ( α σ − (1) , . . . , α σ − ( l ) ) , where α, β and σ are the LLT data as-sociated to λ in Definition 5.4.2.The operator D b ,...,b l on the left hand side is a Negut element in E , as defined in § D b ,...,b l · satisfies (58) .Proof. We prove the equivalent identity(134) ω ( D b ,...,b l · 1) = X λ t a ( λ ) q dinv p ( λ ) G ν ( λ ) ( X ; q − ) . By Corollary 3.7.2 and Lemma 4.1.6, both sides of (134) involve only Schur functions s λ ( X )indexed by partitions such that ℓ ( λ ) ≤ l . It therefore suffices to prove that (134) holdswhen evaluated in l variables x , . . . , x l . After doing this and using the formula (58) fromCorollary 3.7.2, the desired identity becomes(135) ( H b ) pol = X λ t a ( λ ) q dinv p ( λ ) G ν ( λ ) ( x , . . . , x l ; q − ) . This is the same identity (24) that was mentioned in the introduction to § 3. We now proveit using Theorem 5.3.1.Let b ′ i = ⌊ s − p ( i − ⌋ − ⌊ s − pi ⌋ . As in Remark 5.3.2, this is the number of south stepsalong x = i − δ ′ under our given line, where δ ′ starts at(0 , ⌊ s ⌋ ) and extends forever. For i < l we have b ′ i = b i . On the line x = l − ⌊ r ⌋ , however,the path δ stops at ( l − , δ ′ may extend below the x -axis, giving b l ≤ b ′ l .We now apply Theorem 5.3.1 with b ′ i in place of b i , u = 0, and v = b ′ l − b l , and then takethe polynomial part on both sides of (118). This gives the same left hand side as in (135).On the right hand side, by Corollary 4.5.7, only those terms survive for which the index a satisfies ( a l − , . . . , a , ≤ ( b l , . . . , b ) + (0 , a l − , . . . , a ) in each coordinate, that is, for which(136) a l − ≤ b l and a i ≤ a i +1 + b i +1 for i = 1 , . . . , l − . Now, (136) is precisely the condition for there to exist a (unique) lattice path λ from(0 , ⌊ s ⌋ ) to ( ⌊ r ⌋ , 0) such that a i is the number of lattice squares in the i -th column (definedby x ∈ [ i − , i ]) of the region between λ and the highest path δ . Moreover, when (136)holds, the LLT data for λ , as in Definition 5.4.2, are given by(137) β = ( b l , . . . , b ) + (0 , a l − , . . . , a ) ,α = ( a l − , . . . , a , , and σ ∈ S l such that σ (1) , . . . , σ ( l ) are in the same relative order as c l , . . . , c , where c i =[ s − p ( i − L σ (( b l ,...,b )+(0 ,a l − ,...,a )) / ( a l − ,...,a , ( x ; q ) pol = q dinv p ( λ ) G ν ( λ ) ( x ; q − ) . G / ( X ; q ) = s + ( q + q ) s + q s + q s area = 0 dinv p = 4 G / ( X ; q ) = s + q s area = 1 dinv p = 2 G / ( X ; q ) = s + ( q + q ) s + q s + q s area = 1 dinv p = 3 G / ( X ; q ) = s + q s area = 2 dinv p = 1 G / ( X ; q ) = s + q s + q s area = 2 dinv p = 2 G / ( X ; q ) = s + q s area = 3 dinv p = 1 G / ( X ; q ) = s area = 4 dinv p = 0 Figure 6. An illustration of Theorem 5.5.1 as described in Example 5.5.3. When (136) holds we clearly also have a ( λ ) = | a | . This shows that the polynomial part ofthe right hand side in (118) is the same as the right hand side of (135). (cid:3) Remark . The preceding argument also goes through with u > λ that start at a higher point (0 , n ) on the y axis, with n = ⌊ s ⌋ + u , go directly south to(0 , ⌊ s ⌋ ), and then continue below the given line to ( l − , 0) as before.The corresponding modifications to (133) are (i) the index b on the left hand side is thenumber of south steps in λ on the y axis including the extension to (0 , n ), and (ii) the rowin ν ( λ ) corresponding to south steps in λ on the y axis is also extended accordingly. Example . Figure 6 illustrates Theorem 5.5.1 for s ≈ . r ≈ . 31. We have( c , c , c , c ) ≈ ( . , . , . , . 70) and σ = (1 , , , (2 , , , λ are shownalong with the corresponding statistics and LLT polynomials G ν ( λ ) ( X ; q ) = G σ ( β ) /σ ( α ) ( X ; q ).The highest path δ is the one at the top left in the figure and ( b , b , b , b ) = (1 , , , l = 4 variables, is then(139) ω ( D , , , · x ) = H (1 , , , ( x ) pol = σ (cid:0) x x x (1 − q t x /x )(1 − q t x /x )(1 − q t x /x ) Q ≤ i Definition 5.5.4. For b ∈ Z l , the generalized q, t -Catalan number C b ( q, t ) is the coefficientof the single row Schur function s ( | b | ) ( X ) in ω ( D b · b = 1 l , C b ( q, t ) is the q, t -Catalan number introduced by Garsia and the secondauthor [9]. The generalized q, t -Catalan numbers have been studied in [12]—see § Corollary 5.5.5. With b = ( b , . . . , b l ) , r , s , and p = s/r as in Theorem 5.5.1, C b ( q, t ) = X λ t a ( λ ) q dinv p ( λ ) , (140) where the sum is over lattice paths λ from (0 , ⌊ s ⌋ ) to ( ⌊ r ⌋ , lying weakly below the linethrough (0 , r ) and ( s, . Relation to previous shuffle theorems Theorem 5.5.1 is formulated a little differently than the classical and ( km, kn ) shuffletheorems in [4, 15], although these also have an algebraic side and a combinatorial sideresembling ours. We now explain how to recover them from our version by transformingeach side of (133) into its counterpart in the ( km, kn ) and classical shuffle conjectures.For the ( km, kn ) shuffle conjecture, we take the line y + px = s in (125) to be a perturbationof the line from (0 , kn ) to ( km, s = kn and r slightly larger than km . Our perturbedline has the same lattice points and paths under it as the line from (0 , kn ) to ( km, − p , where p = n/m − ǫ for a small ǫ > 0. The classical shuffle conjectures in[15] are the special cases of the ( km, kn ) conjecture with n = 1. For these we perturb theline from (0 , k ) to ( km, 0) in the same way. Note that for our chosen line we have l = km + 1,and every lattice path λ under it has b l = 0 south steps at x = km .The classical shuffle conjecture was formulated in [15, Conjecture 6.2.2] as the identity(141) ∇ m e k = X λ X P ∈ SSYT (( λ +(1 k )) /λ ) t a ( λ ) q dinv m ( P ) x P , where the sum is over lattice paths λ below the bounding line, and dinv m ( P ) is a statisticdefined in [15], attached to each labelling P of the south steps in λ by non-negative integersstrictly increasing from south to north along each vertical run. The left hand side of (141)is e k [ − M X m, ] · D b ,...,b l · ν ( λ ) = σ ( β/α ), apply Proposition 4.1.4to replace ω G ν ( λ ) ( X ; q − ) with q − I ( ν ( λ ) R ) G ν ( λ ) R ( X ; q ). Then writing out G ν ( λ ) R ( X ; q ) term byterm with tableaux on the tuple ν ( λ ) R of one-column diagrams using Definition 4.1.1 gives(142) D b ,...,b l · X λ X T ∈ SSYT( ν ( λ ) R ) t a ( λ ) q dinv p ( λ ) − I ( ν ( λ ) R )+inv( T ) x T , where inv( T ) is defined in § By the construction, boxes in each column of ν ( λ ) R , from top to bottom, correspondto south steps u in a vertical run in λ , from north to south. Semistandard tableaux T ∈ SSYT( ν ( λ ) R ) therefore biject with labellings P T : { south steps in λ } → N such that P T isstrictly increasing from south to north on each vertical run in λ ; namely, P T ∈ SSYT(( λ +(1 k )) /λ ). Changing (142) to instead sum over labellings, we can match the right hand sidesof (142) and (141) by showing that for p = 1 /m − ǫ ,(143) dinv m ( P T ) = dinv p ( λ ) − I ( ν ( λ ) R ) + inv( T ) . For any super tableau T , [15, Corollary 6.4.2] implies that dinv m ( P T ) = e λ + inv( T ) for anoffset e λ not depending on T . For the tableau T with all entries ¯1, [15, Lemma 6.3.3] givesthat dinv m ( P T ) = b m ( λ ), where we note that b m ( λ ) defined in [15, (100)] is simply dinv p ( λ )with p = 1 /m − ǫ . Therefore, e λ = dinv m ( P T ) − inv( T ) = dinv p ( λ ) − I ( ν ( λ ) R ) by (67).In fact, there is a direct correspondence between the combinatorics of dinv m ( P ) for paths,as defined in [15], and that of triples in negative tableaux on a tuple of single-row shapes, asconsidered in § Proposition 6.1.1. Let λ be a lattice path from (0 , k ) to ( km, , lying weakly below thebounding line y + p x = k with p = 1 /m − ǫ . Let α , β , σ be the LLT data associated to λ for this p . There is a weight-preserving bijection from labellings P ∈ SSYT(( λ + (1 k )) /λ ) tonegative tableaux S ∈ SSYT − ( β/α ) such that (144) dinv m ( P ) = h σ ( S ) . Proof. labelling P = P T ∈ SSYT(( λ + (1 k )) /λ ) corresponds naturally to a semistandardtableau T ∈ SSYT( ν ( λ ) R ). Their statistics are related by (143), into which we can substitutedinv p ( λ ) = h σ ( β/α ) by Proposition 5.4.4. The bijection T T R in the proof of Proposi-tion 4.1.4 satisfies inv( T ) − I ( ν ( λ ) R ) = − inv( T R ). Hence, dinv m ( P T ) = h σ ( β/α ) − inv( T R ).To complete the bijection, take S = T R ◦ σ . Then h σ ( β/α ) − inv( T R ) = h σ ( S ) byLemma 4.5.3, proving (144). (cid:3) See Figure 7 for an example with m = 1 and p = 1 − ǫ . Note that these values give σ = w in the LLT data.Next we turn to the non-compositional ( km, kn ) shuffle conjecture from [4]. Its symmetricfunction side is precisely the Schiffmann algebra operator expression that we denote here by e k [ − M X m,n ] · 1. By Corollary 3.7.3, this agrees with the left hand side D b ,...,b l · km, kn ) shuffle conjecture can be written as in [4, § X u X π ∈ Park( u ) t area( u ) q dinv( u )+tdinv( π ) − maxtdinv(u) F ides( π ) ( x ) . Here u encodes a north-east lattice path lying above the line from (0 , 0) to ( km, kn ), Park( u )encodes the set of standard Young tableaux on a tuple of columns corresponding to verticalruns in the path encoded by u , and F γ ( x ) is a Gessel fundamental quasi-symmetric function. SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 39 Figure 7. Example of the bijection P = P T ↔ T ↔ T R ↔ S = T R ◦ σ inProposition 6.1.1, with m = 1, p = 1 − ǫ , σ = w . Letters in S are ordered1 > > · · · . We see dinv ( P ) = h w ( S ) = 6.532 6 31 4 P = S = −∞ ∞ −∞ ∞ −∞ ∞−∞ ∞ −∞ ∞−∞ ∞−∞ ∞ u correspond to a lattice path λ under the line from (0 , kn ) to ( km, π ( j ) with kn + 1 − π ( j ) so theresulting standard tableau on ν ( λ ) R has columns increasing upwards, as it should, insteadof decreasing. Using [4, Definition 7.1] and taking account the modification to π , we cantranslate notation in (145) as follows: area( u ) = a ( λ ), tdinv( π ) = i ( π ), maxtdinv( π ) = I ( ν ( λ ) R ), and dinv( u ) = dinv p ( λ ), where p = n/m − ǫ .Finally, the definition of ides( π ) becomes the descent set of π relative to the readingorder on ν ( λ ) R . This implies that expanding F ides( π ) ( x ) into monomials gives a sum withsemistandard tableaux T in place of standard tableaux π and x T in place of F ides( π ) ( x ). Afterthese substitutions, (145) coincides with the right hand side of (142).7. A positivity conjecture ω ( D b · 1) = H q,t (cid:18) x b Q i (1 − q t x i /x i +1 ) (cid:19) pol is q, t Schur positive when b i is the number of south steps along x = i − x and y axes. Computational evidence leadsus to conjecture that (146) is q, t Schur positive under a more general geometric conditionon b .Let C be a convex curve (meaning that the region above C is convex) in the first quadrantwith endpoints on the positive x and y axes. Let δ be the highest lattice path strictly below C , and let b i be the number of south steps in δ along x = i − i = 1 , . . . , l , where C meets the x -axis at ( r, 0) with l − < r ≤ l .Algebraically, this means that there is a sequence of real numbers s ≥ s ≥ · · · ≥ s l = 0with weakly decreasing differences s i − − s i ≥ s i − s i +1 , such that b i = ⌈ s i − ⌉ − ⌈ s i ⌉ for i = 1 , . . . , l − b l = ⌈ s l − ⌉ − 1. Note that if δ is the highest lattice path weakly below a curve C ′ , then δ is also the highest path strictly below a curve C slightly above C ′ . Thecondition on δ is therefore more general than it would be with “weakly” in place of “strictly.” Conjecture 7.1.1. When b i is the number of south steps along x = i − in the highest latticepath below a convex curve, as above, the symmetric function in (146) is a linear combinationof Schur functions with coefficients in N [ q, t ] . At q = 1, the q -Kostka coefficients reduce to K λ,µ (1) = K λ,µ = h s λ , h µ i . Hence, the Hall-Littlewood symmetrization operator reduces to H q ( x µ ) pol | q =1 = h µ ( x ) if µ i ≥ i , andotherwise H q ( x µ ) pol = 0. At q = 1, the factors containing t in (46) cancel, so H q,t reducesto the same thing as H q .It follows that (146) specializes at q = 1 to(147) ω ( D b · | q =1 = X a ,...,a l − ≥ t | a | h b +( a ,a − a ,...,a l − − a l − , − a l − ) , with the convention that h µ = 0 if µ i < i . As in Theorem 5.5.1, the index b + ( a , a − a , . . . , a l − − a l − , − a l − ) is non-negative precisely when it is the sequence b ( λ )of lengths of south runs in a lattice path λ lying below the path δ whose south runs aregiven by b . Here a i is the number of lattice squares in column i between λ and δ , so | a | isthe area a ( λ ) enclosed between the two paths. This gives a combinatorial expression(148) ω ( D b · | q =1 = X λ t a ( λ ) h b ( λ ) , for (146) at q = 1, which is positive in terms of complete homogeneous symmetric functions h λ , hence t Schur positive. We may conjecture that when the hypothesis of Conjecture 7.1.1holds, ω ( D b · 1) is given by some Schur positive combinatorial q -analog of (148), but itremains an open problem to formulate such a conjecture precisely.Of course, (148) cannot be considered evidence for Conjecture 7.1.1, since (148) holds forany b ≥ 0, whether the convexity hypothesis is satisfied or not.7.2. Relation to previous conjectures. The generalized q, t -Catalan numbers C b ( q, t ) = h s ( | b | ) ( X ) , ω ( D b · i from Definition 5.5.4 coincide with the functions denoted F ( b , . . . , b l )in [12], where several equivalent expressions for them were obtained. To see that C b ( q, t ) = F ( b , . . . , b l ), one can compare the formula in Proposition 7.2.1, below, with the equationjust before (2.6) in [12]. It was also shown in [12] that this quantity does not depend on b ,hence the notation F ( b , . . . , b l ).Conjecture 7.1.1 implies a conjecture of Negut, announced in [12], which asserts that C b ( q, t ) ∈ N [ q, t ] when b ≥ · · · ≥ b l . Conjecture 7.1.1 is stronger than Negut’s conjecture intwo ways: the weight b is generalized from a partition to the highest path below a convexcurve, and the coefficient of s ( | b | ) ( X ) in ω ( D b · 1) is generalized to the coefficient of any Schurfunction. SHUFFLE THEOREM FOR PATHS UNDER ANY LINE 41 Proposition 7.2.1. The generalized q, t -Catalan number C b ( q, t ) has the following descrip-tion as a series coefficient: C b ( q, t ) = h z − b i l Y i =1 − z − i l − Y i =1 − q t z i /z i +1 Y i From (50) we have ω ( D b · 1) = h z i z b Q l − i =1 (1 − q t z i /z i +1 ) Y i Non-symmetric macdonald polynomials and demazure-lusztig operators , 2016,1602.05153.[2] Per Alexandersson and Mehtaab Sawhney, Properties of non-symmetric Macdonald polynomials at q = 1 and q = 0, Ann. Comb. (2019), no. 2, 219–239.[3] F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for someremarkable operators in the theory of symmetric functions , Methods Appl. Anal. (1999), no. 3, 363–420, Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part III.[4] Francois Bergeron, Adriano Garsia, Emily Sergel Leven, and Guoce Xin, Compositional ( km, kn ) -shuffleconjectures , Int. Math. Res. Not. IMRN (2016), no. 14, 4229–4270.[5] Igor Burban and Olivier Schiffmann, On the Hall algebra of an elliptic curve, I , Duke Math. J. (2012), no. 7, 1171–1231.[6] Erik Carlsson and Anton Mellit, A proof of the shuffle conjecture , J. Amer. Math. Soc. (2018), no. 3,661–697.[7] B. L. Feigin and A. I. Tsymbaliuk, Equivariant K -theory of Hilbert schemes via shuffle algebra , KyotoJ. Math. (2011), no. 4, 831–854.[8] Amy M. Fu and Alain Lascoux, Non-symmetric Cauchy kernels for the classical groups , J. Combin.Theory Ser. A (2009), no. 4, 903–917.[9] A. M. Garsia and M. Haiman, A remarkable q, t -Catalan sequence and q -Lagrange inversion , J. AlgebraicCombin. (1996), no. 3, 191–244.[10] , Some natural bigraded S n -modules and q, t -Kostka coefficients , Electron. J. Combin. (1996),no. 2, Research Paper 24, approx. 60 pp. (electronic), The Foata Festschrift.[11] A. M. Garsia, M. Haiman, and G. Tesler, Explicit plethystic formulas for Macdonald q, t -Kostka co-efficients , S´em. Lothar. Combin. (1999), Art. B42m, 45 pp. (electronic), The Andrews Festschrift(Maratea, 1998).[12] Eugene Gorsky, Graham Hawkes, Anne Schilling, and Julianne Rainbolt, Generalized q, t -Catalan num-bers , Algebr. Comb. (2020), no. 4, 855–886.[13] I. Grojnowski and M. Haiman, Affine Hecke algebras and positivity of LLT and Macdonald polynomials ,Unpublished manuscript, 2007.[14] J. Haglund, M. Haiman, and N. Loehr, A combinatorial formula for Macdonald polynomials , J. Amer.Math. Soc. (2005), no. 3, 735–761 (electronic).[15] J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov, A combinatorial formula for thecharacter of the diagonal coinvariants , Duke Math. J. (2005), no. 2, 195–232.[16] J. Haglund, J. Morse, and M. Zabrocki, A compositional shuffle conjecture specifying touch points ofthe Dyck path , Canad. J. Math. (2012), no. 4, 822–844. [17] Mark Haiman, Macdonald polynomials and geometry , New perspectives in algebraic combinatorics(Berkeley, CA, 1996–97), Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge,1999, pp. 207–254.[18] , Vanishing theorems and character formulas for the Hilbert scheme of points in the plane , Invent.Math. (2002), no. 2, 371–407.[19] Tatsuyuki Hikita, Affine Springer fibers of type A and combinatorics of diagonal coinvariants , Adv.Math. (2014), 88–122.[20] Bogdan Ion, Standard bases for affine parabolic modules and nonsymmetric Macdonald polynomials , J.Algebra (2008), no. 8, 3480–3517.[21] Nai Huan Jing, q -hypergeometric series and Macdonald functions , J. Algebraic Combin. (1994), no. 3,291–305.[22] Alain Lascoux, Double crystal graphs , Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000),Progr. Math., vol. 210, Birkh¨auser Boston, Boston, MA, 2003, pp. 95–114.[23] Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Ribbon tableaux, Hall-Littlewood functions,quantum affine algebras, and unipotent varieties , J. Math. Phys. (1997), no. 2, 1041–1068.[24] George Lusztig, Affine Hecke algebras and their graded version , J. Amer. Math. Soc. (1989), no. 3,599–635.[25] I. G. Macdonald, Symmetric functions and Hall polynomials , second ed., The Clarendon Press, OxfordUniversity Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications.[26] , Affine Hecke algebras and orthogonal polynomials , Cambridge Tracts in Mathematics, vol. 157,Cambridge University Press, Cambridge, 2003.[27] Anton Mellit, Toric braids and (m,n)-parking functions , Preprint, arXiv:1604.07456, 2016.[28] Andrei Negut, The shuffle algebra revisited , Int. Math. Res. Not. IMRN (2014), no. 22, 6242–6275.[29] Olivier Schiffmann, On the Hall algebra of an elliptic curve, II , Duke Math. J. (2012), no. 9,1711–1750.[30] Olivier Schiffmann and Eric Vasserot, The elliptic Hall algebra and the K -theory of the Hilbert schemeof A , Duke Math. J. (2013), no. 2, 279–366.(Blasiak) Dept. of Mathematics, Drexel University, Philadelphia, PA Email address : [email protected] (Haiman) Dept. of Mathematics, University of California, Berkeley, CA Email address : [email protected] (Morse) Dept. of Mathematics, University of Virginia, Charlottesville, VA Email address : [email protected] (Pun) Dept. of Mathematics, University of Virginia, Charlottesville, VA Email address : [email protected] (Seelinger) Dept. of Mathematics, University of Virginia, Charlottesville, VA Email address ::