aa r X i v : . [ m a t h . R A ] S e p GREEDY BASES IN RANK 2 GENERALIZED CLUSTER ALGEBRAS
DYLAN RUPEL
To the memory of Andrei Zelevinsky
Abstract.
In this note we extend the notion of greedy bases developed by Lee, Li, and Zelevinsky torank two generalized cluster algebras, i.e. binomial exchange relations are replaced by polynomial exchangerelations. In the process we give a combinatorial construction in terms of a refined notion of compatiblepairs on a maximal Dyck path.
Contents
1. Introduction 11.1. Organization 21.2. Related and Future Work 2Acknowledgements 22. Rank 2 Generalized Cluster Algebras and their Greedy Bases 32.1. Structure of Rank 2 Generalized Cluster Algebras 32.2. Greedy Elements in Rank 2 Generalized Cluster Algebras 62.3. Combinatorial Construction of Greedy Elements 83. Structure of Maximal Dyck Paths 134. Combinatorics of Graded Compatible Pairs: Shadows 144.1. Unbounded Gradings 144.2. Bounded Gradings 194.3. Magnitudes of Bounded Gradings 215. Appendix: Multinomial Coefficients 23References 251.
Introduction
At the heart of the theory of cluster algebras is the hope for a combinatorially defined basis of a slightlycomplicated inductively defined algebra. This is motivated by the conjectural relationship with the dualcanonical basis: the cluster monomials should be contained in this basis. Any such “good basis” of a clusteralgebra should satisfy two main conditions: it should be independent of the choice of an initial cluster andit should contain all cluster monomials.This note grew out of an attempt to understand one such “good basis”, namely the greedy basis, of arank 2 cluster algebra. This basis consists of indecomposable positive elements with a rich combinatorialdescription. Our main goal in this project was to understand these combinatorics and to see in exactlywhat generality such a basis should exist. To our surprise this basis is extremely general in the sense that itexists for all rank 2 generalized cluster algebras defined by arbitrary monic, palindromic polynomial exchangerelations with non-negative integer coefficients.Our main theorem is the following analogue of the main result from [LLZ], we refer the reader to thesections below for precise definitions.
Date : September 12, 2018. D. RUPEL
Theorem 1.1.
Let k denote an ordered field with positive cone Π and suppose P , P ∈ Π[ z ] are monic,palindromic polynomials. Denote by k the Z -subalgebra of k generated by the coefficients of P and P . Write A ( P , P ) for the associated generalized cluster algebra over k . (a) For each ( a , a ) ∈ Z there exists a (unique) greedy element x [ a , a ] ∈ A ( P , P ) . (b) Each greedy element is an indecomposable positive element of A ( P , P ) . (c) The greedy elements x [ a , a ] for ( a , a ) ∈ Z form a k -basis of the generalized cluster algebra A ( P , P ) , which we call the greedy basis . (d) The greedy basis is independent of the choice of initial cluster. (e)
The greedy basis contains all cluster monomials.
Organization.
The paper is organized as follows. Section 2 defines the generalized cluster algebra,establishes certain analogues of structural results from the classical theory of cluster algebras, and introducesthe greedy basis of a rank 2 generalized cluster algebra. This section also contains the proofs of our mainresults, some of them as corollaries of purely combinatorial statements contained in section 4. In section 3we introduce the underlying structure for the combinatorics we seek to understand, namely maximal Dyckpaths in a lattice rectangle of arbitrary size. Section 4 develops the combinatorics of graded compatiblepairs generalizing the combinatorics extensively studied in [LLZ]. In the appendix, section 5, we collect andprove auxiliary results related to multinomial coefficients necessary for working with arbitrary powers ofpolynomials.The organization is chosen to allow the section on generalized cluster algebras to be read almost completelyindependently of the combinatorial sections. In particular if one is willing to accept the technical results ofsection 4, especially Proposition 4.20 and Proposition 4.22, then section 2 and the proof of the main theoremcan be understood on their own.1.2.
Related and Future Work.
In [CS] the authors use a special class of generalized cluster algebras tounderstand the Teichm¨uller theory of hyperbolic orbifold surfaces. There they conjecture the positivity forthe initial cluster expansion of all generalized cluster variables. Theorem 1.1, in particular parts (b) and (e),establishes the positivity of the initial cluster expansion of all generalized cluster variables, verifying theirconjecture in the rank 2 case. Analogous to the inductive proof of positivity for skew-symmetric clusteralgebras of arbitrary rank given in [LS2], it seems likely that one could prove the positivity for generalizedcluster algebras of arbitrary rank inductively from our results presented here.Chekhov and Shapiro also offer a combinatorial description of their generalized cluster variables by pass-ing to the universal cover of the orbifold surface and applying the combinatorics of T -paths [S] with anappropriate coloring. In the case of a disk with two orbifold points (of arbitrary order) and one markedpoint on the boundary one obtains a rank 2 generalized cluster algebra. In this case one may describe anexplicit bijection between T -paths in the universal cover and our combinatorial theory of graded compatiblepairs, perhaps this bijection will be expounded upon in a future work.One of the most powerful tools in the study of classical cluster algebras is the categorification usingthe representation theory of valued quivers, for rank 2 cluster algebras this is established in [CZ] and[R1]. Here the non-initial cluster monomials exactly correspond to rigid representations of the quiver. Itwould be interesting to find a generalization of the representation theory of valued quivers for which rigidrepresentations correspond to non-initial generalized cluster monomials in A ( P , P ). In this case it seemslikely that the Caldero-Chapoton algebras defined in [CLS] would be a useful tool.Non-commutative analogues of cluster recursions have been studied in [LS1, R2] and a combinatorialconstruction was given. Polynomial generalizations have also been studied in [U], though only Laurentnessis established and only for monic, palindromic polynomials P = P . In a forthcoming work [R3] we willuse the combinatorics developed here to establish the Laurentness and positivity of rank 2 non-commutativegeneralized cluster variables. Acknowledgements.
The author would like to thank Arkady Berenstein, Kyungyong Lee, Li Li, ThomasMcConville, and Andrei Zelevinsky for useful discussions related to this project.Many of these ideas were worked out while the author was a postdoctoral research fellow in the ClusterAlgebras program at the Mathematical Sciences Research Institute. The author would like to thank theMSRI for their hospitality and support. The author would also like to thank the organizers of the ClusterAlgebras program for giving him the opportunity to work in such a stimulating environment. ank 2 Generalized Greedy Bases 3 Rank 2 Generalized Cluster Algebras and their Greedy Bases
Fix any field k of characteristic zero. Let P , P ∈ k [ z ] be arbitrary monic palindromic polynomials ofdegree d and d respectively, where a degree d polynomial P ( z ) is palindromic if P ( z ) = z d P ( z − ). Considerthe ring k ( x , x ) of rational functions in commuting variables x and x . We inductively define rationalfunctions x k ∈ k ( x , x ) for k ∈ Z by the rule:(2.1) x k +1 x k − = ( P ( x k ) if k is even; P ( x k ) if k is odd.Write k for the Z -subalgebra of k generated by the coefficients of P and P . Define the generalized clusteralgebra A ( P , P ) to be the k -subalgebra of k ( x , x ) generated by the set { x k } k ∈ Z of “generalized clustervariables” which we usually refer to simply as cluster variables. For k ∈ Z the k th clusters in A ( P , P ) isthe pair { x k , x k +1 } of neighboring cluster variables. Example 2.1.
We will take P ( z ) = 1 + z + z and P ( z ) = 1 + z + z + z . Then the first few clustervariables are given by x = x − (1 + x + x ); x = x − x − [ x + x (1 + x + x ) + x (1 + x + x ) + (1 + x + x ) ]; x = x − x − [ x + x (2 + x ) + x (3 + 4 x + 4 x + x ) + x (4 + 9 x + 14 x + 11 x + 6 x + x )+ 3 x (1 + x + x ) + 2 x (1 + x + x ) + (1 + x + x ) ] . Structure of Rank 2 Generalized Cluster Algebras.
It will be convenient to introduce Laurentpolynomial subalgebras T k = k [ x ± k , x ± k +1 ] ⊂ k ( x , x ) for each k ∈ Z . Then we may formulate our firststructural result on rank 2 generalized cluster algebras. Theorem 2.2 (Laurent Phenomenon) . The generalized cluster algebra A ( P , P ) is contained in T k for each k ∈ Z . Before presenting the proof we need to introduce some additional notation. Write P ( z ) = ρ + ρ z + · · · + ρ d z d and P ( z ) = ̺ + ̺ z + · · · + ̺ d z d , since these are monic and palindromic the coefficients satisfy: ρ = ρ d = 1 = ̺ = ̺ d ; ρ t = ρ d − t for 0 ≤ t ≤ d ; ̺ t = ̺ d − t for 0 ≤ t ≤ d .We learned the following proof of Laurentness for rank 2 cluster algebras from Arkady Berenstein. Proof.
We will prove for each m ∈ Z the containment x m ∈ T k . This is accomplished by induction byconsidering the element x d m +1 x m +4 as follows (without loss of generality assume m is odd): x d m +1 x m +4 = x d m +1 P ( x m +3 ) x m +2 = x d m +1 P ( x m +3 ) − P ( x m +1 ) x m +2 + P ( x m +1 ) x m +2 = d P t =0 ( ρ t x d m +1 x tm +3 − ρ d − t x d − tm +1 ) x m +2 + x m = d P t =1 ρ t ( x tm +1 x tm +3 − x d − tm +1 x m +2 + x m , where the last equality used that P was palindromic. Using that P was assumed monic, we see that x tm +1 x tm +3 − P t ( x m +2 ) − ∈ x m +2 k [ x m +2 ] for each t ≥ d P t =1 ρ t ( x tm +1 x tm +3 − x d − tm +1 x m +2 ∈ k [ x m +1 , x m +2 ] . It follows that x m ∈ k [ x m +1 , x m +2 , x m +3 , x m +4 ] for each m ∈ Z . By a similar calculation one may prove themembership x m ∈ k [ x m − , x m − , x m − , x m − ]. By inductively “shifting the viewing window” we see that x m ∈ k [ x k − , x k , x k +1 , x k +2 ] ⊂ T k for each k ∈ Z . (cid:3) D. RUPEL
Remark 2.3. If k is an ordered field it makes sense to talk about positive elements, we will make thisprecise in Section 2.2. In this case, when P and P have positive coefficients it is not apparent from thisproof of Laurentness that the generalized cluster variables can be expressed as Laurent polynomials withpositive coefficients. Establishing this positivity will be one of the central results of this note. It follows from the proof of Theorem 2.2 that the rank 2 generalized cluster algebra A ( P , P ) is equal toeach of its lower bound algebras k [ x k − , x k , x k +1 , x k +2 ] for k ∈ Z . This allows us to identify a relatively simple k -basis of A ( P , P ), indeed for each ( a , a ) ∈ Z we define the standard monomial z k [ a , a ] ∈ A ( P , P )in the k th cluster by z k [ a , a ] = x [ a ] + k − x [ − a ] + k x [ − a ] + k +1 x [ a ] + k +2 , where we write [ a ] + = max( a, cluster monomials to be the set (cid:8) z k [ a , a ] (cid:9) a ,a ∈ Z ≤ ,k ∈ Z Theorem 2.4.
For each k ∈ Z the set of all standard monomials (cid:8) z k [ a , a ] (cid:9) a ,a ∈ Z forms a k -basis of A ( P , P ) .Proof. Since A ( P , P ) = k [ x k − , x k , x k +1 , x k +2 ] we know that A ( P , P ) is spanned over k by all monomialsof the form x b k − x b k x b k +1 x b k +2 where each b i is a nonnegative integer. Using the exchange relations (2.1) wemay eliminate any factors of the form x k − x k +1 or x k x k +2 , in particular we see that the standard monomials (cid:8) z k [ a , a ] (cid:9) a ,a ∈ Z span A ( P , P ).It only remains to show that the set of standard monomials is linearly independent over k . For this wenote that the smallest monomial appearing in the k th cluster expansion of z k [ a , a ] is x − a k x − a k +1 , but the setof monomials { x − a k x − a k +1 } a ,a ∈ Z is linearly independent over k in T k . It follows that the standard monomialsare linearly independent in A ( P , P ) ⊂ T k . (cid:3) In fact there is a stronger version of the Laurent phenomenon which states that A ( P , P ) is exactly theset of all universally Laurent elements of k ( x , x ), i.e. A ( P , P ) consists of all rational functions whichare Laurent when expressed in terms of any given cluster { x k , x k +1 } . Moreover, as the next result states,the check for membership in A ( P , P ) can be restricted to checking Laurentness for any three consecutiveclusters. Theorem 2.5 (Strong Laurent Phenomenon) . For any m ∈ Z we have (2.2) A ( P , P ) = \ k ∈ Z T k = m +1 \ k = m − T k . Proof.
To prove this we establish an analogue of the equality of upper and lower bounds for acyclic clusteralgebras from [BFZ]. More precisely, the result will follow if we can establish the equality(2.3) k [ x m − , x m , x m +1 , x m +2 ] = T m − ∩ T m ∩ T m +1 . Indeed, A ( P , P ) has already been identified with the left hand side in the course of proving Theorem 2.2.The containment of the left hand side of (2.3) into the right hand side follows from the Laurent Phe-nomenon. Our goal is to establish the opposite containment. We accomplish this by decomposing T m bydecomposing T m = k [ x − m , x − m +1 ] + k [ x m , x ± m +1 ] + k [ x ± m , x m +1 ]and proving the containment upon intersection of each summand with T m − ∩ T m +1 .Without loss of generality we will assume m is even. We begin by establishing the equality(2.4) T m − ∩ k [ x − m , x − m +1 ] ∩ T m +1 = k if 2 ≤ d d ; k [ x m − x m +2 ] if d = d = 1; k [ x m − , x d m − x m +2 ] if d = 0; k [ x m − x d m +2 , x m +2 ] if d = 0.Notice that each ring on the right hand side of (2.4) is contained in the left hand side. Indeed, by the LaurentPhenomenon each of these rings is contained in T m − ∩ T m +1 . When d = d = 1 we have x m − x m +2 = ( x m x − m +1 + x − m +1 )( x − m x m +1 + x − m ) = 1 + x − m + x − m +1 + x − m x − m +1 ∈ k [ x − m , x − m +1 ] . ank 2 Generalized Greedy Bases 5 When d = 0 we have x m − = x − m +1 and x d m − x m +2 = x − d m +1 P ( x m +1 ) x − m = P ( x − m +1 ) x − m ∈ k [ x − m , x − m +1 ] , where we used that P is palindromic. Finally when d = 0 we have x m +2 = x − m and x m − x d m +2 = x − m +1 P ( x m ) x − d m = x − m +1 P ( x − m ) ∈ k [ x − m , x − m +1 ] . Now we check in each case that the intersection on the left hand side of (2.4) is contained in the desiredring on the right. Consider an element y ∈ k [ x − m , x − m +1 ], then y may be written in the form(2.5) y = X s,t ≥ c s,t x − sm x − tm +1 , where c s,t ∈ k is zero for all but finitely many s and t . Applying the identity x m +1 = P ( x m ) x m − we get y = X t ≥ P s ≥ c s,t x − sm P t ( x m ) x tm − = X t ≥ P s ≥ c s,t x − sm P t ( x − m ) x − d tm x tm − , where the second equality used that P is palindromic. It follows from the membership y ∈ T m − that P s ≥ c s,t z s P t ( z ) ∈ k [ z ] for all t ≥
0. Denote this polynomial by C t , i.e.(2.6) y = X t ≥ C t ( x − m ) x − d tm x tm − = X t ≥ C t ( x − m ) P t ( x m ) x − d tm x − tm +1 = X t ≥ C t ( x − m ) P t ( x − m ) x − tm +1 . By a similar calculation with (2.5) and the identity x m = P ( x m +1 ) x m +2 we get(2.7) y = X s ≥ C ′ s ( x − m +1 ) x − d sm +1 x sm +2 = X s ≥ C ′ s ( x − m +1 ) P s ( x − m +1 ) x − sm for some polynomials C ′ s ∈ k [ z ].If d = 0, we have x − m = x m +2 and the first equality in (2.6) becomes y = X t ≥ C t ( x m +2 ) x d tm +2 x tm − ∈ k [ x m − x d m +2 , x m +2 ] . Similarly, if d = 0 we have x − m +1 = x m − and the first equality in (2.7) becomes y = X s ≥ C ′ s ( x m − ) x d sm − x sm +2 ∈ k [ x m − , x d m − x m +2 ] . Now we consider the cases d d = 0, first we introduce some more notation. Denote by σ ( t ) the largestindex so that c σ ( t ) ,t = 0 in the expansion (2.5), then using the last equality in (2.6) we must have σ ( t ) ≥ d t for each t ≥
0. Similarly denote by τ ( s ) the largest index so that c s,τ ( s ) = 0 in (2.5), then the last equalityin (2.7) gives τ ( s ) ≥ d s for each s ≥ t is the largest so that c s,t = 0 for some s . Then c σ ( t ) ,τ ( σ ( t )) = 0 and τ (cid:0) σ ( t ) (cid:1) ≥ d σ ( t ) ≥ d d t .For d d ≥ t >
0, this implies τ (cid:0) σ ( t ) (cid:1) > t contradicting the maximality of t . Thus we must have t = 0, by a similar argument we see that s = 0 when d d ≥
2. It follows that y ∈ k .Now suppose d d = 1. Then we have τ (cid:0) σ ( t ) (cid:1) ≥ σ ( t ) ≥ t . Since t was maximal each of these inequalitiesmust be an equality. We establish the membership y ∈ k [ x m − x m +2 ] by a simple induction on the maximalvalue t such that c t,t = 0. Indeed, when t = 0 the claim is clear. When t >
0, the element y − c t,t x tm − x tm +2 is contained in k [ x m − x m +2 ] by the induction hypothesis, it follows that y ∈ k [ x m − x m +2 ]. This completesthe proof of the equality (2.4).The final step in the proof of Theorem 2.5 is to show the equality T m − ∩ k [ x m , x ± m +1 ] = k [ x m − , x m , x m +1 ] ⊂ T m +1 . The inclusion “ ⊇ ” is clear, thus we aim to establish the inclusion “ ⊆ ”. Indeed, any y ∈ k [ x m , x ± m +1 ] can bewritten in the form y = N X t = − N c t x tm +1 , D. RUPEL for some positive integer N where c t ∈ k [ x m ] for each t . Making the substitution x m +1 = P ( x m ) x m − we get y = N X t =0 c t P t ( x m ) x − tm − + N X t =1 c − t P t ( x m ) x tm − . If in addition we assume y ∈ T m − , we must have c − t P t ( x m ) ∈ k [ x ± m ] for 1 ≤ t ≤ N . But P ( x m ) has constantterm 1, and so we must actually have c − t P t ( x m ) ∈ k [ x m ] for 1 ≤ t ≤ N . Thus we get the containment y = N X t =0 c t x tm +1 + N X t =1 c − t P t ( x m ) x tm − ∈ k [ x m − , x m , x m +1 ]as desired. By a similar calculation we get k [ x ± m , x m +1 ] ∩ T m +1 = k [ x m , x m +1 , x m +2 ] ⊂ T m − .We are now ready to complete the proof: T m − ∩ T m ∩ T m +1 = T m − ∩ (cid:0) k [ x − m , x − m +1 ] + k [ x m , x ± m +1 ] + k [ x ± m , x m +1 ] (cid:1) ∩ T m +1 = T m − ∩ k [ x − m , x − m +1 ] ∩ T m +1 + k [ x m − , x m , x m +1 ] + k [ x m , x m +1 , x m +2 ] ⊂ k [ x m − , x m , x m +1 , x m +2 ] , as desired. This complete the proof of Theorem 2.5. (cid:3) Greedy Elements in Rank 2 Generalized Cluster Algebras.
In this section we introduce greedyelements in a rank 2 generalized cluster algebra. For this we need to work over a field with an inherentnotion of positivity.
Definition 2.6. A prepostive cone Π ⊂ k \ {− } satisfies the following closure properties: • For any a, b ∈ Π , we have a + b, ab ∈ Π ; • For any a ∈ k , we have a ∈ Π .A prepositive cone is called positive if k = Π ∪ − Π . A field k together with a positive cone Π is called an ordered field . We define a total order ≤ on k by declaring for a, b ∈ k that a ≤ b if b − a ∈ Π . From now on we assume k is ordered with positive cone Π and P , P ∈ Π[ z ]. Define the semiring k ≥ = k ∩ Π, this is the Z ≥ -subsemiring of k generated by the coefficients of P and P . For each k ∈ Z define the positive semiring T ≥ k = k ≥ [ x ± k , x ± k +1 ] ⊂ T k , the set of positive elements of T k . We are interestedin those elements of A ( P , P ) which are universally positive in the following sense. Definition 2.7.
An nonzero element x ∈ A ( P , P ) is called universally positive if its expression as aLaurent polynomial in any cluster { x k , x k +1 } only contains nonnegative coefficients, i.e. x ∈ T k ∈ Z T ≥ k . Apositive element is called indecomposable if it cannot be written as a sum of two nonzero positive elements. Our aim in this note is to establish the existence of a universally positive indecomposable basis of A ( P , P )called the greedy basis . To describe this basis we need to introduce the following notion of a pointed elementof T k . Definition 2.8.
Let ( a , a ) ∈ Z . An element x ∈ T k is said to be pointed at ( a , a ) in the k th cluster ifit can be written in the form x = x − a k x − a k +1 X p,q ≥ c ( p, q ) x pk x qk +1 , where c ( p, q ) ∈ k and c (0 ,
0) = 1 . We will simply refer to an element as “pointed” if it is pointed in the initial cluster { x , x } .Consider a collection (cid:8) x [ a , a ] (cid:9) a ,a ∈ Z where x [ a , a ] is pointed at ( a , a ) ∈ Z . Call such a collectionof pointed elements “bounded” if x [ a , a ] = z [ a , a ] is a standard monomial for ( a , a ) ∈ Z \ Z > . Thenext result claims that any complete bounded collection of pointed elements in A ( P , P ) forms a k -basis. Proposition 2.9.
Suppose there exists a complete bounded collection (cid:8) x [ a , a ] (cid:9) a ,a ∈ Z ⊂ A ( P , P ) where x [ a , a ] is pointed at ( a , a ) ∈ Z . Then (cid:8) x [ a , a ] (cid:9) a ,a ∈ Z forms a k -basis of A ( P , P ) . ank 2 Generalized Greedy Bases 7 Proof.
By definition the smallest monomial appearing in a pointed element x [ a , a ] is x − a x − a , but themonomials { x − a k x − a k +1 } a ,a ∈ Z are linearly independent over k in T . It follows that the set { x [ a , a ] } a ,a ∈ Z is linearly independent in A ( P , P ) ⊂ T .To finish we only need to show that these pointed elements span A ( P , P ). Suppose x ∈ A ( P , P ) andwrite x = P b ,b ∈ Z c b ,b x − b x − b for some c b ,b ∈ Z . Define the “negative support” S ( x ) of x by S ( x ) = { ( b , b ) ∈ Z > : c b ,b = 0 } . Moreover, write m ( x ) = max { b + b : ( b , b ) ∈ S ( x ) } ∪ { } and let M ( x ) ⊂ S ( x ) denote those pairs( b , b ) ∈ S ( x ) for which b + b = m ( x ) is maximal. Notice that the element x ′ = x − X ( b ,b ) ∈ M ( x ) c b ,b x [ b , b ]satisfies m ( x ′ ) < m ( x ). Iterating this process of subtracting off maximal points we may produce an element x ′′ ∈ A ( P , P ) with m ( x ′′ ) = 0, i.e. S ( x ′′ ) = ∅ . Then by Theorem 2.4 and using that the collection (cid:8) x [ a , a ] (cid:9) a ,a ∈ Z is bounded we may write x ′′ = X ( a ,a ) ∈ Z \ Z > d a ,a z [ a , a ] = X ( a ,a ) ∈ Z \ Z > d a ,a x [ a , a ] , (cid:0) d a ,a ∈ k (cid:1) where we note that z [ b , b ] for ( b , b ) ∈ Z > does not appear by the construction of x ′′ . It followsthat x ′′ , and hence x , is contained in the k -span of (cid:8) x [ a , a ] (cid:9) a ,a ∈ Z . Thus we see that the collection (cid:8) x [ a , a ] (cid:9) a ,a ∈ Z spans A ( P , P ). (cid:3) In view of Theorem 2.5, it is natural to look at elements that are positive in three consecutive clusters.The following result gives precise conditions on the coefficients c ( p, q ) for a pointed element to be positive inthe initial cluster and its two immediate neighbors. The key insight of [LLZ] is that knowing this restrictedpositivity can be enough to know an element is universally positive, but more on that later. Though thisresult holds in greater generality we restrict attention to elements pointed in the initial cluster. Proposition 2.10.
Suppose x ∈ T is pointed at ( a , a ) ∈ Z and x ∈ T ≥ ∩ T ≥ ∩ T ≥ is positive whenexpanded in each of three consecutive clusters. Then the coefficients c ( p, q ) in the initial cluster expansionof x satisfy the following inequality in k for all ( p, q ) ∈ Z ≥ : c ( p, q ) ≥ max " p X k =1 X ( k ,...,k d ) ⊢ k ( − k − c ( p − k − k − · · · − d k d , q ) ̺ k · · · ̺ k d d (cid:18) a − q + k − a − q − , k , . . . , k d (cid:19) + , " q X ℓ =1 X ( ℓ ,...,ℓ d ) ⊢ ℓ ( − ℓ − c ( p, q − ℓ − ℓ − · · · − d ℓ d ) ρ ℓ · · · ρ ℓ d d (cid:18) a − p + ℓ − a − p − , ℓ , . . . , ℓ d (cid:19) + ! , where ρ i and ̺ i denote the coefficients of the polynomials P and P respectively and where [ a ] + = max( a, . Before proceeding with the proof we refer the reader to the appendix, section 5, for our notations andconventions related to multinomial coefficients.
Proof.
Consider the initial cluster expansion of x :(2.8) x = x − a x − a X p,q ≥ c ( p, q ) x p x q . To establish the second inequality we will make the substitution x = P ( x ) x in (2.8), we leave the substitution x = P ( x ) x in (2.8), and thus the verification of the first inequality, to the reader. Applying Corollary 5.6we may expand x p − a = (cid:16) P ( x ) x (cid:17) p − a as x p − a = X ℓ ≥ X ( ℓ ,...,ℓ d ) ⊢ ℓ ( − ℓ ρ ℓ · · · ρ ℓ d d (cid:18) a − p + ℓ − a − p − , ℓ , . . . , ℓ d (cid:19) x ℓ +2 ℓ + ··· + d ℓ d x a − p , D. RUPEL where we used that ρ = 1 to slightly simplify the expression. Substituting into the initial cluster expansion(2.8) of x , we see that the coefficient of x q − a x a − p in the expansion of x as an element of T is given by(2.9) X ℓ ≥ X ( ℓ ,...,ℓ d ) ⊢ ℓ ( − ℓ c ( p, q − ℓ − ℓ − · · · − d ℓ d ) ρ ℓ · · · ρ ℓ d d (cid:18) a − p + ℓ − a − p − , ℓ , . . . , ℓ d (cid:19) ≥ , which must be non-negative by the membership x ∈ T ≥ . Note that the summand is zero for any ℓ > q and is equal to c ( p, q ) for ℓ = 0. Solving the inequality (2.9) for c ( p, q ) and remembering that c ( p, q ) ≥ (cid:3) We will be interested in certain greedy pointed elements x [ a , a ] ∈ T for ( a , a ) ∈ Z which are “mini-mally positive” in the sense that the bound in Proposition 2.10 is sharp. Definition 2.11.
For ( a , a ) ∈ Z define the greedy pointed element x [ a , a ] ∈ T whose pointed expansioncoefficients c ( p, q ) are given by the following “greedy recursion”: c ( p, q ) = max " p X k =1 X ( k ,...,k d ) ⊢ k ( − k − c ( p − k − k − · · · − d k d , q ) ̺ k · · · ̺ k d d (cid:18) a − q + k − a − q − , k , . . . , k d (cid:19) + , " q X ℓ =1 X ( ℓ ,...,ℓ d ) ⊢ ℓ ( − ℓ − c ( p, q − ℓ − ℓ − · · · − d ℓ d ) ρ ℓ · · · ρ ℓ d d (cid:18) a − p + ℓ − a − p − , ℓ , . . . , ℓ d (cid:19) + ! (2.10) for each nonzero ( p, q ) ∈ Z ≥ . Remark 2.12.
One might more naturally call the elements x [ a , a ] “frugal” since they spend as little aspossible to be positive, however the originals were coined “greedy” and we will adhere to this terminology. Clearly, the greedy element x [ a , a ] is unique and indecomposable once it is well-defined, i.e. onceit is known that only finitely many of the coefficients c ( p, q ) are nonzero. To establish this we will give acombinatorial construction in the next section. This combinatorial expression will also imply a more pleasantrecursion which eliminates the need for the maximum. Proposition 2.13.
Fix ( a , a ) ∈ Z and define c ( p, q ) by the recursion (2.10) . Then for each nonzero ( p, q ) ∈ Z ≥ the expansion coefficient c ( p, q ) admits the following closed form: c ( p, q ) = (cid:20) p P k =1 P ( k ,...,k d ) ⊢ k ( − k − c ( p − k − k − · · · − d k d , q ) ̺ k · · · ̺ k d d (cid:0) a − q + k − a − q − ,k ,...,k d (cid:1)(cid:21) + if a q ≤ a p ; (cid:20) q P ℓ =1 P ( ℓ ,...,ℓ d ) ⊢ ℓ ( − ℓ − c ( p, q − ℓ − ℓ − · · · − d ℓ d ) ρ ℓ · · · ρ ℓ d d (cid:0) a − p + ℓ − a − p − ,ℓ ,...,ℓ d (cid:1)(cid:21) + if a q ≥ a p . (2.11)2.3. Combinatorial Construction of Greedy Elements.
The results of this section rely on the nota-tion and results presented in Section 4 which may be read and understood independently from the resultspresented here and above.Fix ( a , a ) ∈ Z . Let D = D [ a ] + , [ a ] + denote a maximal Dyck path with horizontal edges D andvertical edges D , see Section 3 for details. Write C = C [ a ] + , [ a ] + for the set of compatible pairs in D ,see Definition 4.1. We begin by introducing the coefficients of the combinatorial construction of the greedyelement x [ a , a ]. Definition 2.14.
For each compatible pair ( S , S ) ∈ C we define the coefficient c S ,S = ρ ℓ · · · ρ ℓ d d ̺ k · · · ̺ k d d , where ℓ r and k r are the cardinalities of the sets { h ∈ D : S ( h ) = r } and { v ∈ D : S ( v ) = r } respectively.We will also write c S = ρ ℓ · · · ρ ℓ d d and c S = ̺ k · · · ̺ k d d with notation as above, so that c S ,S = c S c S . ank 2 Generalized Greedy Bases 9 Theorem 2.15.
For any ( a , a ) ∈ Z the greedy element x [ a , a ] can be computed via (2.12) x [ a , a ] = x − a x − a X ( S ,S ) ∈C c S ,S x | S | x | S | . Example 2.16.
We continue Example 2.1 and will use the compatible pairs given in Example 4.3. For P ( z ) = 1 + z + z and P ( z ) = 1 + z + z + z we may easily match the cluster variable x with theright hand side of (2.12) for ( a , a ) = (5 , . Indeed, one may associate conveniently factorized expressionsto the compatible pairs presented in Example 4.3, for example the following compatible pairs contribute tosummands containing x : x (1 + x + x ) 012 0 0 20 0 2 x (1 + x + x ) 012 01 0 10 0 3 x (1 + x + x )(1 + x ) ,which clearly agrees with the coefficient of x in x . The expressions associated to all other compatible pairscan be found in Example 4.3. To establish the combinatorial construction of greedy elements we will need some preliminary results, butfirst a definition. Using the symmetry of the exchange relations (2.1) we see that the generalized clusteralgebra A ( P , P ) admits reflection automorphisms σ p ( p ∈ Z ) defined by σ p ( x k ) = x p − k . One easilychecks that the reflections { σ p } p ∈ Z generate a (possibly infinite) dihedral group and that this group maybe generated by σ and σ alone. In the next result we take (2.12) as the definition of the greedy element x [ a , a ]. Proposition 2.17.
For any ( a , a ) ∈ Z the elements x [ a , a ] defined by (2.12) satisfy the followingsymmetry properties: (2.13) σ ( x [ a , a ]) = x [ a , d [ a ] + − a ] and σ ( x [ a , a ]) = x [ d [ a ] + − a , a ] , where [ a ] + = max( a, .Proof. By symmetry it suffices to establish the second identity in (2.13). We will have a few cases to consider. • Suppose a , a ≤
0. Then D [ a ] + , [ a ] + = D , consists of a single point and we have σ ( x [ a , a ]) = σ ( x − a x − a ) = x − a x − a . Using the identity x = P ( x ) x this becomes x a P ( x ) − a x − a . But notice that the maximal Dyck path D − a , consists of exactly − a consecutive horizontal edges and no vertical edges. In particular, in everycompatible pair ( S , S ) ∈ C − a , the vertical grading S is trivial and every possible horizontal grading S : D → [0 , d ] occurs in C − a , . Thus we see that x [ − a , a ] = x a x − a P ( x ) − a , as claimed. • Suppose a ≤ < a . Then D [ a ] + , [ a ] + consists of exactly a consecutive vertical edges and no horizontaledges. As in the previous case it is easy to see that x [ a , a ] = x − a x − a P ( x ) a . Applying σ and theidentity x = P ( x ) x gives σ ( x [ a , a ]) = x − a x − a P ( x ) a = x a x − a P ( x ) − a P (cid:0) P ( x ) x − (cid:1) a = x − d a + a x − a P ( x ) − a (cid:0) x d + ̺ d − x d − P ( x ) + · · · + ̺ x P ( x ) d − + P ( x ) d (cid:1) a , where we used that P is palindromic to reverse the coefficients. Thus we need to show that P ( x ) − a (cid:0) x d + ̺ d − x d − P ( x ) + · · · + ̺ x P ( x ) d − + P ( x ) d (cid:1) a = X ( S ,S ) ∈C d a − a ,a c S ,S x | S | x | S | . Fix a partition ( k , k , . . . , k d ) ⊢ a . On the left we will expand using (5.1) and take the coefficient of x k +2 k + ··· + d k d in (cid:18) a k , k , . . . , k d (cid:19)(cid:0) P ( x ) d (cid:1) k (cid:0) ̺ x P ( x ) d − (cid:1) k · · · (cid:0) ̺ d − x d − P ( x ) (cid:1) k d − (cid:0) x d (cid:1) k d . On the right we consider the sum P ( S ,S ) c S ,S x | S | x | S | over all compatible pairs such that k r is the cardinalityof the set { v ∈ D : S ( v ) = r } , notice that there are exactly (cid:0) a k ,k ,...,k d (cid:1) such vertical gradings S and wewill always have | S | = k + 2 k + · · · + d k d . Thus to complete the proof it suffices to show X S ∈C ( S ) c S ,S x | S | = P ( x ) − a (cid:0) P ( x ) d (cid:1) k (cid:0) ̺ P ( x ) d − (cid:1) k · · · (cid:0) ̺ d − P ( x ) (cid:1) k d − for each vertical grading S as above, where the summation runs over all horizontal gradings S such that( S , S ) ∈ C d a − a ,a . Using that c S ,S = c S c S and that in this case we have c S = ̺ k · · · ̺ k d − d − , we mayfurther reduce the problem to showing that(2.14) X S ∈C ( S ) c S x | S | = P ( x ) − a P ( x ) d k P ( x ) ( d − k · · · P ( x ) k d − . To see this notice that, by Lemma 4.4, any horizontal edge h outside sh( S ) may be assigned any weight S ( h ) without affecting compatibility. At most S ( v ) = d for every vertical edge v , which leaves − a hori-zontal edges outside the shadow of S , this accounts for the factor P ( x ) − a . Finally for 0 ≤ r ≤ d considerthe k r vertical edges v such that S ( v ) = r . There will be d − r horizontal edges outside sh( S ) for eachsuch vertical edge, this contributes the factor P ( x ) ( d − r ) k r . This completes the proof of (2.14) and thusthe proof of the proposition in this case. • Suppose 0 < a , a . Then by definition x [ a , a ] = x − a x − a P ( S ,S ) ∈C a ,a c S ,S x | S | x | S | . We apply σ and the identity x = P ( x ) x to get σ ( x [ a , a ]) = x − a x − a X ( S ,S ) ∈C a ,a c S ,S x | S | x | S | = x − d a + a x − a P ( x ) − a X ( S ,S ) ∈C a ,a c S ,S x d a −| S | P ( x ) | S | x | S | . In the notation of Section 4.2 notice that | ϕ ∗ d S | = d a − | S | . Thus comparing with the definition (2.12)of x [ d a − a , a ] we see that the claim will follow if we can establish the following identity:(2.15) X S ∈C ( S ) c S ,S x | S | = P ( x ) a −| S | X S ′ ∈C ( ϕ ∗ d S ) c S ′ ,ϕ ∗ d S x | S ′ | for each vertical grading S . According to Lemma 4.4 the number of horizontal edges | D \ sh( S ) | outsidethe shadow of S is a − min( a , | S | ) = [ a − | S | ] + . Similarly the number of horizontal edges | D ′ \ sh( ϕ ∗ d S ) | outside the shadow of ϕ ∗ d S is d a − a − min( d a − a , | ϕ ∗ d S | ) = [ d a − a − | ϕ ∗ d S | ] + = [ | S | − a ] + = [ a − | S | ] + − ( a − | S | ) . Since each of these edges contributes a factor of P ( x ) we can rewrite (2.15) as P ( x ) [ a −| S | ] + X S ∈C rs ( S ) c S ,S x | S | = P ( x ) [ a −| S | ] + X S ′ ∈C rs ( ϕ ∗ d S ) c S ′ ,ϕ ∗ d S x | S ′ | , where C rs ( S ) denotes those horizontal gradings S ∈ C ( S ) such that supp( S ) ⊂ rsh( S ).Since P is palindromic we have c S = c ϕ ∗ d S . From the definition (4.1) of Ω we have c S = c Ω( S ) and | S | = | Ω( S ) | for each S ∈ C rs ( S ). The result then follows by applying the bijection Ω : C rs ( S ) →C rs ( ϕ ∗ d S ) established in Proposition 4.20. (cid:3) The next result says that the combinatorially defined greedy elements are actually elements of the gener-alized cluster algebra.
Corollary 2.18.
For any a , a ∈ Z the elements x [ a , a ] ∈ T defined by (2.12) are contained in A ( P , P ) .Proof. By Proposition 2.17 we have x [ a , a ] = σ ( x [ a , d [ a ] + − a ]) ∈ T and x [ a , a ] = σ ( x [ d [ a ] + − a , a ]) ∈ T . The result is then a consequence of Theorem 2.5. (cid:3) ank 2 Generalized Greedy Bases 11 We conclude this section by showing that the combinatorially defined elements x [ a , a ] satisfy the recur-sion (2.10). More precisely, we will show that they satisfy (2.11) and then appealing to Proposition 2.10 wemay conclude that they satisfy (2.10). Proposition 2.19.
For any a , a ≥ the elements x [ a , a ] ∈ T defined by (2.12) satisfy (2.11) .Proof. By symmetry it suffices to show the second equality, that is(2.16) c ( p, q ) = " q X ℓ =1 X ( ℓ ,...,ℓ d ) ⊢ ℓ ( − ℓ − c ( p, q − ℓ − ℓ − · · · − d ℓ d ) ρ ℓ · · · ρ ℓ d d (cid:18) a − p + ℓ − a − p − , ℓ , . . . , ℓ d (cid:19) + for a q ≥ a p and ( p, q ) = (0 , • Suppose a ≤ p . First notice that ( − ℓ − (cid:0) a − p + ℓ − a − p − ,ℓ ,...,ℓ d (cid:1) is always nonpositive and thus the right handside is zero. Now the inequality a ≤ p together with a q ≥ a p implies q ≥ a . But then Lemma 4.21implies c ( p, q ) = 0 since we may identify p = | S | and q = | S | for some compatible pair ( S , S ) ∈ C a ,a . • Suppose p < a . Then as in (2.9) we may interpret the quantity(2.17) X ℓ ≥ X ( ℓ ,...,ℓ d ) ⊢ ℓ ( − ℓ c ( p, q − ℓ − ℓ − · · · − d ℓ d ) ρ ℓ · · · ρ ℓ d d (cid:18) a − p + ℓ − a − p − , ℓ , . . . , ℓ d (cid:19) as the the coefficient of x q − a x a − p in the expansion of x [ a , a ] as an element of T . Applying σ we mayinterpret this as the coefficient of x a − p x q − a in σ ( x [ a , a ]) = x [ d a − a , a ]. If we denote the expansioncoefficients of x [ d a − a , a ] by c ′ ( p ′ , q ′ ) then we may identify the quantity (2.17) with c ′ ( d a − p, q ). Wewill show that our current assumptions imply this coefficient is zero which implies (2.16).To simplify the notation we will abbreviate a ′ = d a − a and p ′ = d a − p . Denote compatible pairsin D ′ := D [ a ′ ] + ,a by ( S ′ , S ′ ) ∈ C [ a ′ ] + ,a . Then note that c ′ ( p ′ , q ) is nonzero if and only if there existsa compatible pair with | S ′ | = q and | S ′ | = p ′ > a ′ . Moreover, since we assume ( p, q ) = (0 ,
0) we have( | S ′ | , | S ′ | ) = (0 , d a ). Finally, the assumption a q ≥ a p translates into the inequality(2.18) ( d a − a ′ ) | S ′ | ≥ a ( d a − | S ′ | ) . We now match with the results from Proposition 4.22 by splitting into further subcases. First note that d a ≤ a ′ is impossible since we assume a > a ′ ≤
0. Then D ′ consists of a consecutive vertical edges and no horizontal edges so that( | S ′ | , | S ′ | ) is contained in the segment (cid:2) (0 , , (0 , d a ) (cid:1) . Since | S ′ | = 0, the inequality (2.18) becomes d a ≤ | S ′ | which is impossible.— Suppose 0 < d a ′ ≤ a and 0 < a ′ < d a . Then by Proposition 4.22.b the point ( | S ′ | , | S ′ | ) must lie onor below the segment (cid:2) (0 , d a ) , ( d a ′ , d a − d d a ′ ) (cid:3) , which is equivalent to the inequality | S ′ | ≤ − d | S ′ | + d a . Combining this with (2.18) gives the inequalities 0 ≥ | S ′ | and | S ′ | ≥ d a , which are only satisfied for( | S ′ | , | S ′ | ) = (0 , d a ). However we have assumed this is not the case, thus there are no compatible pairswith ( | S ′ | , | S ′ | ) = ( q, p ′ ) and (2.17) is equal to zero.— Suppose 0 < a < d a ′ and 0 < a ′ < d a . Then Proposition 4.22.c says ( | S ′ | , | S ′ | ) must lie strictlybelow the segment (cid:2) (0 , d a ) , ( a , a ′ ) (cid:3) (actually Proposition 4.22 allows ( | S ′ | , | S ′ | ) = (0 , d a ) but we haveexcluded this point), which is equivalent to the inequality | S ′ | < − d a − a ′ a | S ′ | + d a . But notice that this is exactly the complementary inequality to (2.18), i.e.( d a − a ′ ) | S ′ | < a ( d a − | S ′ | ) , thus there are no compatible pairs with ( | S ′ | , | S ′ | ) = ( q, p ′ ) and (2.17) is equal to zero.This completes the proof of Proposition 2.19. (cid:3) Proof of Theorem 2.15.
Consider ( a , a ) ∈ Z . Using Proposition 2.10 and Proposition 2.19 we may con-clude for ( a , a ) ∈ Z ≥ that the combinatorial element defined by (2.12) satisfies (2.10) and thus thatit is actually the greedy element x [ a , a ]. For ( a , a ) ∈ Z < both terms in (2.10) are always zero and D [ a ] + , [ a ] + = D , , thus the combinatorially defined element and the greedy element x [ a , a ] are both equalto the initial cluster monomial x − a x − a .Now suppose a > a <
0. Notice that the first term in (2.10) is always zero for a ≤
0. Using thesecond term in (2.10) we see that c ( p,
0) = 0 for all p > c ( p, q ) = 0 whenever p >
0. Comparingwith (2.8) we may conclude that x [ a , a ] = x − a x [ a , x [ a ,
0] has already identifiedwith the combinatorial element defined by (2.12). Using Proposition 2.17 we may conclude that x [ a ,
0] = x a and thus x [ a , a ] = x − a x a = σ ( x [ − a , a ]) coincides with the combinatorially defined element.The case a < a > (cid:3) We are now ready to complete the proof of our main theorem.
Proof of Theorem 1.1.
The existence of greedy elements is the content of Theorem 2.15, uniqueness andindecomposability follow from the definition (2.10). This gives parts (a) and (b).In the proof of Theorem 2.15 above we have seen that x [ a , a ] = z [ a , a ] is a standard monomial for( a , a ) ∈ Z \ Z > , i.e. the set (cid:8) x [ a , a ] (cid:9) a ,a ∈ Z of greedy elements is a complete bounded collection ofpointed elements in A ( P , P ), thus part (c) follows from Proposition 2.9.To see parts (d) and (e) we note that x [ a , a ] = x − a x − a is an initial cluster monomial for every( a , a ) ∈ Z < . Since any other cluster monomial can be obtained from an initial cluster monomial by areflection σ p for some p ∈ Z (or better, by alternately applying σ and σ ), we see that both part (d) and(e) follow from Proposition 2.17. (cid:3) To complete this section we describe explicitly which greedy elements correspond to cluster monomials.To do so we introduce two-parameter Chebyshev polynomials u k,j ( k, j ∈ Z ) defined recursively by: u − ,j = 0 , u ,j = 1 , u k +1 ,j +1 = d j u k,j − u k − ,j − where d j = ( d , if j is odd; d , if j is even. Remark 2.20.
The two-parameter Chebyshev polynomials can be seen as the components of positive rootsin the rank two root system associated to the Cartan matrix (cid:18) − d − d (cid:19) . Since we will not need thisfact we leave the verification to the reader. Using Proposition 2.17 and the definition of the Chebyshev polynomials we easily see that the clustervariables are given by(2.19) x k = ( x [ u k − , , u k − , ] for k ≥ x [ u − k − , , u − k, ] for k ≤ σ and σ we can transform( x , x ) ( ( x k , x k +1 ) for odd k ;( x k +1 , x k ) for even k .Applying the same sequence of reflections to the initial cluster monomial x [ a , a ] = x − a x − a for ( a , a ) ∈ Z ≤ and applying (2.19) we see that the k th cluster monomial z k [ a , a ] is given by(2.20) z k [ a , a ] = x − a k x − a k +1 = x [ − a u k − , − a u k − , , − a u k − , − a u k − , ] for k ≥ x [ a , a ] for k = 1; x [ − a u − k − , − a u − k − , , − a u − k, − a u − k − , ] for k ≤ ank 2 Generalized Greedy Bases 13 Combining (2.19) and (2.20) we get the following factorization properties for greedy elements correspondingto cluster monomials:(2.21) x [ a u k − , + a u k − , , a u k − , + a u k − , ] = x [ u k − , , u k − , ] a x [ u k − , , u k − , ] a for ( a , a ) ∈ Z ≥ and k ≥ x [ a u − k − , + a u − k − , , a u − k, + a u − k − , ] = x [ u − k − , , u − k, ] a x [ u − k − , , u − k − , ] a for ( a , a ) ∈ Z ≥ and k ≤ Remark 2.21.
Consider any Dyck path D of the form D a u k − , + a u k − , ,a u k − , + a u k − , or D a u − k − , + a u − k − , ,a u − k, + a u − k − , giving rise to a cluster monomial. Using (2.12) , the factorization equations above imply that the “wraparound” of subpaths in D assumed in Section 3 can be dropped when considering compatible pairs. In otherwords, there is a principle of non-interaction between the various subpaths of D of the form D u k − , ,u k − , and D u k − , ,u k − , or D u − k − , ,u − k, and D u − k − , ,u − k − , , i.e. when considering compatible pairs we may view compatible pairs on each such subpath independent ofall other such subpaths of D . Structure of Maximal Dyck Paths
For non-negative integers a , a ∈ Z ≥ , let R = R a ,a denote the rectangle in R with corner vertices(0 ,
0) and ( a , a ). A Dyck path in R is a lattice path in Z ⊂ R beginning at (0 , a , a ), where the path never crosses above the main diagonal of R . We will considerthe maximal Dyck path D = D a ,a which lies closest to (and possibly touches) the main diagonal of R , i.e.any lattice point above D is also above the main diagonal.Label the horizontal and vertical edges of D as D = { h , . . . , h a } and D = { v , . . . , v a } respectively,where edges are labeled by their maximum distance from the vertical and horizontal axes respectively. Wewill call the distance from a horizontal edge to the horizontal axis its height and the distance from a verticaledge to the vertical axis its depth . It will be convenient to fold the rectangle R into a torus and identify thevertices (0 ,
0) and ( a , a ), in this way D becomes a closed loop. Thus we will sometimes allow arbitraryinteger subscripts h j and v j with the understanding that h j = h j + a and v j = v j + a for j ∈ Z .For edges e, e ′ ∈ D we will let ee ′ denote the subpath of D beginning at e and ending at e ′ . We consider ee to be the path which only contains the edge e . Note that when e lies to the North-East of e ′ the path ee ′ will contain the vertex (0 , ≡ ( a , a ) and “wrap around” D . We will write ( ee ′ ) for the set of horizontaledges in the subpath ee ′ and ( ee ′ ) for its set of vertical edges. It will also be convenient to denote by ee ′ , ee ′ , and ee ′ the paths obtained from ee ′ by removing the edge e , removing the edge e ′ , or removing bothedges e and e ′ respectively. Example 3.1.
For ( a , a ) = (5 , the maximal Dyck path D has horizontal edges D = { h , h , h , h , h } and vertical edges D = { v , v } which can be visualized as h h h v h h v .Then ( h v ) = { h , h , h } and ( h v ) = { v , v } , while ( v h ) = { h , h , h } and ( v h ) = { v } . Thepath h h has length 1, while v h = D has length 7. We will often need the following easy but incredibly useful Lemma presented in [LLZ] which preciselydescribes the Dyck path D . Lemma 3.2. (a)
The height of the horizontal edge h j is ⌊ ( j − a /a ⌋ , and so the vertical distance | ( h i h j ) | between h i and h j for ≤ i < j ≤ a is equal to | ( h i h j ) | = ⌊ ( j − a /a ⌋ − ⌊ ( i − a /a ⌋ . (b) The depth of the vertical edge v j is ⌈ ja /a ⌉ , and so the horizontal distance | ( v i v j ) | between v i and v j for ≤ i < j ≤ a is equal to | ( v i v j ) | = ⌈ ja /a ⌉ − ⌈ ia /a ⌉ . In particular, this implies the following result giving a bound on the slope of certain subpaths of D inrelation to the slope of the main diagonal of R . Corollary 3.3.
For any h ∈ D and v j ∈ D with h to the left/below v j we have a (cid:0) | ( hv j ) |− (cid:1) < a | ( hv j ) | .Proof. Suppose h has height j ′ − < j . Then | ( hv j ) | ≥ | ( v j ′ v j ) | + 1 and | ( hv j ) | − j − j ′ . Then byLemma 3.2 we have a | ( hv j ) | ≥ a (cid:0) | ( v j ′ v j ) | + 1 (cid:1) = a (cid:0) ⌈ ja /a ⌉ − ⌈ j ′ a /a ⌉ + 1 (cid:1) > a ( ja /a − j ′ a /a ) = a ( j − j ′ ) = a (cid:0) | ( hv j ) | − (cid:1) . (cid:3) Combinatorics of Graded Compatible Pairs: Shadows
Here we introduce our main combinatorial object of study, namely graded compatible pairs. We will beginwith general results that do not require an upper bound on the gradings and restrict to the bounded casewhen it becomes necessary. Most of the results in this section are generalizations of, or are inspired by,similar results from [LLZ, Section 3]. Let D = D a ,a be any maximal Dyck path with horizontal edges D and vertical edges D .4.1. Unbounded Gradings.Definition 4.1.
Consider functions S : D → Z ≥ and S : D → Z ≥ which we call horizontal (resp. vertical ) gradings . We call the pair ( S , S ) (graded) compatible if for every h ∈ D and v ∈ D there existsan edge e ∈ D so that at least one of the following conditions on the gradings is satisfied: he is a proper subpath of hv and | ( he ) | = X h ′ ∈ ( he ) S ( h ′ );(HGC) ev is a proper subpath of hv and | ( ev ) | = X v ′ ∈ ( ev ) S ( v ′ ) . (VGC) Write C = C a ,a for the set of all such compatible pairs ( S , S ) . Remark 4.2.
For h ∈ D with S ( h ) = 0 the horizontal grading condition (HGC) of Definition 4.1 istrivially satisfied for all v ∈ D by taking e = h . Similarly for v ∈ D with S ( v ) = 0 the vertical gradingcondition (VGC) is trivially satisfied. Example 4.3.
We continue Example 3.1. Here we will consider bounded gradings S : D → { , , } and S : D → { , , , } on the Dyck path D . All compatible pairs are given below, we draw all horizontalgradings compatible with a given vertical grading where for example “ ” written over an edge h means S ( h ) can be either , , or and the pair ( S , S ) will be compatible:
012 012 012 0012 012 0(1 + x + x )
012 012 0 1012 012 0 x (1 + x + x )
012 0 0 2012 012 0 x (1 + x + x ) x (1 + x + x ) ank 2 Generalized Greedy Bases 15
012 012 012 0012 0 1 x (1 + x + x )
012 012 0 1012 0 1 x (1 + x + x )
012 0 0 2012 0 1 x (1 + x + x ) x (1 + x + x )012 012 012 00 0 2 x (1 + x + x )
012 012 0 10 0 2 x (1 + x + x )
012 0 0 20 0 2 x (1 + x + x ) 0 0 0 30 0 2 x
012 012 01 00 0 3 x (1 + x + x ) (1 + x ) 012 01 0 10 0 3 x (1 + x + x )(1 + x ) 01 0 0 20 0 3 x (1 + x ) 0 0 0 30 0 3 x .We will illustrate the compatibility of S = (2 , , , , and S = (1 , , where we have written S = (cid:0) S ( h ) , S ( h ) , S ( h ) , S ( h ) , S ( h ) (cid:1) and S = (cid:0) S ( v ) , S ( v ) (cid:1) . As in Remark 4.2 the condition (HGC) is automatically satisfied for the horizontal edges h , h , h and eachvertical edge. The condition (HGC) cannot be satisfied for h and any vertical edge. The condition (VGC) is satisfied for h and v by taking e = h , while the condition (VGC) is satisfied for h and v by taking e = h . The condition (HGC) cannot be satisfied for h and v , but the condition (VGC) is satisfied bytaking e = h . The condition (HGC) is satisfied for h and v by taking e = v , while the condition (VGC) cannot be satisfied. For a horizontal grading S : D → Z ≥ , write C ( S ) for the set of all vertical gradings S : D → Z ≥ with ( S , S ) ∈ C . Define C ( S ) similarly. We define the magnitude of a horizontal grading S or a verticalgrading S respectively by | S | = P h ∈ D S ( h ) and | S | = P v ∈ D S ( v ). The following result greatly simplifiesthe check for membership in C ( S ) and C ( S ). Lemma 4.4. (a)
For every S : D → Z ≥ there exist subsets rsh( S ) ⊂ sh( S ) ⊂ D such that: (i) for S : D → Z ≥ , S ∈ C ( S ) if and only if S ( v ) = 0 for all v ∈ sh( S ) \ rsh( S ) and one ofthe conditions (HGC) or (VGC) is satisfied for each h ∈ supp( S ) and v ∈ rsh( S ) ; (ii) | sh( S ) | = min( a , | S | ) . (b) For every S : D → Z ≥ there exist subsets rsh( S ) ⊂ sh( S ) ⊂ D such that (i) for S : D → Z ≥ , S ∈ C ( S ) if and only if S ( h ) = 0 for all h ∈ sh( S ) \ rsh( S ) and one ofthe conditions (HGC) or (VGC) is satisfied for each h ∈ rsh( S ) and v ∈ supp( S ) ; (ii) | sh( S ) | = min( a , | S | ) . We will refer to the sets sh( S i ) and rsh( S i ) respectively as the shadow and remote shadow of S i ( i = 1 , S i will be defined in terms ofcertain “local shadows”. Definition 4.5. (a)
For each h ∈ D , define sh( h ; S ) , the local shadow of S at h , as the set of vertical edges ( he ) inthe shortest subpath he of D such that | ( he ) | = P h ′ ∈ ( he ) S ( h ′ ) . Write D ( h ; S ) = he for this minimalpath. If there is no such subpath then we set sh( h ; S ) = D and D ( h ; S ) = D . We define the shadow of S as the union of local shadows: sh( S ) = S h ∈ D sh( h ; S ) . (b) For each v ∈ D , define sh( v ; S ) , the local shadow of S at v , as the set of horizontal edges ( ev ) inthe shortest subpath ev of D such that | ( ev ) | = P v ′ ∈ ( ev ) S ( v ′ ) . Write D ( v ; S ) for this minimal path ev . If there is no such subpath then we set sh( v ; S ) = D and D ( v ; S ) = D . We define the shadow of S as the union of local shadows: sh( S ) = S v ∈ D sh( v ; S ) . Remark 4.6.
When S ( h ) = 0 , we have sh( h ; S ) = ∅ and D ( h ; S ) = hh is the subpath of D consisting ofonly the edge h . A similar claim holds if S ( v ) = 0 . It will be useful to consider certain shadow statistics for subpaths of D which are closely related to thecompatibility conditions for a pair ( S , S ) ∈ C . Definition 4.7.
For a subpath ee ′ ⊂ D define the horizontal shadow statistic f S ( ee ′ ) := −| ( ee ′ ) | + X h ∈ ( ee ′ ) S ( h ) for each horizontal grading S and the vertical shadow statistic f S ( ee ′ ) := −| ( ee ′ ) | + X v ∈ ( ee ′ ) S ( v ) for each vertical grading S . It immediately follows from the definitions that the functions f S and f S satisfy the following additivityproperties with respect to concatenation of paths: f S ( ee ′′′ ) = f S ( ee ′ ) + f S ( e ′′ e ′′′ ) ,f S ( ee ′′′ ) = f S ( ee ′ ) + f S ( e ′′ e ′′′ ) , whenever ee ′′′ = ee ′ ∐ e ′′ e ′′′ . The next lemma relates the minimal shadow paths D ( h ; S ) and D ( v ; S ) tothe shadow statistics. Lemma 4.8. (a)
Suppose h ∈ supp( S ) and write D ( h ; S ) = he . (i) f S ( D ( h ; S )) = 0 . (ii) For any proper subpath he ′ ⊂ D ( h ; S ) we have f S ( he ′ ) > . (iii) For any proper subpath e ′ e ⊂ D ( h ; S ) we have f S ( e ′ e ) < .In particular, one may conclude that the last edge e of D ( h ; S ) is vertical. (b) Suppose v ∈ supp( S ) and write D ( v ; S ) = ev . (i) f S ( D ( v ; S )) = 0 . (ii) For any proper subpath e ′ v ⊂ D ( v ; S ) we have f S ( e ′ v ) > . (iii) For any proper subpath ee ′ ⊂ D ( v ; S ) we have f S ( ee ′ ) < .In particular, one may conclude that the first edge e of D ( v ; S ) is horizontal.Proof. We will prove (a), the proof of (b) is similar. Recalling the definition of D ( h ; S ) establishes (i).Consider the value of f S ( he ′ ) as e ′ traverses the path D ( h ; S ) from left to right. When e ′ = h , we have f S ( hh ) = S ( h ) > h ∈ supp( S ). As e ′ moves to the right, f S ( he ′ ) will either increase, stay constant,or decrease by 1. It follows that f S ( he ′ ) will remain positive until it first reaches zero, at which point wemust have e ′ = e by minimality. This proves (ii), (iii) is an immediate consequence of (i) and (ii) using theadditivity property of f S . (cid:3) The next easy result is an immediate consequence of Lemma 4.8 but will be useful in the proofs ofProposition 4.20 and Proposition 4.22 below.
Corollary 4.9.
Suppose h ∈ supp( S ) and let v ∈ D ( h ; S ) be any vertical edge. Then (HGC) is notsatisfied for h and v , however for any S ∈ C ( S ) the condition (VGC) is satisfied for h and v . In particular, D ( v ; S ) is a proper subpath of hv for each S ∈ C ( S ) .Proof. By Lemma 4.8 we have f S ( he ) > e ∈ hv and thus (HGC) cannot be satisfied. Since S ∈ C ( S ) the condition (VGC) must be satisfied. (cid:3) We can now understand the relationships between the various local shadows associated to a horizontalgrading S or a vertical grading S . Lemma 4.10. (a)
Let S be a horizontal grading. For any horizontal edges h, h ′ ∈ D , the local shadows sh( h ; S ) and sh( h ′ ; S ) are either disjoint or one is contained in the other. ank 2 Generalized Greedy Bases 17 (b) Let S be a vertical grading. For any vertical edges v, v ′ ∈ D , the local shadows sh( v ; S ) and sh( v ′ ; S ) are either disjoint or one is contained in the other.Proof. We will prove (1), the proof of (2) is similar. The proof will be in terms of the local shadow paths D ( h ; S ) and D ( h ′ ; S ).For h = h ′ the claim is trivial. If either D ( h ; S ) = D or D ( h ′ ; S ) = D the claim is again trivially true,thus we assume that both D ( h ; S ) and D ( h ′ ; S ) are proper subpaths of D .Suppose D ( h ; S ) and D ( h ′ ; S ) intersect but not in a proper containment. Then either h ∈ D ( h ′ ; S )or h ′ ∈ D ( h ; S ), without loss of generality assume h ′ ∈ D ( h ; S ) = he . Then e ∈ D ( h ′ ; S ) since there isnot a proper containment of subpaths. Denote by e ′ the edge of D immediately preceding h ′ . It followsfrom Lemma 4.8 that f S ( he ′ ) > f S ( h ′ e ) >
0. But notice that the path he is just the concatenationof he ′ and h ′ e , in particular adding the two inequalities above gives f S ( he ) = f S ( he ′ ) + f S ( h ′ e ) > D ( h ; S ). (cid:3) Definition 4.11. (a)
The remote shadow of S is the set rsh( S ) obtained from sh( S ) by removing for each d ∈ [1 , a ] the(up to) S ( h d ) vertical edges of depth d immediately following h d . By considering the definition ofcompatibility one may alternatively described rsh( S ) ⊂ sh( S ) as the subset consisting of all verticaledges v for which there exists a vertical grading S ∈ C ( S ) with S ( v ) = 0 . (b) The remote shadow of S is the set rsh( S ) obtained from sh( S ) by removing for each ℓ ∈ [1 , a ] the(up to) S ( v ℓ ) horizontal edges of height ℓ − immediately preceding v ℓ . By considering the definition ofcompatibility one may alternatively describe rsh( S ) ⊂ sh( S ) as the subset consisting of all horizontaledges h for which there exists a horizontal grading S ∈ C ( S ) with S ( h ) = 0 . We are now ready to prove Lemma 4.4. We will explain part (a), part (b) follows by a similar argument.
Proof of Lemma 4.4.a.
Fix a horizontal grading S . We have constructed the sets rsh( S ) and sh( S ). Thusit remains to check that they satisfy the desired conditions (i) and (ii). Suppose S ∈ C ( S ). From thedefinition of rsh( S ) we must have S ( v ) = 0 for all v ∈ sh( S ) \ rsh( S ). Since S and S are compatible wesee that either (HGC) or (VGC) is satisfied for every pair h ∈ D and v ∈ D , in particular for h ∈ supp( S )and v ∈ rsh( S ). This proves the forward implication of (i).Now suppose S : D → Z ≥ satisfies S ( v ) = 0 for all v ∈ sh( S ) \ rsh( S ) and one of the conditions(HGC) or (VGC) is satisfied for each h ∈ supp( S ) and v ∈ rsh( S ). As in Remark 4.2 we know that, when S ( h ) = 0, (HGC) is satisfied for h and each v ∈ D . Thus to check compatibility we may restrict attentionto h ∈ supp( S ).Consider the path D ( h ; S ) = he . Using the definition of sh( S ) as a union of local shadows we seethat any v ∈ D \ sh( S ) lies outside the path he , in other words the edge e is contained in the path hv .Thus (HGC) is satisfied for h and v using exactly this edge e . Thus we have further reduced the check forcompatibility to h ∈ supp( S ) and v ∈ sh( S ).Furthermore since S ( v ) = 0 for v ∈ sh( S ) \ rsh( S ), it again follows from Remark 4.2 that (VGC) istrivially satisfied for all h ∈ supp( S ) and v ∈ sh( S ) \ rsh( S ). Thus we only need one of the conditions(HGC) or (VGC) for h ∈ supp( S ) and v ∈ rsh( S ), but this is guaranteed by assumption. Thus S ∈ C ( S ),establishing the reverse implication of (i).We now move on to proving (ii). If there exists an edge h ∈ D with sh( h ; S ) = D then f S ( he ) ≥ e . In particular, when e is the edge immediately preceding h in D we have | sh( S ) | = a = | ( he ) | ≤ P h ′ ∈ ( he ) S ( h ′ ) = | S | . This establishes (ii) in this case.Now suppose sh( h ; S ) = D for all h ∈ D . It follows from Lemma 4.10 that for any fixed horizontalgrading S we can find horizontal edges η , . . . , η p ∈ supp( S ) (labeled in the natural order along the Dyckpath D ) such that • sh( η i ; S ) ∩ sh( η j ; S ) = ∅ for i = j ; • sh( S ) = sh( η ; S ) ∐ · · · ∐ sh( η p ; S ) is a partition of sh( S ). Define vertical edges ν , . . . , ν p by D ( η j ; S ) = η j ν j . Then each horizontal edge h ∈ supp( S ) is containedin one of the maximal local shadows sh( η j ; S ) and thus we see that | sh( S ) | = | sh( η ; S ) | + · · · + | sh( η p ; S ) | = p X j =1 | ( η j ν j ) | = p X j =1 X h ′ ∈ ( η j ν j ) S ( h ′ ) = | S | , Since sh( S ) ⊂ D , we have | S | = | sh( S ) | ≤ a . This completes the proof of (ii). (cid:3) It will be useful to partition the remote shadows according to which local shadow contains a given edge.
Definition 4.12. (a)
For < j ≤ a and < d ≤ a , denote by rsh( S ) j ; d the set of v ∈ rsh( S ) of depth d suchthat v ∈ sh( h j ; S ) and h j is the first edge before v with this property (i.e. the path h j v is shortestpossible). Define the local remote shadow of the edge h j as rsh( h j ; S ) = ` d ∈ [1 ,a ] rsh( S ) j ; d . (b) For < k ≤ a and ≤ ℓ < a , denote by rsh( S ) k ; ℓ the set of h ∈ rsh( S ) of height ℓ such that h ∈ sh( v k ; S ) and v k is the first edge after h with this property (i.e. the path hv k is shortest possible).Define the local remote shadow of the edge v k as rsh( v k ; S ) = ` ℓ ∈ [0 ,a − rsh( S ) k ; ℓ . Remark 4.13.
For a horizontal grading S there are only finitely many j with rsh( h j ; S ) = ∅ . Thus thereexists a sequence ≤ j < · · · < j n ≤ a such that rsh( h j t ; S ) = ∅ for each t ∈ [1 , n ] and there is a partition rsh( S ) = rsh( h j ; S ) ∐ · · · ∐ rsh( h j n ; S ) . The next lemma gives precise conditions describing when the remote shadows rsh( S ) j ; d and rsh( S ) j ; ℓ are non-empty. First note that rsh( S ) d ; d = ∅ for every d and rsh( S ) ℓ +1; ℓ = ∅ for every ℓ . Lemma 4.14. (a)
Let < d, j ≤ a with j = d . Then rsh( S ) j ; d = ∅ if and only if f S ( hh d +1 ) < < f S ( h j h ) for everyhorizontal edge h ∈ ( h j h d +1 ) . (b) Let ≤ ℓ < a and < j ≤ a with j = ℓ + 1 . Then rsh( S ) j ; ℓ = ∅ if and only if f S ( v ℓ v ) < 0. Let ℓ ≤ d be the largest so that f S ( h ℓ h d +1 ) ≥ 0. If h ℓ / ∈ supp( S ) then f S ( h ℓ +1 h d +1 ) ≥ f S ( h ℓ h d +1 ) ≥ h ℓ . Thus we must have S ( h ℓ ) = 0, inparticular D ( h ℓ ; S ) is a non-trivial subpath of D . If h ℓ h d +1 is a subpath of D ( h ℓ ; S ), then v ∈ D ( h ℓ ; S )contradicting the minimality of the path h j v . Thus D ( h ℓ ; S ) must be a proper subpath of h ℓ h d +1 .Write e ′ for the first edge after D ( h ℓ ; S ), since f S (cid:0) D ( h ℓ ; S ) (cid:1) = 0 additivity gives f S ( e ′ h d +1 ) ≥ 0. If D ( h ℓ ; S ) does not contain h d , we may move to the right from e ′ until reaching a horizontal edge h ℓ ′ whichwill satisfy f S ( h ℓ ′ h d +1 ) ≥ f S ( e ′ h d +1 ) ≥ 0, this again contradicts the maximality of h ℓ . Thus D ( h ℓ ; S )must contain h d . But then every edge in the path e ′ h d +1 is vertical which is absurd since f S ( e ′ h d +1 ) ≥ f S ( hh d +1 ) < h ∈ ( h j h d +1 ) .Now suppose rsh( S ) j ; d = ∅ . This can happen in one of two ways: • the local shadow sh( h j ; S ) is “too short”; • for each v ∈ sh( h j ; S ) of depth d there exists j ′ = j ′ ( v ) so that the edge h j ′ is closer to v than h j with v ∈ sh( h j ′ ; S ).The local shadow sh( h j ; S ) is “too short” if sh( h j ; S ) does not contain any vertical edges of depth d .Then we have f S ( h j e ) = 0 where D ( h j ; S ) = h j e with e a vertical edge to the left of h d . Taking h j ′ ∈ h j h d to be the first horizontal edge to the right of e will give f S ( h j h j ′ ) ≤ 0, establishing the reverse implicationin this case.Thus we may assume for each v ∈ sh( h j ; S ) of depth d the existence of a number j ′ > j such that v ∈ sh( h j ′ ; S ). Take v to be the last edge of depth d in sh( h j ; S ). By Lemma 4.8.a.ii we have f S ( h j h j ′ ) > ank 2 Generalized Greedy Bases 19 and f S ( h j ′ v ) ≥ 0, so by additivity we have f S ( h j v ) > 0. This inequality implies that a vertical edge of depth d which comes after v is also contained in sh( h j ; S ), but such cannot exist by the choice of v . This implies h j ′ h d +1 = h j ′ v and so f S ( h j ′ h d +1 ) = f S ( h j ′ v ) ≥ 0, completing the proof of the reverse implication. (cid:3) Remark 4.15. The inequalities above imply that when rsh( S ) j ; d = ∅ the set rsh( S ) j ′ ; d ′ will be emptywhenever j < j ′ < d < d ′ . Now we are able to give an explicit formula for the sizes of the remote shadows. Lemma 4.16. (a) Let < d, j ≤ a with d = j and suppose f S ( hh d +1 ) < < f S ( h j h ) for every horizontal edge h ∈ ( h j h d +1 ) . Then | rsh( S ) j ; d | = min h ∈ ( h j h d +1 ) min (cid:0) − f S ( hh d +1 ) , f S ( h j h ) (cid:1) . (b) Let ≤ ℓ < a and < j ≤ a with ℓ = j and suppose f S ( v ℓ v ) < < f S ( vv j ) for every vertical edge v ∈ ( v ℓ v j ) . Then | rsh( S ) j ; ℓ | = min v ∈ ( v ℓ v j ) min (cid:0) − f S ( v ℓ v ) , f S ( vv j ) (cid:1) .Proof. As always, we will prove (a) and leave the adaptation of the proof to (b) for the reader.By Lemma 4.14 the remote shadow rsh( S ) j ; d is non-empty. Consider any element v ∈ rsh( S ) j ; d . Since v has depth d we have ( h j h d ) ≤ ( h j v ) ≤ ( h j h d +1 ) . For each h ∈ ( h j h d +1 ) we have v / ∈ sh( h ; S ) but v ∈ sh( h j ; S ) so that f S ( hv ) < f S ( h j v ) ≥ f S gives( h j h ) + X h ′ ∈ ( hh d +1 ) S ( h ′ ) < ( h j h ) + ( hv ) = ( h j v ) ≤ X h ′ ∈ ( h j h d +1 ) S ( h ′ )for every h ∈ ( h j h d +1 ) . Combining the two displayed sequences of inequalities above we getmax h ∈ ( h j h d +1 ) ( h j h ) + X h ′ ∈ ( hh d +1 ) S ( h ′ ) < ( h j v ) ≤ min ( h j h d +1 ) , X h ′ ∈ ( h j h d +1 ) S ( h ′ ) for each v ∈ rsh( S ) j ; d . Thus the number of possible v ∈ rsh( S ) j ; d is equal to | rsh( S ) j ; d | = min ( h j h d +1 ) , X h ′ ∈ ( h j h d +1 ) S ( h ′ ) − max h ∈ ( h j h d +1 ) ( h j h ) + X h ′ ∈ ( hh d +1 ) S ( h ′ ) = min ( h j h d +1 ) , X h ′ ∈ ( h j h d +1 ) S ( h ′ ) + min h ∈ ( h j h d +1 ) − ( h j h ) − X h ′ ∈ ( hh d +1 ) S ( h ′ ) = min h ∈ ( h j h d +1 ) min ( hh d +1 ) − X h ′ ∈ ( hh d +1 ) S ( h ′ ) , − ( h j h ) + X h ′ ∈ ( h j h ) S ( h ′ ) = min h ∈ ( h j h d +1 ) min (cid:0) − f S ( hh d +1 ) , f S ( h j h ) (cid:1) . (cid:3) Bounded Gradings. Fix a vertical grading S and suppose r ≥ max { S ( v ) : v ∈ D } ∪ { l a a m } .Write D ′ = D ra − a ,a with horizontal edges D ′ = { h ′ , . . . , h ′ ra − a } and vertical edges D ′ = { v ′ , . . . , v ′ a } .We identify the vertical edges of D and the vertical edges of D ′ via a map ϕ r : D ′ → D given by ϕ r ( v ′ j ) = v a +1 − j . Define the vertical grading ϕ ∗ r S : D ′ → Z ≥ by the rule ϕ ∗ r S ( v ′ j ) = r − S ( v a +1 − j ), itis well-defined (non-negative) since S ( v j ) is always bounded above by r . Lemma 4.17. For any v ′ i , v ′ j ∈ D ′ we have f ϕ ∗ r S ( v ′ i v ′ j ) = − f S ( v a − j v a − i ) . Proof. This follows from a direct calculation using the definitions and Lemma 3.2: f ϕ ∗ r S ( v ′ i v ′ j ) = X v ′ ∈ ( v ′ i v ′ j ) ϕ ∗ r S ( v ′ ) − | ( v ′ i v ′ j ) | = X v ′ ∈ ( v ′ i v ′ j ) ϕ ∗ r S ( v ′ ) − (cid:0) ⌈ j ( ra − a ) /a ⌉ − ⌈ i ( ra − a ) /a ⌉ (cid:1) = X v ∈ ( v a − j v a − i ) (cid:0) r − S ( v ) (cid:1) − ⌈ jr − ja /a ⌉ + ⌈ ir − ia /a ⌉ = − X v ∈ ( v a − j v a − i ) S ( v ) − ⌈− ja /a ⌉ + ⌈− ia /a ⌉ = − X v ∈ ( v a − j v a − i ) S ( v ) + ⌈ ( a − i ) a /a ⌉ − ⌈ ( a − j ) a /a ⌉ = − X v ∈ ( v a − j v a − i ) S ( v ) + | ( v a − j v a − i ) | = − f S ( v a − j v a − i ) . (cid:3) Corollary 4.18. Suppose ≤ ℓ < a and < j ≤ a with ℓ = j . Then we have | rsh( S ) j ; ℓ | = | rsh( ϕ ∗ r S ) a − ℓ ; a − j | . Proof. According to Lemma 4.16 and Lemma 4.17 we have | rsh( S ) j ; ℓ | = min v ∈ ( v ℓ v j ) min ( − f S ( v ℓ v ) , f S ( vv j ))= min v ′ ∈ ( v ′ a − j v ′ a − ℓ ) min (cid:0) f ϕ ∗ r S ( v ′ v ′ a − ℓ ) , − f ϕ ∗ r S ( v ′ a − j v ′ ) (cid:1) = | rsh( ϕ ∗ r S ) a − ℓ ; a − j | . (cid:3) Thus for each ℓ and j we may define a bijection θ j ; ℓ : rsh( S ) j ; ℓ → rsh( ϕ ∗ r S ) a − ℓ ; a − j which preservesthe natural left-to-right order. Following Lemma 4.4 we will define C rs ( S ) ⊂ C ( S ) to be the subset ofhorizontal gradings S compatible with S whose support is contained in the remote shadow of S , i.e.supp( S ) ⊂ rsh( S ). Now we can define a map Ω : C rs ( S ) → C rs ( ϕ ∗ r S ) as follows:(4.1) Ω( S )( h ′ ) = ( h ′ ∈ D ′ \ rsh( ϕ ∗ r S ); S ( h ) if h ′ = θ j ; ℓ ( h ) for h ∈ rsh( S ) j ; ℓ .Note that Ω admits an obvious inverse map. The following result is as much of an analogue of Lemma 4.17as one can hope for, however it is the essential ingredient to show that Ω is well-defined, i.e. that the pair(Ω( S ) , ϕ ∗ r S ) is compatible. Lemma 4.19. Suppose h ′ = θ j ; ℓ ( h ) for h ∈ rsh( S ) j ; ℓ ∩ supp( S ) . Then f Ω( S ) ( h ′ v ′ a − ℓ ) = f S ( hv j ) .Proof. Since h ∈ rsh( S ) j ; ℓ it follows from Lemma 4.10 that each horizontal edge in hv j is contained in somersh( S ) j ′ ; ℓ ′ where ℓ ≤ ℓ ′ < j ′ ≤ j . Using this we may partition the edges in supp( S ) ∩ ( hv j ) to get: f S ( hv j ) = X h ′′ ∈ ( hv j ) S ( h ′′ ) − ( hv j ) = X h ′′ ∈ ( hv ℓ +1 ) S ( h ′′ ) + X h ′′ ∈ ( v ℓ +1 v j ) S ( h ′′ ) − ( v ℓ +1 v j ) = X h ′′ ∈ ( hv ℓ +1 ) ∩ rsh( S ) j ; ℓ S ( h ′′ ) + X j ′ : ℓ Let S : D → Z ≥ be a horizontal grading. Then S ∈ C rs ( S ) if and only if Ω( S ) ∈C rs ( ϕ ∗ r S ) .Proof. We will prove the forward implication. The reverse implication can be obtained by a similar argumentwith Ω − .To check compatibility it suffices to consider h ′ ∈ supp(Ω( S )) ⊂ rsh( ϕ ∗ r S ). Suppose h ′ = θ j ; ℓ ( h ) for h ∈ rsh( S ) j ; ℓ ∩ supp( S ). The containment h ′ ∈ rsh( ϕ ∗ r S ) a − ℓ ; a − j implies for any vertical edge v ′ a − ℓ ′ with ℓ < ℓ ′ ≤ j that D ( v ′ a − ℓ ′ ; ϕ ∗ r S ) is a proper subpath of h ′ v ′ a − ℓ ′ , i.e. the condition (VGC) is satisfiedfor h ′ and v ′ a − ℓ ′ . We claim that D ( h ′ ; Ω( S )) is a subpath of h ′ v ′ a − ℓ which implies the condition (HGC) issatisfied for h ′ and each other vertical edge v ′ a − ℓ ′ with ℓ ′ ≤ ℓ or ℓ ′ > j .Indeed, since h ∈ D ( v j ; S ) each horizontal edge h ′′ ∈ ( hv j ) is also contained in D ( v j ; S ). ThenCorollary 4.9 implies we may partition hv j into shadow paths D ( h ′′ ; S ), horizontal edges outside the supportof S , and the remaining vertical edges. Thus we have f Ω( S ) ( h ′ v ′ a − ℓ ) = f S ( hv j ) ≤ 0. In particular, arguingas in the proof of Lemma 4.8 we see that there exists v ′ ∈ ( h ′ v ′ a − ℓ ) so that f Ω( S ) ( h ′ v ′ ) = 0, i.e. D ( h ′ ; Ω( S ))is a subpath of h ′ v ′ a − ℓ as desired. (cid:3) Magnitudes of Bounded Gradings. Fix nonnegative integers d , d ≥ 0. In this section we makeexplicit the possibilities for the pair of magnitudes ( | S | , | S | ) associated to a compatible pair ( S , S ) ∈ C a ,a .We assume throughout that S ( h ) ≤ d for each h ∈ D and S ( v ) ≤ d for each v ∈ D . Lemma 4.21. For any compatible pair ( S , S ) ∈ C a ,a we must have | S | < a or | S | < a .Proof. Indeed, we will show that | S | ≥ a and | S | ≥ a is impossible. Thus we assume that | S | ≥ a andwill deduce | S | < a , the other case can be proven by a similar argument.Notice that under this assumption Lemma 4.4 says | sh( S ) | = a , in other words sh( S ) = D . Itfollows that each vertical edge v ∈ D is contained in some maximal local shadow. Let sh( h ; S ) be sucha maximal local shadow. By Corollary 4.9, for each vertical edge v ∈ sh( h ; S ) the maximal shadow path D ( v ; S ) is a proper subpath of hv . It follows that h / ∈ sh( S ) and thus the inequality | S | < a follows fromLemma 4.4. (cid:3) Proposition 4.22. Let ( S , S ) ∈ C a ,a be a bounded compatible pair. (a) If ≤ d a ≤ a , then ( | S | , | S | ) is contained in the (possibly degenerate) trapezoid with cornervertices (0 , , ( d a , , (0 , d a ) , and ( d a − d d a , d a ) . (b) If ≤ d a ≤ a , then ( | S | , | S | ) is contained in the (possibly degenerate) trapezoid with cornervertices (0 , , ( d a , , (0 , d a ) , and ( d a , d a − d d a ) . (c) If < a < d a and < a < d a , then ( | S | , | S | ) is contained in the non-convex quadrilateral withcorner vertices (0 , , ( d a , , (0 , d a ) , and ( a , a ) with the convention that the boundary segments (cid:2) (0 , , ( d a , (cid:3) and (cid:2) (0 , , (0 , d a ) (cid:3) are included, while the boundary segments (cid:0) ( d a , , ( a , a ) (cid:3) and (cid:0) (0 , d a ) , ( a , a ) (cid:3) are excluded. (0 , 0) ( d a , ,d a ) ( d a − d d a ,d a ) | S || S | Case (a) (0 , 0) ( d a , ,d a ) ( d a ,d a − d d a ) | S || S | Case (b) (0 , 0) ( d a , ,d a )( a ,a ) | S || S | Case (c) Proof. There will be many cases to consider, we begin by handling the easier degenerate cases. • Suppose a = a = 0. Then the maximal Dyck path D is a single vertex, i.e. D = D = ∅ , so that( | S | , | S | ) = (0 , • Suppose a > a = 0. Then the maximal Dyck path D consists of a consecutive horizontal edges and novertical edges. It follows that ( | S | , | S | ) is contained in the segment (cid:2) (0 , , ( d a , (cid:3) . This verifies (a) inthis degenerate case. • Suppose a > a = 0. Then the maximal Dyck path D consists of a consecutive vertical edges and nohorizontal edges. It follows that ( | S | , | S | ) is contained in the segment (cid:2) (0 , , (0 , d a ) (cid:3) . This verifies (b) inthis degenerate case. • Suppose a ≥ d a > 0. We first note that | S | ≤ d a and | S | ≤ d a . Thus it only remains to justify theboundary segment (cid:2) ( d a , , ( d a − d d a , d a ) (cid:3) . In the present case we have rsh( S ) = ∅ , thus accordingto Lemma 4.4 we have | sh( S ) | = | S | and S ( h ) = 0 for each h ∈ sh( S ). It follows that | S | ≤ d a − d | S | and thus ( | S | , | S | ) lies on or to the left of the segment. This complete the proof of (a). • Suppose a ≥ d a > 0. By a similar argument as above we have | S | ≤ d a − d | S | , from which (b)follows. • Suppose 0 < a < d a and 0 < a < d a . First note that | S | ≤ d a and | S | ≤ d a . Thus we onlyneed to justify the boundary segments (cid:2) ( d a , , ( a , a ) (cid:3) and (cid:2) (0 , d a ) , ( a , a ) (cid:3) .Following Lemma 4.21 we will assume 0 < | S | < a and justify the boundary segment (cid:2) ( d a , , ( a , a ) (cid:3) ,the other boundary segment follows by a similar argument under the assumption 0 < | S | < a . Note thatthe condition that ( | S | , | S | ) lies strictly to the left of this segment is equivalent to the inequality(4.2) | S | < − d a − a a | S | + d a . This translates into the inequality a | S | < d a (cid:0) a −| S | (cid:1) + a | S | , where we note that the quantity a −| S | is exactly the number of edges outside the shadow of S . Thus it suffices to show(4.3) a | S | < a | S | for any S ∈ C rs ( S ).Since sh( S ) ⊂ D is a proper subset, Lemma 4.10 implies that sh( S ) is a disjoint union of maximal localshadows each of which is a proper subset of D . Let sh( v ; S ) denote such a maximal local shadow and write D ( v ; S ) = ev for the corresponding minimal shadow path. Then, by Corollary 4.9, for any h ∈ ( ev ) thelocal shadow path D ( h ; S ) is a proper subset of ev . It follows that a (cid:12)(cid:12)(cid:12) S | ( ev ) (cid:12)(cid:12)(cid:12) ≤ a (cid:16) | ( ev ) | − (cid:17) < a | ( ev ) | = a (cid:12)(cid:12)(cid:12) S | ( ev ) (cid:12)(cid:12)(cid:12) , (4.4) ank 2 Generalized Greedy Bases 23 where the second inequality follows from Corollary 3.3 and the last equality is the condition (VGC) for theedges e and v .Since S ( h ) = 0 for any horizontal edge h outside the shadow of S and S ( v ) = 0 for any vertical edgeoutside of a minimal shadow path, (4.4) implies (4.3). This completes the proof of the final case and thusconcludes the proof of Proposition 4.22. (cid:3) Appendix: Multinomial Coefficients In this section we collect certain lesser-known analogues for multinomial coefficients of well-known iden-tities involving binomial coefficients.The binomial coefficients describe the coefficients when expanding powers of binomials, generalizing thiswe define multinomial coefficients (cid:0) nk ,...,k r (cid:1) by(5.1) ( x + · · · + x r ) n = X ( k ,...,k r ) ⊢ n (cid:18) nk , . . . , k r (cid:19) x k · · · x k r r , where ( k , . . . , k r ) ⊢ n denotes a partition of the positive integer n into r positive parts, i.e. k , . . . , k r ∈ Z ≥ with k + · · · + k r = n . By convention, for n ≥ (cid:0) nk ,...,k r (cid:1) = 0 whenever k i < i . Wewill discuss n < Lemma 5.1. Suppose n ≥ . For any partition ( k , . . . , k r ) ⊢ n we have the “Pascal identity” (cid:18) nk , . . . , k r (cid:19) = (cid:18) n − k − , . . . , k r (cid:19) + · · · + (cid:18) n − k , . . . , k r − (cid:19) . Proof. We apply the definition of (cid:0) nk ,...,k r (cid:1) and expand in two different ways:( x + · · · + x r ) n = ( x + · · · + x r )( x + · · · + x r ) n − = ( x + · · · + x r ) X ( k ,...,k r ) ⊢ n − (cid:18) n − k , . . . , k r (cid:19) x k · · · x k r r = X ( k ,...,k r ) ⊢ n (cid:20)(cid:18) n − k − , . . . , k r (cid:19) + · · · + (cid:18) n − k , . . . , k r − (cid:19)(cid:21) x k · · · x k r r . The coefficient of x k · · · x k r r in ( x + · · · + x r ) n is thus described by (cid:0) n − k − ,...,k r (cid:1) + · · · + (cid:0) n − k ,...,k r − (cid:1) and also (cid:0) nk ,...,k r (cid:1) , so these must be equal. (cid:3) Using this recursive description of the multinomial coefficients one may derive an expression in terms offactorials, again analogous to the well-known formula for binomial coefficients. Lemma 5.2. Suppose n ≥ . For any partition ( k , . . . , k r ) ⊢ n we have (cid:18) nk , . . . , k r (cid:19) = n ! k ! · · · k r ! . Proof. We will work by induction, when n = 0 there is only the trivial partition (0 , . . . , ⊢ r 0, where ⊢ r denotes a partition into r parts. On both sides of the desired equality we have 1, establishing the base ofthe induction. Suppose n > k , . . . , k r ) ⊢ n . Using Lemma 5.1 and the induction hypothesiswe have (cid:18) nk , . . . , k r (cid:19) = (cid:18) n − k − , . . . , k r (cid:19) + · · · + (cid:18) n − k , . . . , k r − (cid:19) = k ( n − k ! · · · k r ! + · · · + k r ( n − k ! · · · k r != ( k + · · · + k r ) ( n − k ! · · · k r != n ! k ! · · · k r ! , where we remark for the reader that there is no conflict in the second equality with our convention fornonnegative n that (cid:0) nk ,...,k r (cid:1) = 0 whenever k i < i , for example if k = 0 then both (cid:0) n − k − ,...,k r (cid:1) and k n − k ! ··· k r ! are zero. (cid:3) Corollary 5.3. For n, k ≥ and ( k , . . . , k r ) ⊢ k we have the following identity relating binomial coefficientsand multinomial coefficients (cid:18) nk (cid:19)(cid:18) kk , . . . , k r (cid:19) = (cid:18) nn − k, k , . . . , k r (cid:19) . Proof. This is an immediate consequence of Lemma 5.2 and the well-known identity (cid:0) nk (cid:1) = n ! k !( n − k )! . (cid:3) Recall that we can make sense of (cid:0) nk (cid:1) for k ≥ n ∈ Z , i.e. for n ≥ (cid:0) − nk (cid:1) = ( − k (cid:0) n + k − k (cid:1) . Thus we can make sense of (cid:0) nk ,k ,...,k r (cid:1) for n, k ∈ Z by applying Corollary 5.3. Moreprecisely, when n, k < (cid:18) nk , k , . . . , k r (cid:19) = (cid:18) nn − k (cid:19)(cid:18) n − k k , . . . , k r (cid:19) = ( − n − k (cid:18) − k − n − k (cid:19)(cid:18) n − k k , . . . , k r (cid:19) , where − k − ≥ 0. By our conventions, the binomial coefficient (cid:0) − k − n − k (cid:1) is zero for n − k < 0, in particularthe multinomial coefficient (cid:0) nk ,k ,...,k r (cid:1) is zero whenever n < k .Our primary goal in this section is to understand how to use multinomial coefficients to expand as powerseries the negative powers of a polynomial in a single variable. The following lemma is slightly more generalbut will serve our purposes here. Lemma 5.4. For any n > and c ∈ k × we have (5.2) ( c + x + · · · + x r ) − n = X k ≥ X ( k ,...,k r ) ⊢ k ( − k (cid:18) n + k − n − , k , . . . , k r (cid:19) c − n − k x k · · · x k r r . Proof. We apply the standard binomial formula with negative exponents and then the definition of themultinomial coefficients to get( c + x + · · · + x r ) − n = c − n (cid:0) x + · · · + x r ) /c (cid:1) − n = X k ≥ ( − k (cid:18) n + k − k (cid:19) c − n − k ( x + · · · + x r ) k = X k ≥ X ( k ,...,k r ) ⊢ k ( − k (cid:18) n + k − k (cid:19)(cid:18) kk , . . . , k r (cid:19) c − n − k x k · · · x k r r = X k ≥ X ( k ,...,k r ) ⊢ k ( − k (cid:18) n + k − n − , k , . . . , k r (cid:19) c − n − k x k · · · x k r r , where the last equality follows from Corollary 5.3. (cid:3) ank 2 Generalized Greedy Bases 25 Remark 5.5. Notice that the result of Lemma 5.4 makes sense for any n ∈ Z . Indeed, applying the definitionof the standard (positive) multinomial coefficients to ( c + x + · · · + x r ) n for n ≥ we get ( c + x + · · · + x r ) n = X ( k ,k ,...,k r ) ⊢ n (cid:18) nk , k , . . . , k r (cid:19) c k x k · · · x k r r = n X k =0 X ( k ,...,k r ) ⊢ k (cid:18) nn − k, k , . . . , k r (cid:19) c n − k x k · · · x k r r = X k ≥ X ( k ,...,k r ) ⊢ k (cid:18) nk (cid:19)(cid:18) kk , . . . , k r (cid:19) c n − k x k · · · x k r r = X k ≥ X ( k ,...,k r ) ⊢ k ( − k (cid:18) − n + k − k (cid:19)(cid:18) kk , . . . , k r (cid:19) c n − k x k · · · x k r r = X k ≥ X ( k ,...,k r ) ⊢ k ( − k (cid:18) − n + k − − n − , k , . . . , k r (cid:19) c n − k x k · · · x k r r , which agrees with (5.2) as claimed. To complete our discussion of multinomial coefficients consider a polynomial P ( z ) ∈ k [ z ] of degree d andwrite P ( z ) = ρ + ρ z + · · · + ρ d z d . Corollary 5.6. 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