Catalan functions and k -Schur positivity
aa r X i v : . [ m a t h . C O ] A p r CATALAN FUNCTIONS AND k -SCHUR POSITIVITY JONAH BLASIAK, JENNIFER MORSE, ANNA PUN, AND DANIEL SUMMERS
Abstract.
We prove that graded k -Schur functions are G -equivariant Euler character-istics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. Weexpose a new miraculous shift invariance property of the graded k -Schur functions andresolve the Schur positivity and k -branching conjectures in the strongest possible termsby providing direct combinatorial formulas using strong marked tableaux. Introduction
We resolve conjectures made in [23], [7], and [22] which were inspired by problemson Macdonald polynomials. These remarkable polynomials form a basis for the ringof symmetric functions over the field Q ( q, t ). Their study over the last three decadeshas generated an impressive body of research, a prominent focus being the Macdonaldpositivity conjecture: the Schur expansion coefficients of the (Garsia) modified Macdonaldpolynomials H µ ( x ; q, t ) lie in N [ q, t ]. This was proved by Haiman [13] using geometry ofHilbert schemes, yet many questions arising in this study remain unanswered.Lapointe, Lascoux, and Morse [23] considerably strengthened the Macdonald positivityconjecture. They constructed a family of functions and conjectured (i) they form a basisfor the space Λ k = span Q ( q,t ) { H µ ( x ; q, t ) } µ ≤ k , (ii) they are Schur positive, and (iii) theexpansion of H µ ( x ; q, t ) ∈ Λ k in this basis has coefficients in N [ q, t ]. The problem of Schurexpanding Macdonald polynomials thus factors into the two positivity problems (ii) and(iii). Because the intricate construction of these functions lacked in mechanism for proof,many conjecturally equivalent candidates have since been proposed. Informally, all thesecandidates are now called k -Schur functions .At the forefront of k -Schur investigations is the conjectured branching property :the k + 1-Schur expansion of a k -Schur function has coefficients in N [ t ] . (1.1)Since a k -Schur function reduces to a Schur function for large k , the iteration of branchingimplies (ii). However, every effort to prove that even one of the k -Schur candidates satisfies(i) and (1.1), or even (i) and (ii), over the last decades has failed.In contrast, the ungraded case has been more tractable. The k -Schur concept is inter-esting even when t = 1 despite needing generic t for Macdonald polynomial applications.The existence of an ungraded family of k -Schur functions satisfying (the t = 1 versionsof) (i), (ii), and (1.1) was established for functions s ( k ) λ ( x ) defined in [26] using chains inweak order of the affine symmetric group b S k +1 . It was proven that they form a basis Key words and phrases. k -Schur functions, Schur positivity, branching rule, spin, strong tableaux,generalized Kostka polynomials.Authors were supported by NSF Grants DMS-1600391 (J. B.) and DMS-1833333 (J. M.). for Λ k | t =1 with the Gromov-Witten invariants for the quantum cohomology of the Grass-mannian included in their structure constants [27]. Lam established that the basis s ( k ) λ ( x ) represents Schubert classes in the homology of the affine Grassmannian Gr SL k +1 of SL k +1 [19] and gave a geometric proof of (1.1) at t = 1: the branching property re-flects that the image of a Schubert class is a positive sum of Schubert classes under aninclusion H ∗ (Gr SL k +1 ) → H ∗ (Gr SL k +2 ) [20]. Additionally, an algorithm for computing thebranching coefficients using an intricate equivalence relation is worked out in [21, 22].Building off the work on the ungraded case, a candidate { s ( k ) λ ( x ; t ) } for k -Schur functionswas proposed in [21] by attaching a nonnegative integer called spin to strong markedtableaux , certain chains in the strong (Bruhat) order of b S k +1 . It was conjectured thatthey satisfy the desired properties (i)–(iii) and shown [21] that they are equal to the weakorder k -Schur functions s ( k ) λ ( x ) at t = 1.In a different vein, Li-Chung Chen and Mark Haiman [7] conjectured that k -Schur func-tions are a subclass of a family of symmetric functions indexed by pairs (Ψ , γ ) consisting ofan upper order ideal Ψ of positive roots (of which there are Catalan many) and a weight γ ∈ Z ℓ . These Catalan (symmetric) functions can be defined by a Demazure-operatorformula, and are equal to GL ℓ -equivariant Euler characteristics of vector bundles on theflag variety by the Borel-Weil-Bott theorem. Chen-Haiman [7] investigated their Schurexpansions and conjectured a positive combinatorial formula when γ ≥ γ ≥ · · · . Pa-nyushev [30] studied similar questions and proved a cohomological vanishing theorem toestablish Schur positivity of a large subclass of Catalan functions.Catalan functions are the modified Hall-Littlewood polynomials when the ideal of rootsΨ consists of all the positive roots and γ is a partition. Here their Schur expansion coef-ficients are the Kostka-Foulkes polynomials (Lusztig’s t -analog of weight multiplicities intype A), which have been extensively studied from algebraic, geometric, and combinatorialperspectives (see, e.g., [29, 14, 8]). In the case that Ψ consists of the roots above a blockdiagonal matrix, the Schur expansion coefficients are the generalized Kostka polynomials investigated by Broer, Shimozono-Weyman, and others [5, 34, 32, 31, 33, 18].Chen-Haiman constructed an ideal of roots associated to each partition λ with λ ≤ k and conjectured that the associated Catalan functions { s ( k ) λ ( x ; t ) } are k -Schur functions [7].A key discovery in our work is an elegant set of ideals of roots which we show yields thesame family { s ( k ) λ ( x ; t ) } . From there, we prove • (Chen-Haiman conjecture) The polynomials s ( k ) λ ( x ; t ) are the k -Schur functions s ( k ) λ ( x ; t ). • ( k -Schur branching) The coefficients of (1.1) are P t spin( T ) over certain skew strongtableaux T ; this settles Conjecture 1.1 of [22] in the strongest possible terms. • (Schur positive basis) The functions s ( k ) λ ( x ; t ) are a Schur positive basis of Λ k . This re-solves step (ii) in the route for finding the Schur expansion of Macdonald polynomialsand finally establishes that a k -Schur candidate satisfies (i), (ii), and (1.1). • (Dual Pieri rule) The polynomials s ( k ) λ ( x ; t ) satisfy a vertical-strip defining rule. • (Shift invariance) s ( k ) λ ( x ; t ) = e ⊥ ℓ s ( k +1) λ +1 ℓ ( x ; t ). This powerful new discovery followseasily from our definition of s ( k ) λ ( x ; t ) and, together with the dual Pieri rule, it imme-diately yields our k -Schur branching rule. ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 3 Main results
We work in the ring Λ = Q ( t )[ h , h , . . . ] of symmetric functions in infinitely manyvariables x = ( x , x , . . . ), where h d = h d ( x ) = P i ≤···≤ i d x i · · · x i d . The Schur functions s λ indexed by partitions λ form a basis for Λ. Schur functions may be defined moregenerally for any γ ∈ Z ℓ by the following version of the Jacobi-Trudi formula: s γ = s γ ( x ) = det( h γ i + j − i ( x )) ≤ i,j ≤ ℓ ∈ Λ , (2.1)where by convention h ( x ) = 1 and h d ( x ) = 0 for d < s γ , as well as the modified Hall-Littlewoodpolynomials and generalizations thereof studied by Broer and Shimozono-Weyman [5, 34].They can be described geometrically in terms of cohomology of vector bundles on the flagvariety (see Theorem 3.1), or algebraically as follows: Definition 2.1.
Fix a positive integer ℓ . A root ideal is an upper order ideal of the poset∆ + ℓ = ∆ + := { ( i, j ) | ≤ i < j ≤ ℓ (cid:9) with partial order given by ( a, b ) ≤ ( c, d ) when a ≥ c and b ≤ d . Given a root ideal Ψ ⊂ ∆ + ℓ and γ ∈ Z ℓ , the associated Catalan function is H (Ψ; γ )( x ; t ) := Y ( i,j ) ∈ Ψ (cid:0) − tR ij (cid:1) − s γ ( x ) ∈ Λ , (2.2)where the raising operator R ij acts on the subscripts of the s γ by R ij s γ = s γ + ǫ i − ǫ j (a discussion of raising operators is given in § kℓ = { ( µ , . . . , µ ℓ ) ∈ Z ℓ : k ≥ µ ≥ · · · ≥ µ ℓ ≥ } denote the set of partitionscontained in the ℓ × k -rectangle and Par k the set of partitions µ with µ ≤ k . The trailingzeros are a useful bookkeeping device for elements of Par kℓ , whereas for Par k we adoptthe more common convention: each µ ∈ Par k is identified with ( µ , . . . , µ ℓ ( µ ) , i ) for any i ≥
0, where the length ℓ ( µ ) is the number of nonzero parts of µ . Definition 2.2.
For µ ∈ Par kℓ , define the root ideal∆ k ( µ ) = { ( i, j ) ∈ ∆ + ℓ | k − µ i + i < j } , (2.3)and the Catalan function s ( k ) µ ( x ; t ) := H (∆ k ( µ ); µ ) = ℓ Y i =1 ℓ Y j = k +1 − µ i + i (cid:0) − tR ij (cid:1) − s µ ( x ) . (2.4)We will soon see (Theorem 2.4) that the s ( k ) µ ( x ; t ) are the k -Schur functions, whichproves a conjecture of Chen-Haiman. This is a consequence of four fundamental proper-ties of these Catalan functions described in the next theorem. Their statement requiresthe definition of strong marked tableaux, which are certain saturated chains in the strongBruhat order for the affine symmetric group b S k +1 , together with some extra data. Theprecise definition is most readily given in terms of partitions arising in modular represen-tation theory called k + 1-cores. Combinatorial examples are provided in § diagram of a partition λ is the subset of boxes { ( r, c ) ∈ Z ≥ × Z ≥ | c ≤ λ r } in the plane, drawn in English (matrix-style) notation so that rows (resp. columns) are JONAH BLASIAK, JENNIFER MORSE, ANNA PUN, AND DANIEL SUMMERS increasing from north to south (resp. west to east). Each box has a hook length whichcounts the number of boxes below it in its column and weakly to its right in its row. A k + 1 -core is a partition with no box of hook length k + 1. There is a bijection p [25] fromthe set of k + 1-cores to Par k mapping a k + 1-core κ to the partition λ whose r -th row λ r is the number of boxes in the r -th row of κ having hook length ≤ k .A strong cover τ ⇒ κ is a pair of k + 1-cores such that τ ⊂ κ and | p ( τ ) | + 1 = | p ( κ ) | .A strong marked cover τ r = ⇒ κ is a strong cover τ ⇒ κ together with a positive integer r which is allowed to be the smallest row index of any connected component of the skewshape κ/τ . Let η = ( η , η , . . . ) ∈ Z ∞≥ with m = | η | := P i η i finite. A strong markedtableau T of weight η is a sequence of strong marked covers κ (0) r == ⇒ κ (1) r == ⇒ · · · r m == ⇒ κ ( m ) such that r v i +1 ≥ r v i +2 ≥ · · · ≥ r v i + η i for all i ≥
1, where v i := η + · · · + η i − . A vertical strong marked tableau is defined the same way except we require each subsequence r v i +1 < r v i +2 < · · · < r v i + η i to be strictly increasing rather than weakly decreasing. Wewrite inside( T ) = p ( κ (0) ) and outside( T ) = p ( κ ( m ) ). The set of strong marked tableaux(resp. vertical strong marked tableaux) T of weight η with outside( T ) = µ is denotedSMT kη ( µ ) (resp. VSMT kη ( µ )).The spin of a strong marked cover τ r = ⇒ κ is defined to be c · ( h −
1) + N , where c isthe number of connected components of the skew shape κ/τ , h is the height (number ofrows) of each component, and N is the number of components entirely contained in rows > r . For a (vertical) strong marked tableau T , spin( T ) is defined to be the sum of thespins of the strong marked covers comprising T .For f ∈ Λ, let f ⊥ be the linear operator on Λ that is adjoint to multiplication by f with respect to the Hall inner product, i.e., h f ⊥ ( g ) , h i = h g, f h i for all g, h ∈ Λ. Theorem 2.3.
Fix a positive integer ℓ . The Catalan functions { s ( k ) µ } k ≥ , µ ∈ Par kℓ , satisfythe following properties: (horizontal dual Pieri rule) h ⊥ d s ( k ) µ = X T ∈ SMT k ( d ) ( µ ) t spin( T ) s ( k )inside( T ) for all d ≥ e ⊥ d s ( k ) µ = X T ∈ VSMT k ( d ) ( µ ) t spin( T ) s ( k )inside( T ) for all d ≥ s ( k ) µ = e ⊥ ℓ s ( k +1) µ +1 ℓ ; (2.7)(Schur function stability) if k ≥ | µ | , then s ( k ) µ = s µ . (2.8)Theorem 2.3 has several powerful consequences. Foremost is that the s ( k ) µ are k -Schurfunctions. More precisely, we adopt the following definition of the k -Schur functions from[21, § µ ∈ Par k , let s ( k ) µ ( x ; t ) = X η ∈ Z ∞≥ , | η | = | µ | X T ∈ SMT kη ( µ ) t spin( T ) x η . (2.9) ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 5 Among the several conjecturally equivalent definitions, this one has the advantage thatits t = 1 specializations { s ( k ) µ ( x ; 1) } are known to agree [21] with a quite different lookingcombinatorial definition using weak tableaux from [26], and are Schubert classes in thehomology of the affine Grassmannian Gr SL k +1 of SL k +1 [19]. Theorem 2.4.
For µ ∈ Par kℓ , the k -Schur function s ( k ) µ ( x ; t ) is the Catalan function s ( k ) µ ( x ; t ) .Proof. The homogeneous symmetric function basis for Λ consists of h λ = Q i h λ i , as λ ranges over partitions. For any partition λ of size | µ | , (2.5) implies h s ( k ) µ , h λ i = h h ⊥ λ s ( k ) µ , i = X T ∈ SMT kλ ( µ ) t spin( T ) , where we have also used (2.8) to obtain s ( k )inside( T ) = s ( k )0 ℓ = 1 for every T in the sum. Thebasis of monomial symmetric functions { m ν ( x ) } is dual to the homogenous basis and thus s ( k ) µ ( x ; t ) = X partitions λ of | µ | X T ∈ SMT kλ ( µ ) t spin( T ) m λ ( x ) . (2.10)Let η ∈ Z ∞≥ with | η | = | µ | and let λ be the partition obtained by sorting η . Since the h ⊥ d pairwise commute, again using (2.5) and (2.8) we obtain X T ∈ SMT kη ( µ ) t spin( T ) = ( h ⊥ η h ⊥ η · · · ) s ( k ) µ = ( h ⊥ λ h ⊥ λ · · · ) s ( k ) µ = X T ∈ SMT kλ ( µ ) t spin( T ) . (2.11)Thus the right sides of (2.10) and (2.9) agree, as desired. (cid:3) Theorem 2.4 has several important consequences; we give informal statements now,deferring their precise versions for later.
Corollary 2.5. (1)
The k -Schur functions s ( k ) µ ( x ; t ) defined by (2.9) are symmetric functions. (2) The k -Schur functions s ( k ) µ ( x ; t ) are equal to Catalan functions defined by Chen-Haiman via a different root ideal than our ∆ k ( µ ) (see Section 10). (3) For µ ∈ Par kℓ , s ( k ) µ ( x ; t ) is the GL ℓ -equivariant Euler characteristic of a vectorbundle on the flag variety determined by µ and k . (4) Homology Schubert classes of Gr SL k +1 are equal to the ungraded ( t = 1 ) versionof this Euler characteristic.Proof. Theorem 2.4 allows us to work interchangeably with s ( k ) µ ( x ; t ) and s ( k ) µ ( x ; t ) andwe do so from now on without further mention. Symmetry follows directly from thedefinition of the Catalan functions. We match s ( k ) µ to the Catalan functions studied byChen-Haiman in Theorem 10.4 to settle (2). Statement (3) is proved in Theorem 3.1, andthis in turn implies (4) using [21, Theorem 4.11] and [19, Theorem 7.1]. (cid:3) JONAH BLASIAK, JENNIFER MORSE, ANNA PUN, AND DANIEL SUMMERS
The realization of the k -Schur functions as the subclass (2.4) of Catalan functions ishighly lucrative. One of the most striking outcomes is Property (2.7); overlooked in priorinvestigations of k -Schur functions, it fully resolves k -branching: The coefficients in the k + 1 -Schur expansion of a k -Schur function are none otherthan the structure coefficients appearing in the vertical dual Pieri rule (2.6) . Theorem 2.6 ( k -Schur branching rule) . For µ ∈ Par kℓ , the expansion of the k -Schurfunction s ( k ) µ into k + 1 -Schur functions is given by s ( k ) µ = X T ∈ VSMT k +1( ℓ ) ( µ +1 ℓ ) t spin( T ) s ( k +1)inside( T ) . (2.12)The Schur expansion of a k -Schur function can then be achieved by incrementallyiterating (2.12) until k is large enough to apply (2.8). Interestingly, a more elegantformula can be derived by a different combination of (2.6), (2.7), and (2.8). Theorem 2.7 ( k -Schur into Schur) . Let µ ∈ Par kℓ and set m = max( | µ | − k, . TheSchur expansion of the k -Schur function s ( k ) µ is given by s ( k ) µ = X T ∈ VSMT k + m ( ℓm ) ( µ + m ℓ ) t spin( T ) s inside( T ) . (2.13) Proof.
Apply (2.7) m times to obtain s ( k ) µ = ( e ⊥ ℓ ) m s ( k + m ) µ + m ℓ . (2.14)The vertical dual Pieri rule (2.6) then gives the ( k + m )-Schur function decompositionand (2.8) ensures this is the Schur function decomposition by the careful choice of m . (cid:3) See Example 2.13 for Theorem 2.6 and Example 2.14 and Figure 1 for Theorem 2.7.Recall from the introduction that the original k -Schur candidate of [23] (as well assubsequent candidates) conjecturally satisfies (i)–(ii), i.e., forms a Schur positive basisfor the space Λ k = span Q ( q,t ) { H µ ( x ; q, t ) } µ ≤ k . We settle this conjecture for the k -Schurfunctions of (2.9), thereby giving the first proof that a k -Schur candidate satisfies (i)–(ii).This follows from Theorem 2.7 together with the next result which additionally refinesstatement (i) to give bases for subspaces Λ kℓ which depend on ℓ as well as k . Theorem 2.8.
For any positive integers ℓ and k , the k -Schur functions { s ( k ) µ ( x ; t ) } µ ∈ Par kℓ form a basis for the space Λ kℓ = span Q ( t ) { H µ ( x ; t ) | µ ∈ Par kℓ } ⊂ Λ . Here H µ ( x ; t ) = H µ ( x ; 0 , t ) are the modified Hall-Littlewood polynomials (denoted Q ′ µ ( x ; t ) in [29, p. 234]). Note that Q ( q, t ) ⊗ Q ( t ) ( P ℓ Λ kℓ ) = Λ k since Λ k has the alternativedescription Λ k = span Q ( q,t ) { H µ ( x ; t ) | µ ∈ Par k } ; this follows from the fact that the Schurfunctions and Macdonald’s integral forms J µ ( x ; q, t ) are related by a triangular change ofbasis—see, e.g., Equations 6.6, 2.18, and 2.20 of [24].We also give three intrinsic descriptions of the k -Schur functions. Studies of ungraded k -Schur functions have been served well by the characterization of s ( k ) µ ( x ; 1) as the unique ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 7 symmetric functions satisfying (2.5) and (2.8) at t = 1. We can now show that the k -Schurfunctions s ( k ) µ ( x ; t ) of (2.9) have this characterization (without the t = 1 specialization),as well as a new one coming from the shift invariance property (2.7). Corollary 2.9.
The k -Schur functions { s ( k ) µ } k ≥ , µ ∈ Par kℓ are the unique family of symmetricfunctions satisfying the following subsets of properties (2.5) – (2.8) . (1) (2.5) and (2.8) ; (2) (2.6) and (2.8) ; (3) (2.6) for d = ℓ , (2.7) , and (2.8) .Proof. The proof of Theorem 2.4 establishes (1). By a similar argument, (2.6) and (2.8)imply s ( k ) µ ( x ; t ) = X partitions λ of | µ | X T ∈ VSMT kλ ( µ ) t spin( T ) ω ( m λ ( x )) , (2.15)where ω is the involution on Λ defined by ω ( e d ) = h d for d ≥
0. This establishes (2). Theproperties in (3) determine a unique family of symmetric functions by Theorem 2.7. (cid:3)
Outline.
The bulk of our paper is devoted to developing machinery to prove thevertical dual Pieri rule (2.6). The remaining results are fairly straightforward; we prove(2.7), (2.8), and Theorem 2.8 in Section 4, and (2.5) in § • We prove that e ⊥ d ( H (Ψ; γ )) = P S ⊂ [ ℓ ] , | S | = d H (Ψ; γ − ǫ S ) (Lemma 4.11), which isour starting point for evaluating the left side of (2.6). • To handle the terms H (Ψ; γ − ǫ S ) in this sum, we prove a k -Schur straighteningrule (Theorem 7.12) which shows that analogs of the s ( k ) µ indexed by nonpartitionsare equal to 0 or to a power of t times s ( k ) ν for partition ν . • Miraculously, the combinatorics arising in this rule exactly matches that of strongcovers (Proposition 8.10). • To prove the k -Schur straightening rule, we develop several tools for working withCatalan functions including a recurrence which expresses a Catalan function asthe sum of two such polynomials with similar root ideals (Proposition 5.6). • To prove (2.6) by induction on d , we must prove a stronger statement in whichthe right side of (2.6) is replaced by a sum over tableaux which are marked onlyin rows ≤ m , and the left side is an algebraically defined generalization of e ⊥ d s ( k ) µ .2.2. Combinatorial examples.Example 2.10.
The 5-core κ = 53221 and its image p ( κ ) = 32221 ∈ Par are κ = p ( κ ) = , JONAH BLASIAK, JENNIFER MORSE, ANNA PUN, AND DANIEL SUMMERS where the boxes of κ are labeled by their hook lengths.A (vertical) strong marked tableau T = ( κ (0) r == ⇒ κ (1) r == ⇒ · · · r m == ⇒ κ ( m ) ) is drawnby filling each skew shape κ ( i ) /κ ( i − with the entry i and starring the entry in position( r i , κ ( i ) r i ), for all i ∈ [ m ]. (This is really the standardization of T , but suffices for theexamples in this paper as we will always specify the weight η separately.) Strong markedcovers are drawn this way too, regarding them as strong marked tableaux of weight (1). Example 2.11.
Let k = 4. For τ = 663331111 and κ = 665443221, p ( τ ) = 332221111and p ( κ ) = 222222221. Thus τ ⇒ κ is a strong cover and it has two distinct markings: ⋆
11 1 1 ⋆
111 111 τ = ⇒ κ τ = ⇒ κ spin = 4 spin = 5 Remark 2.12.
Although strong marked covers are typically marked by the content of thenortheastmost box of a connected component of κ/τ , it is equivalent (and more naturalfor us) to use row indices.Given a k + 1-core κ and λ = p ( κ ), the k -skew diagram of λ denoted k -skew( λ ), isthe subdiagram of κ consisting of boxes with hook length ≤ k . Hence the row lengths of k -skew( λ ) are given by λ itself. Example 2.13.
According to Theorem 2.6, the expansion of s (3)22221 into 4-Schur functionsis obtained by summing t spin( T ) s (4)inside( T ) over the set VSMT (33332) of vertical strongmarked tableaux given below. Note that 86532 = p − (33332) is the outer shape of eachdiagram on the first line. T ⋆ ⋆
42 3 ⋆ ⋆ ⋆ ⋆ ⋆
41 3 ⋆
52 4 ⋆ ⋆ ⋆ ⋆
41 3 3 ⋆
52 4 ⋆ ⋆ ⋆ ⋆
43 3 ⋆ ⋆ ⋆ k -skew(inside( T ))inside( T ) 3222 3321 33111 22221spin( T ) 2 2 2 0 s (3)22221 = t s (4)3222 + t s (4)3321 + t s (4)33111 + s (4)22221 . ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 9 ⋆ ⋆ ⋆ A A A B C ⋆ ⋆ A A A⋆ B C ⋆ ⋆ A A A B⋆ C ⋆ ⋆ B C⋆ ⋆ ⋆ ⋆ A A A B C ⋆ ⋆ A A A⋆ B C ⋆ ⋆ A A A B⋆ C ⋆ ⋆ B C⋆ ⋆ ⋆ ⋆ A A A B C ⋆ ⋆ A A A⋆ B C ⋆ ⋆ A A A B⋆ C ⋆ ⋆ B C⋆ ⋆ ⋆ ⋆ A A A B C ⋆ ⋆ A A A⋆ B C ⋆ ⋆ A A A B⋆ C ⋆ ⋆ B C⋆ ⋆ ⋆ ⋆ A A A B C ⋆ ⋆ A A A⋆ B C ⋆ ⋆ A A A B⋆ C ⋆ ⋆ B C⋆ ⋆ ⋆ ⋆ A B B B B C ⋆ ⋆ A⋆ B B B B C ⋆ ⋆ B B B B⋆ C ⋆ ⋆ C⋆ ⋆ ⋆ ⋆ A B B B B C ⋆ ⋆ A⋆ B B B B C ⋆ ⋆ B B B B⋆ C ⋆ ⋆ C⋆ ⋆ ⋆ ⋆ A B B B B C ⋆ ⋆ A⋆ B B B B C ⋆ ⋆ B B B B⋆ C ⋆ ⋆ C⋆ ⋆ ⋆ ⋆ A B B B B C ⋆ ⋆ A⋆ B B B B C ⋆ ⋆ B B B B⋆ C ⋆ ⋆ C⋆ ⋆ ⋆ ⋆ A B B B B C ⋆ ⋆ A⋆ B B B B C ⋆ ⋆ B B B B⋆ C ⋆ ⋆ C⋆ Spin6543210Figure 1: According to Theorem 2.7 with k = 1, ℓ = 4, the Schur expansion of the 1-Schurfunction s (1)1111 (also equal to the modified Hall-Littlewood polynomial H ) is obtainedby summing t spin( T ) s inside( T ) over the set VSMT , , (4 , , ,
4) of vertical strong markedtableaux T given above. We have written A, B, C in place 10 , ,
12. Note that T havingweight ℓ m = (4 , ,
4) means that the skew shapes (strong covers) labeled by 1 , , . . . , ℓ have stars in rows 1 , , . . . , ℓ , as do the next ℓ skew shapes, and so on. Example 2.14.
We compute the Schur expansion of s (4)3321 using the (proof of) Theo-rem 2.7: it is given by the sum t spin( T ) s inside( T ) over the set VSMT , , (6654) of verticalstrong marked tableaux given below. Note that the m in Theorem 2.7 is a convenientchoice, but often a smaller m suffices; the 7-Schur expansion of s (4)3321 is already the Schurexpansion, so m = 3 (hence k + m = 7) suffices. ⋆ ⋆ ⋆ B ⋆ ⋆ A⋆ C ⋆ ⋆ B⋆ ⋆ ⋆ A C⋆ ⋆ ⋆ ⋆ B ⋆ ⋆ A⋆ C ⋆ ⋆ B⋆ ⋆ ⋆ A C⋆ ⋆ ⋆ ⋆ B ⋆ ⋆ A A A A⋆ C ⋆ ⋆ B⋆ ⋆ ⋆ C⋆ ⋆ ⋆ ⋆ B ⋆ ⋆ A A A A⋆ C ⋆ ⋆ B⋆ ⋆ ⋆ C⋆ ⋆ ⋆ ⋆ B ⋆ ⋆ A A A A⋆ C ⋆ ⋆ B⋆ ⋆ ⋆ C⋆ ⋆ ⋆ ⋆ B ⋆ ⋆ A A A A⋆ C ⋆ ⋆ B⋆ ⋆ ⋆ C⋆ Spin3210 s (4)3321 = t s + t ( s + s ) + t ( s + s ) + s . Catalan functions as G -equivariant Euler characteristics We first review the geometric description of Catalan functions from [7, 30], and thensummarize prior work on these polynomials. This serves to provide context for our resultsbut is not necessary for the remainder of the paper.Let G = GL ℓ ( C ), B ⊂ G the standard lower triangular Borel subgroup, and H ⊂ B thesubgroup of diagonal matrices. The character group of H (integral weights) are identifiedwith Z ℓ via the correspondence sending γ ∈ Z ℓ to the character H → C × given bydiag( z , . . . , z ℓ ) z γ · · · z γ ℓ ℓ .The character of a G -module M is defined bych( M ) = X partitions λℓ ( λ ) ≤ ℓ dim Hom( V λ , M ) s λ ∈ Λ , (3.1)where V λ denotes the irreducible G -module of highest weight λ .Given a B -module N , let G × B N denote the homogeneous G -vector bundle on G/B with fiber N above B ∈ G/B , and let L G/B ( N ) denote the locally free O G/B -module of its
ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 11 sections. For γ ∈ Z ℓ , let C γ denote the one-dimensional B -module of weight γ . Considerthe adjoint action of B on the Lie algebra n of strictly lower triangular matrices. The B -stable (or “ad-nilpotent”) ideals of n are in bijection with root ideals via the map sendingthe root ideal Ψ to the B -submodule, call it N Ψ , of n with weights { ǫ j − ǫ i | ( i, j ) ∈ Ψ } .Note that the dual N ∗ Ψ has weights given by Ψ.The Catalan functions appear naturally as certain G -equivariant Euler characteristics,as the following result shows. Theorem 3.1 ([7, 30]) . Let (Ψ , γ ) be an indexed root ideal. Let S j N ∗ Ψ denote the j -thsymmetric power of the B -module N ∗ Ψ . Then H (Ψ; γ ) = X i,j ≥ ( − i t j ch (cid:16) H i (cid:0) G/B, L G/B ( S j N ∗ Ψ ⊗ C γ ) (cid:1)(cid:17) . (3.2) Proof.
This is a consequence of the Borel-Weil-Bott theorem and follows from a straight-forward extension of the argument going from Equation 2.1 to Equation 2.4 in [34]. (cid:3)
Remark 3.2. (1) A result essentially the same as this one is proved in [30, Theorem 3.8] by adaptinga proof [15] for the case Ψ = ∆ + .(2) For the precise statement here, we have followed the conventions of [34], which conve-niently handles a duality in Borel-Weil-Bott. Note that had we chosen to use the up-per triangular Borel subgroup B ′ instead, Borel-Weil-Bott implies that χ G/B ′ ( γ ) := P i ≥ ( − i ch (cid:0) H i ( G/B ′ , L G/B ′ ( C γ )) (cid:1) = s ( γ ℓ ,...,γ ) , while χ G/B ( γ ) = H ( ∅ ; γ ) = s γ .See [17, §
3] for a nice explanation of this duality.(3) A version of (3.2) in fact holds for any B -module N , with the left side replaced bya raising operator formula over the multiset of weights of N . However, restrictingto the N ∗ Ψ is natural from the geometric perspective; [30] allows more general (butnot arbitrary) B -modules N than the N ∗ Ψ considered here.(4) By our definition (2.1) of s λ , s λ = 0 for weakly decreasing λ ∈ Z ℓ with λ ℓ <
0, soboth sides of (3.2) record only the polynomial representations. This differs from[34, 30], which give versions of (3.2) without the polynomial truncation.
Conjecture 3.3 ([7, Conj. 5.4.3]) . For any partition γ and root ideal Ψ , the cohomologyin (3.2) vanishes for i > and hence H (Ψ; γ ) is a Schur positive symmetric function. As previously mentioned, the Catalan functions H (∆ + ; µ ) for partition µ are the mod-ified Hall-Littlewood polynomials (see Proposition 4.6). In this case, the Schur expansioncoefficients are the Kostka-Foulkes polynomials, which have been extensively studied (see,e.g., [29, 14, 8]). Another well-studied class of Catalan functions are the parabolic Hall-Littlewood polynomials —the case Ψ = ∆( η ) for some η ∈ Z r ≥ , where∆( η ) := (cid:8) α ∈ ∆ + | η | above the block diagonal with block sizes η , . . . , η r (cid:9) . For example, ∆(1 , ,
2) = . Broer proved that in the case Ψ = ∆ + , the cohomology in (3.2) vanishes for i > γ i − γ j ≥ − i < j (see [4, Theorem 2.4] and [3, Proposition 2(iii)]). Broer posedConjecture 3.3 in the parabolic case Ψ = ∆( η ) (see, e.g., [34, Conjecture 5]) and proved itin the subcase γ is a dominant sequence of rectangles , meaning that γ = ( a η , a η , . . . , a η r r )for some a ≥ a ≥ · · · ≥ a r [5, Theorem 2.2].Panyushev proved that the cohomology in (3.2) vanishes for i > γ − ρ + P ( i,j ) ∈ ∆ + \ Ψ ǫ i − ǫ j is weakly decreasing, where ρ = ( ℓ − , ℓ − , . . . ,
0) (see [30,Theorem 3.2] for the full statement of which this is a special case); this includes the casewhere γ is a partition with distinct parts and Ψ is any root ideal as well as many instanceswhere γ is not a partition.A natural combinatorial problem arising here is to find a positive combinatorial formulafor the Schur expansion of H (Ψ; γ ) when γ is a partition. Shimozono-Weyman posed a(still open) conjecture that the Schur expansion of H (∆( η ); γ ) when γ is a partition canbe described using an intricate combinatorial procedure called katabolism [34]. Progresshas been made in the case γ is a dominant sequence of rectangles: the Schur expan-sion was described by Schilling-Warnaar [31] and Shimozono [32] (independently) usinga cocyclage poset on Littlewood-Richardson tableaux; by Shimozono [33] using affineDemazure crystals; and by A. N. Kirillov-Schilling-Shimozono [18] using rigged configura-tions. Chen-Haiman conjectured a generalization of the Shimozono-Weyman katabolismformula to any root ideal Ψ and partition γ [7, Conjecture 5.4.3].4. Catalan functions
We elaborate on the definition of the Catalan functions H (Ψ; γ ) and give anotherdescription of these polynomials using Hall-Littlewood vertex operators. We then prove(2.7), (2.8), and Theorem 2.8, along the way establishing some basic facts about Catalanfunctions which may have further applications beyond this paper.4.1. Notation.
Throughout the paper we use the following notation/conventions: Wewrite [ a, b ] for the interval { i ∈ Z | a ≤ i ≤ b } and [ n ] := [1 , n ]. For i ∈ [ ℓ ], we write ǫ i ∈ Z ℓ for the weight with a 1 in position i and 0’s elsewhere, and for a set S ⊂ [ ℓ ],denote ǫ S = P i ∈ S ǫ i . We often omit set braces on singleton sets to avoid clutter.4.2. Indexed root ideals.
We regard the set ∆ + ℓ = ∆ + := (cid:8) ( i, j ) | ≤ i < j ≤ ℓ (cid:9) aslabels for the set of positive roots of the root system of type A ℓ − (though for brevity werefer to elements of ∆ + as roots as well). For α = ( i, j ) ∈ ∆ + ℓ , we write ε α = ǫ i − ǫ j ∈ Z ℓ for the corresponding positive root (not to be confused with ǫ { i,j } = ǫ i + ǫ j ). As previouslymentioned, a root ideal is an upper order ideal of the poset ∆ + with partial order givenby ( a, b ) ≤ ( c, d ) when a ≥ c and b ≤ d . We also work with the complement ∆ + \ Ψ, alower order ideal of ∆ + .An indexed root ideal of length ℓ is a pair (Ψ , γ ) consisting of a root ideal Ψ ⊂ ∆ + ℓ anda weight γ ∈ Z ℓ .Given an indexed root ideal (Ψ , γ ) of length ℓ , we represent the Catalan function H (Ψ; γ ) by the ℓ × ℓ grid of boxes (labeled by matrix-style coordinates as in Figure 2), ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 13 , , , , , , , , , , Ψ = { (1 , , (1 , , (1 , , (2 , , (2 , , (3 , , (3 , } ∆ + \ Ψ = { (1 , , (2 , , (4 , } Figure 2: For ℓ = 5, a root ideal Ψ and its complement ∆ + \ Ψ:with the boxes of Ψ shaded and the entries of γ written along the diagonal. For example,with Ψ as in Figure 2 and γ = 33411, H (Ψ; γ ) = . This is the most convenient way we have found to compute with these polynomials.4.3.
Catalan functions.
Recall from (2.1) that for any γ ∈ Z ℓ , the Schur function s γ isdefined to be det( h γ i + j − i ). The s γ are expressed in the basis of Schur functions indexedby partitions by the following straightening rule : Proposition 4.1 (Schur function straightening) . For any γ ∈ Z ℓ , s γ ( x ) = ( sgn( γ + ρ ) s sort( γ + ρ ) − ρ ( x ) if γ + ρ has distinct nonnegative parts, otherwise,where ρ = ( ℓ − , ℓ − , . . . , , sort( β ) denotes the weakly decreasing sequence obtained bysorting β , and sgn( β ) denotes sign of the shortest permutation taking β to sort( β ) .Proof. This is a consequence of the Jacobi-Trudi formula s λ = det( h λ i + j − i ) for partitions λ and the row interchange property of the determinant. (cid:3) Corollary 4.2.
For a fixed degree d ∈ Z , the Schur function s γ is nonzero for only finitelymany of the weights { γ ∈ Z ℓ : | γ | = d } . Example 4.3.
For ℓ = 4 and γ = (3 , , , s γ ( x ) = 0 since γ + ρ = (6 , , ,
5) doesnot have distinct parts. For γ = (4 , , , s ( x ) = s ( x ) since γ + ρ = (7 , , , γ + ρ ) = +1, and sort( γ + ρ ) − ρ = (9 , , , − (3 , , ,
0) = (6 , , , , γ ) be an indexed root ideal. In (2.2), we defined the Catalan function H (Ψ; γ )using raising operators R ij . Raising operators do not give rise to well-defined operatorson Λ (for example R s = s = 0, but s = 0), so they should be thought of as actingon the subscripts γ rather than the s γ themselves. A precise way to interpret (2.2) is H (Ψ; γ )( x ; t ) = ˜ π (cid:18) Y ( i,j ) ∈ Ψ (cid:0) − tz i /z j (cid:1) − z γ (cid:19) , (4.1) where the map ˜ π is defined by first letting π : Q [ z ± , . . . , z ± ℓ ] → Q [ h , h , . . . ] be the linearmap determined by z γ s γ , and then ˜ π : ( Q [ z ± , . . . , z ± ℓ ])[[ t ]] → ( Q [ h , h , . . . ])[[ t ]] isits natural extension, given by P i ≥ f i t i P i ≥ π ( f i ) t i for any f i ∈ Q [ z ± , . . . , z ± ℓ ]. Itfollows from Corollary 4.2 that the right side of (4.1) actually lies in the degree | γ | partof the ring of symmetric functions Λ = Q ( t )[ h , h , . . . ]. Remark 4.4.
The map π is essentially the Demazure operator corresponding to thelongest element of S ℓ , though the Demazure operator is typically defined as a map from Q [ z ± , . . . , z ± ℓ ] → Q [ z ± , . . . , z ± ℓ ] S ℓ (see, e.g., [34, § Example 4.5.
With ℓ = 4, µ = 3321, and Ψ = { (1 , , (2 , , (1 , } , we have H (Ψ; µ ) = (1 − tR ) − (1 − tR ) − (1 − tR ) − s = s + t ( s + s + s ) + t ( s + s + s )+ t ( s − + s + s ) + t ( s − + s − )= s + t ( s + s ) + t ( s + s ) + t s . Proposition 4.1 is used to truncate the series to terms s α with α + ρ ∈ Z ℓ ≥ for the secondequality and it is used again for the third to give s = s = s − = s − = 0 and s − = − s .This demonstrates the direct way to obtain the Schur expansion from the definition ofCatalan functions. In contrast, Example 2.14 illustrates the Schur expansion of H (Ψ; µ ) = s (4)3321 obtained via Theorem 2.7; the former involves cancellation whereas the latter ismanifestly positive.4.4. Compositional Hall-Littlewood polynomials.
A useful alternative descriptionof the Catalan functions involves Garsia’s version [10] of Jing’s Hall-Littlewood vertexoperators [16]. These are the symmetric function operators defined for any m ∈ Z by H m = X i,j ≥ ( − i t j h m + i + j ( x ) e ⊥ i h ⊥ j ∈ End(Λ) (4.2)(see [37, Definition 2.5.2] for this formula for H m ). Now for γ ∈ Z ℓ , define H γ = H γ H γ · · · H γ ℓ and H γ ( x ; t ) = H γ · . (4.3)When µ is a partition, H µ ( x ; t ) is the modified Hall-Littlewood polynomial [10, Theo-rem 2.1]. For general γ , the H γ ( x ; t ) are known as compositional Hall-Littlewood polyno-mials . They played a key role in the recent proof [6] of the Shuffle Conjecture [12]. As wenow show, any Catalan function can be conveniently expressed in terms of compositionalHall-Littlewood polynomials, making them central to our work as well. Proposition 4.6.
The compositional Hall-Littlewood polynomials are Catalan functionsfor the root ideal ∆ + . That is, for any γ ∈ Z ℓ , H (∆ + ; γ ) = Y ( i,j ) ∈ ∆ + (cid:0) − tR ij (cid:1) − s γ = H γ · H γ . Proof.
This follows from Proposition 7 and Remark 2 of [35]. (cid:3)
ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 15 Proposition 4.7.
For any indexed root ideal (Ψ , γ ) , H (Ψ; γ )( x ; t ) = Y ( i,j ) ∈ ∆ + \ Ψ (1 − t R ij ) H γ ( x ; t ) , (4.4) where the raising operator R ij acts on the subscripts of the H γ by R ij H γ = H γ + ǫ i − ǫ j . The right side of (4.4) can be expressed more formally asΦ (cid:18) Y ( i,j ) ∈ ∆ + \ Ψ (cid:0) − tz i /z j (cid:1) z γ (cid:19) , (4.5)where Φ is the linear map Q [ z ± , . . . , z ± ℓ ][ t ] → Λ determined by t i z γ t i H γ . Proof.
It follows from Proposition 4.6 that Φ is equal to the composition ˜ π ◦ D , where D : Q [ z ± , . . . , z ± ℓ ][ t ] → Q [ z ± , . . . , z ± ℓ ][[ t ]] is the linear map given by left multiplication by Q ( i,j ) ∈ ∆ + (cid:0) − t z i /z j (cid:1) − . Using the description (4.1) of the Catalan function H (Ψ; γ ), wehave H (Ψ; γ ) = ˜ π (cid:18) Y ( i,j ) ∈ Ψ (1 − t z i /z j ) − z γ (cid:19) = ˜ π (cid:18) Y ( i,j ) ∈ ∆ + (1 − t z i /z j ) − Y ( i,j ) ∈ ∆ + \ Ψ (1 − t z i /z j ) z γ (cid:19) = ˜ π ◦ D (cid:18) Y ( i,j ) ∈ ∆ + \ Ψ (1 − t z i /z j ) z γ (cid:19) , which agrees with (4.5). (cid:3) The Garsia-Jing operator H m at t = 1 reduces to multiplication by h m ( x ) (this isequivalent to the identity (9.14) arising in a proof later on). Hence for any γ ∈ Z ℓ , H γ ( x ; 1) = h γ ( x ) = h γ ( x ) h γ ( x ) · · · h γ ℓ ( x ) , where h m ( x ) = 0 if m <
0. This yields the following explicit description of the Catalanfunctions at t = 1 in terms of homogeneous symmetric functions. Corollary 4.8.
For any indexed root ideal (Ψ , γ ) , H (Ψ , γ )( x ; 1) = Y ( i,j ) ∈ ∆ + \ Ψ (1 − R ij ) h γ ( x ) . Example 4.9.
The Catalan function from Example 4.5 at t = 1 is computed usingCorollary 4.8 as follows: for µ = 3321 and ∆ + \ Ψ = { (1 , , (2 , , (3 , } , H (Ψ; µ )( x ; 1) = (1 − R )(1 − R )(1 − R ) h = h − h − h − h + h + h + h − h = h − h − h + h = s + s + s + s + s + s . Proof of Property (2.7) . We first give relations satisfied by the Garsia-Jing and e ⊥ d operators and then deduce a general result about the action of e ⊥ d on Catalan functions. Lemma 4.10.
Let d ∈ Z ≥ , m ∈ Z , and γ ∈ Z ℓ . Then e ⊥ d H m = H m e ⊥ d + H m − e ⊥ d − , (4.6) e ⊥ d H γ = X S ⊂ [ ℓ ] , | S |≤ d H γ − ǫ S e ⊥ d −| S | . (4.7) Proof.
The first relation is [11, Equation 5.37] and the second follows from the first by astraightforward induction on ℓ . (cid:3) Lemma 4.11.
For d ∈ Z ≥ and (Ψ , γ ) any indexed root ideal of length ℓ , e ⊥ d ( H (Ψ; γ )) = X S ⊂ [ ℓ ] , | S | = d H (Ψ; γ − ǫ S ) . (4.8) Proof.
Proposition 4.7 allows us to express any Catalan function in terms of compositionalHall-Littlewood polynomials. The result then follows from Lemma 4.10, (4.3), and thefact that e ⊥ i (1) = 0 for i > (cid:3) In particular, letting d = ℓ in the lemma gives e ⊥ ℓ H (Ψ; γ ) = H (Ψ; γ − ℓ ). In turn, wehave e ⊥ ℓ s ( k +1) µ +1 ℓ = s ( k ) µ since the root ideals ∆ k ( µ ) and ∆ k +1 ( µ + 1 ℓ ) are equal.4.6. Proof of Property (2.8) . The Catalan functions of length ℓ contain those of length ℓ − Proposition 4.12.
For an indexed root ideal (Ψ , γ ) of length ℓ with γ ℓ = 0 , H (Ψ; γ ) = H ( ˆΨ; ˆ γ ) , where ˆΨ = { ( i, j ) ∈ Ψ | j < ℓ } and ˆ γ = ( γ , . . . , γ ℓ − ) .Proof. Using the description of the Catalan functions from Proposition 4.7, we have H (Ψ; γ ) = Y ( i,j ) ∈ ∆ + \ Ψ (1 − t R ij ) H γ = Y ( h,ℓ ) ∈ ∆ + ℓ \ Ψ (1 − t R hℓ ) Y ( i,j ) ∈ ∆ + ℓ − \ ˆΨ (1 − t R ij ) H γ = H ( ˆΨ; ˆ γ ) . The last equality uses that H · H m · m < α ∈ Z ℓ with α ℓ = 0, we have Q ( h,ℓ ) ∈ ∆ + ℓ \ Ψ (1 − t R hℓ ) H α = H ( α ,...,α ℓ − ) . (cid:3) Property (2.8) now follows easily: let µ ∈ Par kℓ with | µ | ≤ k . We need to show s ( k ) µ = s µ .Proposition 4.12 allows us to reduce to the case µ ℓ >
0. Then we have k ≥ | µ | ≥ µ i + ℓ − i for all i ∈ [ ℓ ]. Hence the root ideal ∆ k ( µ ) = { ( i, j ) ∈ ∆ + | k − µ i + i < j } is empty and s ( k ) µ = H (∆ k ( µ ); µ ) = s µ follows.4.7. Proof of Theorem 2.8.
Define the dominance partial order D on Z ℓ by γ D δ if γ + · · · + γ i ≥ δ + · · · + δ i for all i ∈ [ ℓ ]. Lemma 4.13.
For γ ∈ Z ℓ and k = max( γ ) , H γ ∈ span Q ( t ) (cid:8) H λ | λ ∈ Par kℓ and λ D γ (cid:9) . ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 17 Proof.
Since H m · m <
0, the result is a consequence of the stronger claim: H γ ∈ span Q ( t ) (cid:8) H λ | λ ∈ Z ℓ ≤ k is weakly decreasing and λ D γ (cid:9) . This follows from repeated application of the identity [10, Theorem 2.2] H m H n = t H m +1 H n − + t H n H m − H n − H m +1 . (4.9)with m < n , noting that for n = m + 1, we must rearrange to obtain H m H m +1 = t H m +1 H m , rather than apply (4.9) directly. (cid:3) The proof of Theorem 2.8 now goes as follows: by Proposition 4.7, for any µ ∈ Par kℓ , s ( k ) µ = ℓ Y i =1 k − µ i + i Y j = i +1 (1 − t R ij ) H µ . (4.10)Consider H γ arising from such a successive application of raising operators to H µ . Then γ i is obtained by adding some amount not exceeding k − µ i to µ i . Hence γ ∈ Z ℓ ≤ k , implying H γ ∈ span Q ( t ) (cid:8) H λ | λ ∈ Par kℓ and λ D γ (cid:9) by Lemma 4.13. Since each application of araising operator strictly increases dominance order, γ D µ . It follows that s ( k ) µ ∈ Λ kℓ andthe transition matrix expressing { s ( k ) µ } µ ∈ Par kℓ in terms of { H λ } λ ∈ Par kℓ is upper unitriangularwith respect to dominance order, implying that the former is a basis for Λ kℓ .5. Recurrences for the Catalan functions
Computations with Catalan functions are facilitated by recurrences which express aCatalan function as the sum of two Catalan functions with similar indexed root ideals.The bounce graph of a root ideal, defined below, is the natural combinatorial objectarising in these computations.5.1.
Bounce graphs.
We say that α ∈ Ψ is a removable root of
Ψ if Ψ \ α is a root idealand a root β ∈ ∆ + \ Ψ is addable to
Ψ if Ψ ∪ β is a root ideal. Definition 5.1.
Fix a root ideal Ψ ∈ ∆ + ℓ and x ∈ [ ℓ ]. If there is a removable root ( x, j )of Ψ, then define down Ψ ( x ) = j ; otherwise, down Ψ ( x ) is undefined. Similarly, if there isa removable root ( i, x ) of Ψ, then define up Ψ ( x ) = i ; otherwise, up Ψ ( x ) is undefined. Definition 5.2.
The bounce graph of a root ideal Ψ ⊂ ∆ + ℓ is the graph on the vertex set[ ℓ ] with edges ( r, down Ψ ( r )) for each r ∈ [ ℓ ] such that down Ψ ( r ) is defined. The bouncegraph of Ψ is a disjoint union of paths called bounce paths of Ψ.For each vertex r ∈ [ ℓ ], distinguish bot Ψ ( r ) (resp. top Ψ ( r )) to be the maximum (resp.minimum) element of the bounce path of Ψ containing r . For a, b ∈ [ ℓ ] in the same bouncepath of Ψ with a ≤ b , we definepath Ψ ( a, b ) = ( a, down Ψ ( a ) , down ( a ) , . . . , b ) , i.e., the list of indices in this path lying between a and b . We also set downpath Ψ ( r ) =path( r, bot Ψ ( r )) and uppath Ψ ( r ) to be the reverse of path(top Ψ ( r ) , r ) for any r ∈ [ ℓ ]. By a slight abuse of notation, we also write path Ψ ( a, b ), downpath Ψ ( r ), and uppath Ψ ( r ) forthe corresponding sets of indices. For b = down m Ψ ( a ), the bounce from a to b is B Ψ ( a, b ) := | path Ψ ( a, b ) | − m. Example 5.3.
Examples of downpath, uppath, and bounce for the root ideal Ψ below:path Ψ (2 ,
8) = 2 , , Ψ (3) = 3 , Ψ (10) = 10 , , , , B Ψ (2 ,
8) = 2, B Ψ (1 ,
10) = 4, B Ψ (3 ,
6) = 1, and B Ψ (3 ,
3) = 0 . Definition 5.4.
A root ideal Ψ is said to have a wall in rows r, r + 1 if rows r and r + 1 of Ψ have the same length, a ceiling in columns c, c + 1 if columns c and c + 1 of Ψ have the same length, and a mirror in rows r, r + 1 if Ψ has removable roots ( r, c ), ( r + 1 , c + 1) for some c > r + 1. Example 5.5.
The root ideal Ψ in the previous example has a ceiling in columns 2 ,
3, incolumns 3 ,
4, and in columns 8 ,
9, a wall in rows 6 ,
7, in rows 7 ,
8, and in rows 9 ,
10, anda mirror in rows 2 ,
3, in rows 3 ,
4, and in rows 4 , Recurrences for the Catalan functions.Proposition 5.6.
Let (Ψ , µ ) be an indexed root ideal. For any root β addable to Ψ , H (Ψ; µ ) = H (Ψ ∪ β ; µ ) − t H (Ψ ∪ β ; µ + ε β ) . (5.1) For any removable root α of Ψ , H (Ψ; µ ) = H (Ψ \ α ; µ ) + t H (Ψ; µ + ε α ) . (5.2) Proof.
The first identity (5.1) follows directly from Proposition 4.7 : H (Ψ; µ ) = Y ( i,j ) ∈ ∆ + \ Ψ (1 − t R ij ) H µ = (1 − t R β ) Y ( i,j ) ∈ ∆ + \ (Ψ ∪ β ) (1 − t R ij ) H µ = Y ( i,j ) ∈ ∆ + \ (Ψ ∪ β ) (1 − t R ij ) H µ − t Y ( i,j ) ∈ ∆ + \ (Ψ ∪ β ) (1 − t R ij ) H µ + ε β . The second identity (5.2) is then obtained by applying (5.1) with Ψ = Ψ \ α and β = α . (cid:3) We also record a convenient application of the recurrence (5.2) obtained by iterating italong a downpath.
ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 19 Corollary 5.7.
Let (Ψ , µ ) be an indexed root ideal of length ℓ and m ∈ [ ℓ ] . Then H (Ψ; µ ) = X z ∈ downpath Ψ ( m ) t B Ψ ( m,z ) H (Ψ z ; µ + ǫ m − ǫ z ) , (5.3) where Ψ z := Ψ \ { ( z, down Ψ ( z )) } for z = bot Ψ ( m ) and Ψ bot Ψ ( m ) := Ψ .Proof. The proof is by induction on | downpath Ψ ( m ) | . The base case | downpath Ψ ( m ) | = 1is clear. Now suppose | downpath Ψ ( m ) | > m ′ = down Ψ ( m ). The desired resultis obtained by expanding H (Ψ; µ ) on the root ( m, m ′ ) using the recurrence (5.2) and thenapplying the inductive hypothesis: H (Ψ; µ ) = H (Ψ m ; µ ) + t H (Ψ; µ + ǫ m − ǫ m ′ )= H (Ψ m ; µ ) + t X z ∈ downpath Ψ ( m ′ ) t B Ψ ( m ′ ,z ) H (Ψ z ; µ + ǫ m − ǫ m ′ + ǫ m ′ − ǫ z )= X z ∈ downpath Ψ ( m ) t B Ψ ( m,z ) H (Ψ z ; µ + ǫ m − ǫ z ) . (cid:3) Mirror lemmas
We first give a natural generalization of Schur function straightening to Catalan func-tions (Lemma 6.1) and then deduce two
Mirror Lemmas . The first gives sufficient condi-tions for a Catalan function to be zero and the second gives sufficient conditions for twoCatalan functions to be equal. We further show that these lemmas often “commute” withcertain generalizations of the operator e ⊥ d called subset lowering operators.The symmetric group S ℓ acts on the ring Q [ z ± , z ± , . . . , z ± ℓ ] by permuting variables.The simple reflections τ , . . . , τ ℓ − ∈ S ℓ act on the basis of Laurent monomials { z γ } γ ∈ Z ℓ by τ i z γ = z τ i γ , where τ i γ = ( γ , . . . , γ i − , γ i +1 , γ i , γ i +2 , . . . ). This action extends in thenatural way to an action on Q [ z ± , z ± , . . . , z ± ℓ ][[ t ]]. We also consider the action of S ℓ onsubsets Ψ ⊂ [ ℓ ] × [ ℓ ] given by τ i Ψ = { ( τ i ( a ) , τ i ( b )) | ( a, b ) ∈ Ψ } . Lemma 6.1.
Let Ψ ⊂ ∆ + ℓ be a root ideal such that τ i Ψ = Ψ . Then for any γ ∈ Z ℓ , H (Ψ; γ ) + H (Ψ; ǫ i +1 − ǫ i + τ i γ ) = 0 . Proof.
Set f Ψ = Q ( i,j ) ∈ Ψ (cid:0) − t z i /z j (cid:1) − . Recall from (4.1) that the Catalan functions maybe defined in terms of the linear map ˜ π by H (Ψ; γ ) = ˜ π ( f Ψ z γ ). Thus H (Ψ; γ ) + H (Ψ; ǫ i +1 − ǫ i + τ i γ ) = ˜ π (cid:0) f Ψ z γ + f Ψ z ǫ i +1 − ǫ i + τ i γ (cid:1) . We have τ i Ψ = Ψ implies τ i f Ψ = f Ψ (note that Ψ ⊂ ∆ + and τ i Ψ = Ψ imply ( i, i +1) / ∈ Ψ).Hence we obtain ˜ π (cid:0) f Ψ z γ + f Ψ z ǫ i +1 − ǫ i + τ i γ (cid:1) = ˜ π ◦ (1 + z i +1 /z i τ i ) (cid:0) f Ψ z γ (cid:1) , where 1 + z i +1 /z i τ i is regarded as an operator on Q [ z ± , z ± , . . . , z ± ℓ ][[ t ]]. We now claimthat the operator ˜ π ◦ (1 + z i +1 /z i τ i ) : Q [ z ± , z ± , . . . , z ± ℓ ][[ t ]] → Q [ h , h , . . . ][[ t ]] is identi-cally 0, which will complete the proof. It suffices to show that π ◦ (1 + z i +1 /z i τ i )( z δ ) = 0 for any δ ∈ Z ℓ , where π is the map used to define ˜ π (see (4.1)). We have π ◦ (1 + z i +1 /z i τ i )( z δ ) = π (cid:0) z δ + z ǫ i +1 − ǫ i + τ i δ (cid:1) = s δ + s ǫ i +1 − ǫ i + τ i δ = 0 , where the last equality is by the Schur function straightening rule (Proposition 4.1). (cid:3) Lemma 6.2.
Let (Ψ , µ ) be an indexed root ideal of length ℓ and z ∈ [ ℓ − , and suppose Ψ has a ceiling in columns z, z + 1 ; (6.1)Ψ has a wall in rows z, z + 1 ; (6.2) µ z = µ z +1 − . (6.3) Then H (Ψ; µ ) = 0 .Proof. Conditions (6.1)–(6.2) are just another way of saying τ z Ψ = Ψ. By (6.3), ǫ z +1 − ǫ z + τ z µ = µ . Hence the result follows from Lemma 6.1. (cid:3) Example 6.3.
By Lemma 6.2 with z = 2, the following Catalan function is zero: = 0 . Lemma 6.4 (Mirror Lemma I) . Let (Ψ , µ ) be an indexed root ideal of length ℓ , and let y, z, w be indices in the same bounce path of Ψ with ≤ y ≤ z ≤ w < ℓ , satisfying Ψ has a ceiling in columns y, y + 1 ; (6.4)Ψ has a mirror in rows x, x + 1 for all x ∈ path Ψ ( y, up Ψ ( w )) ; (6.5)Ψ has a wall in rows w, w + 1 ; (6.6) µ x = µ x +1 for all x ∈ path Ψ ( y, w ) \ { z } ; (6.7) µ z = µ z +1 − . (6.8) Then H (Ψ; µ ) = 0 .Proof. The proof is by induction on w − y . The base case y = w is Lemma 6.2. Nowassume y < w . By (6.5), the root β = (up Ψ ( w + 1) , w ) is addable to Ψ. So we can expand H (Ψ; µ ) using (5.1) to obtain H (Ψ; µ ) = H (Ψ ∪ β ; µ ) − t H (Ψ ∪ β ; µ + ε β ) . The root ideal Ψ ∪ β has a wall in rows up Ψ ( w ) , up Ψ ( w ) + 1 and a ceiling in columns w, w + 1. Hence, if z = w , we have H (Ψ ∪ β ; µ ) = 0 by the inductive hypothesis and H (Ψ ∪ β ; µ + ε β ) = 0 by Lemma 6.2 ((6.3) holds by ( µ + ε β ) w = ( µ + ε β ) w +1 − z = w , then we have H (Ψ ∪ β ; µ ) = 0 by Lemma 6.2 and H (Ψ ∪ β ; µ + ε β ) = 0 by theinductive hypothesis ((6.8) holds with µ + ε β in place of µ and up Ψ ( w ) in place of z ). (cid:3) Here is another useful variant:
ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 21 Lemma 6.5 (Mirror Lemma II) . Let (Ψ , µ ) be an indexed root ideal of length ℓ , and let y, w be indices in the same bounce path of Ψ with ≤ y ≤ w < ℓ , satisfying (6.4) – (6.6) and µ x = µ x +1 for all x ∈ path Ψ ( y, w ) . (6.9) If Ψ has a removable root α in column y , then H (Ψ; µ ) = H (Ψ \ α ; µ ) . Similarly,if Ψ has a removable root β in row w + 1 , then H (Ψ; µ ) = H (Ψ \ β ; µ ) .Proof. Apply (5.2) with the removable root α to obtain H (Ψ; µ ) = H (Ψ \ α ; µ ) + t H (Ψ; µ + ε α ) = H (Ψ \ α ; µ ) , where the second equality is by Lemma 6.4 applied with indexed root ideal (Ψ , µ + ε α )and z = y ((6.8) holds since ( µ + ε α ) y = ( µ + ε α ) y +1 − β inplace of α gives H (Ψ; µ ) = H (Ψ \ β ; µ ). (cid:3) Example 6.6.
By Lemma 6.5 with y = 2, w = 4, we have α β = α β = α β . The vertical dual Pieri rule gives a combinatorial description of e ⊥ d s ( k ) µ . The proof ofthis rule requires extending it to a combinatorial description of a more general operatoron s ( k ) µ , which we now define. Definition 6.7.
For d ∈ Z ≥ and V ⊂ [ ℓ ], the subset lowering operator L d,V is given by L d,V H (Ψ; µ ) = X S ⊂ V, | S | = d H (Ψ; µ − ǫ S ) , where (Ψ , µ ) is any indexed root ideal of length ℓ .With this notation, Lemma 4.11 says that L d, [ ℓ ] H (Ψ; µ ) = e ⊥ d H (Ψ; µ ) for any d ≥ Remark 6.8.
Just as for raising operators, the subset lowering operators should bethought of as acting on the input µ in H (Ψ; µ ) rather than on the polynomials themselves.They are not in general well-defined operators on symmetric functions: for instance, H ( ∅ ; 12) = 0 but L , { } H ( ∅ ; 12) := H ( ∅ ; 02) = − s = 0. Also see Example 6.11.Despite the fact that L d,V is not a well-defined operator on symmetric functions, itcommutes with raising operators and the recurrences of Proposition 5.6. Moreover, itcommutes with the Mirror Lemmas under some mild assumptions. This means thatif we have any computation involving Catalan functions that only uses the recurrencesand Mirror Lemmas in a controlled way, then we can commute L d,V through this entirecomputation. This is a powerful technique and is crucial to the proof of the vertical dualPieri rule. Proposition 6.9.
Let (Ψ , µ ) be an indexed root ideal. For any root β addable to Ψ , L d,V H (Ψ; µ ) = L d,V H (Ψ ∪ β ; µ ) − t L d,V H (Ψ ∪ β ; µ + ε β ) . (6.10) For any removable root α of Ψ , L d,V H (Ψ; µ ) = L d,V H (Ψ \ α ; µ ) + t L d,V H (Ψ; µ + ε α ) . (6.11) Proof.
This is immediate from the definition of L d,V and Proposition 5.6. (cid:3) Lemma 6.10.
Let (Ψ , µ ) be an indexed root ideal of length ℓ , let z ∈ [ ℓ − , and V ⊂ [ ℓ ] .Suppose this data satisfies (6.1) – (6.3) together with V contains both or neither of z, z + 1 . (6.12) Then L d,V H (Ψ; µ ) = 0 for any d ≥ .Proof. By definition, L d,V H (Ψ; µ ) = X S ⊂ V, | S | = d H (Ψ; µ − ǫ S ) . The terms in the sum such that S contains both or neither of z, z +1 are zero by Lemma 6.2.If V ∩{ z, z +1 } = ∅ , then this accounts for all the terms, and we are done. If { z, z +1 } ⊂ V ,then the remaining terms come in pairs: X S ⊂ V, | S | = d H (Ψ; µ − ǫ S ) = X S ′ ⊂ V \{ z,z +1 }| S ′ | = d − (cid:16) H (Ψ; µ − ǫ S ′ − ǫ z ) + H (Ψ; µ − ǫ S ′ − ǫ z +1 ) (cid:17) . (6.13)By Lemma 6.1 with γ = µ − ǫ S ′ − ǫ z +1 (using that t z γ = γ and µ − ǫ S ′ − ǫ z = ǫ z +1 − ǫ z + t z γ ), H (Ψ; µ − ǫ S ′ − ǫ z ) + H (Ψ; µ − ǫ S ′ − ǫ z +1 ) = 0. Hence the right side of (6.13) is zero. (cid:3) Example 6.11.
By Lemma 6.10 with z = 2, L , { , } := + = 0 . For comparison here is an example in which (6.12) is not satisfied: L , { , } := + = − = − s − ts = 0 . With Lemma 6.10 in hand, we easily obtain generalizations of Lemmas 6.4 and 6.5 tothe L d,V H (Ψ; µ ). Lemma 6.12.
Let (Ψ , µ ) be an indexed root ideal of length ℓ , let y, z, w be indices in thesame bounce path of Ψ with ≤ y ≤ z ≤ w < ℓ , and let V ⊂ [ ℓ ] . Suppose that this datasatisfies (6.4) – (6.8) together with V contains both or neither of x, x + 1 for all x ∈ path Ψ ( y, w ) . (6.14) Then L d,V H (Ψ; µ ) = 0 for any d ≥ .Proof. Repeat the proof of Lemma 6.4 using Lemma 6.10 in place of Lemma 6.2 and(6.10) in place of (5.1). (cid:3)
ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 23 Lemma 6.13.
Let (Ψ , µ ) be an indexed root ideal of length ℓ , let y, w be indices in the samebounce path of Ψ with ≤ y ≤ w < ℓ , let d ≥ , and let V ⊂ [ ℓ ] satisfy (6.4) – (6.6) , (6.9) ,and (6.14) . If Ψ has a removable root α in column y , then L d,V H (Ψ; µ ) = L d,V H (Ψ \ α ; µ ) .Similarly, if Ψ has a removable root β in row w + 1 , then L d,V H (Ψ; µ ) = L d,V H (Ψ \ β ; µ ) .Proof. Repeat the proof of Lemma 6.5 using (6.11) in place of (5.2) and Lemma 6.12 inplace of Lemma 6.4. (cid:3)
Example 6.14.
By Lemma 6.13 with y = 2, w = 4, for any d ≥ V ⊂ { , , , , , } such that V ∩ { , } has size 0 or 2 and V ∩ { , } has size 0 or 2, there holds L d,V α β = L d,V α β = L d,V α β . k -Schur straightening One beautiful consequence of identifying k -Schur functions as a subclass of Catalanfunctions is that we obtain a natural generalization of k -Schur functions to a class ofCatalan functions indexed by the set of weights g Par kℓ := (cid:8) µ ∈ Z ℓ ≤ k | µ + ℓ − ≥ µ + ℓ − ≥ · · · ≥ µ ℓ (cid:9) , (7.1)which contains Par kℓ but many nonpartitions as well. Here we show that these Catalanfunctions obey a k -Schur straightening rule similar to that of ordinary Schur functions,and this plays a crucial role in the proof of the dual Pieri rules.7.1. The k -Schur root ideal. The definition of s ( k ) µ from (2.4) carries over unchangedto this more general setting, which we record for convenience: Definition 7.1.
For µ ∈ g Par kℓ , define the root ideal∆ k ( µ ) = { ( i, j ) ∈ ∆ + ℓ | k − µ i + i < j } , (7.2)and the associated Catalan function s ( k ) µ = H (∆ k ( µ ); µ ) = ℓ Y i =1 ℓ Y j = k +1 − µ i + i (cid:0) − tR ij (cid:1) − s µ . (7.3) Example 7.2.
Here are some examples of the Catalan functions s ( k ) µ for k = 4: s (4)3321 = , s (4)4432320 = , s (4) − = − . The first was computed explicitly in Examples 2.14 and 4.5.
The following is essentially a restatement of the definition of ∆ k , which we will referencefrequently. Proposition 7.3.
For µ ∈ g Par kℓ , the western border of the root ideal ∆ k ( µ ) consists ofthe roots ( i, k + 1 − µ i + i ) for i such that k + 1 − µ i + i ≤ ℓ . Moreover, ( i, k + 1 − µ i + i ) is a removable root of ∆ k ( µ ) if and only if µ i ≥ µ i +1 and k + 1 − µ i + i ≤ ℓ . k -Schur straightening.Lemma 7.4 ( k -Schur straightening I) . Let µ ∈ g Par kℓ , Ψ = ∆ k ( µ ) , and z ∈ [ ℓ − . Suppose y + 1 = up Ψ ( z + 1) is defined; (7.4) µ z = µ z +1 − ; (7.5) µ z +1 ≥ µ z +2 and µ y ≥ µ y +1 . (7.6) Then s ( k ) µ = t s ( k ) µ + ǫ y +1 − ǫ z +1 . (7.7) Proof.
Expand H (Ψ; µ ) using (5.2) with the removable root δ = ( y + 1 , z + 1) to obtain H (Ψ; µ ) = H (Ψ \ δ ; µ ) + t H (Ψ; µ + ε δ ) . (7.8)Using that µ z = µ z +1 − µ y ≥ µ y +1 with the definition of ∆ k shows that Ψ \ δ has awall in rows z, z +1 and a ceiling in columns z, z +1. Hence H (Ψ \ δ ; µ ) = 0 by Lemma 6.2.The root α = ( y + 1 , down Ψ ( y + 1) −
1) = ( y + 1 , z ) is addable to Ψ by the assumption µ y ≥ µ y +1 ; also set B = { β } with β = ( z + 1 , down Ψ ( z + 1)) if this is defined and B = ∅ otherwise. Lemma 6.5, applied to the indexed root ideal (Ψ ∪ α, µ + ε δ ), yields H (Ψ; µ + ε δ ) = H (Ψ ∪ α ; µ + ε δ ) = H (Ψ ∪ α \ B ; µ + ε δ ) = s ( k ) µ + ε δ . For the last equality we are using µ z +1 ≥ µ z +2 to conclude that down Ψ ( z + 1) is definedif and only if the ( z + 1)-st row of Ψ is nonempty. (cid:3) Remark 7.5.
For conditions such as (7.6) arising in this section and the next, cornercases z = ℓ − y = 0 are conveniently handled by defining µ = k and µ ℓ +1 = 0for µ ∈ Par kℓ . The latter is a standard convention, however, we must take care here sincethe definitions of ∆ k ( µ ) and s ( k ) µ = H (∆ k ( µ ); µ ) depend on ℓ ; we still regard ∆ k ( µ ) as asubset of [ ℓ ] × [ ℓ ] not [ ℓ + 1] × [ ℓ + 1]. Example 7.6.
By Lemma 7.4 with z = 2 , y = 0, we have s (4)32321 = = t = t s (4)42221 . Remark 7.7.
It is not difficult to show using Lemma 6.2 that for µ ∈ g Par kℓ , s ( k ) µ = 0 if(7.5) holds and up ∆ k ( µ ) ( z + 1) is undefined. Using this in combination with Lemma 7.4,one can show that for any µ ∈ g Par kℓ , s ( k ) µ is equal to 0 or a power of t times s ( k ) ν for ν ∈ Par kℓ .However, we will not need this general statement. Instead, we focus on the special case ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 25 µ = λ − ǫ z for λ ∈ Par kℓ , where we can give an explicit combinatorial description of when s ( k ) µ is zero and the power of t when it is nonzero. Lemma 7.8.
Let µ ∈ g Par kℓ and Ψ = ∆ k ( µ ) . Let h ∈ Z ≥ and z ∈ [ ℓ − h ] . Suppose that y + 1 = up Ψ ( z + 1) is defined ; (7.9) µ z = µ z +1 −
1; (7.10) µ z +1 = · · · = µ z + h and µ y +1 = · · · = µ y + h ; (7.11) µ z + h ≥ µ z + h +1 and µ y ≥ µ y +1 . (7.12) Then s ( k ) µ = t h s ( k ) µ + ǫ [ y +1 ,y + h ] − ǫ [ z +1 ,z + h ] . (7.13) Proof.
Set µ i = µ + ǫ [ y +1 ,y + i ] − ǫ [ z +1 ,z + i ] for i ∈ [0 , h ]. By h applications of Lemma 7.4, weobtain s ( k ) µ = t s ( k ) µ = · · · = t h s ( k ) µ h . We verify that the hypotheses of Lemma 7.4 are satisfied at each step: it follows fromProposition 7.3 that ( y + i + 1 , k − µ y + i +1 + y + i + 2) = ( y + i + 1 , z + i + 1) (using(7.11) and k + 1 − µ y +1 + y + 1 = z + 1 for the equality) is a removable root of ∆ k ( µ i )for each i ∈ [0 , h − (cid:3) The following definition is forced on us by the combinatorics arising in k -Schur straight-ening. Examples of this definition and Lemma 7.8 are given at the end of the subsection. Definition 7.9.
Let λ ∈ Par kℓ and z ∈ [ ℓ ]. Set µ = λ − ǫ z and Ψ = ∆ k ( µ ). Let c = | uppath Ψ ( z ) | . If z = ℓ or λ z > λ z +1 or up c Ψ ( z + 1) is undefined, then set h = 0;otherwise, set y + 1 = up c Ψ ( z + 1) and let h ∈ [ ℓ − z ] be as large as possible such that µ is constant on each of the intervals [ z +1 , z + h ], [up Ψ ( z ) , up Ψ ( z )+ h ],[up ( z ) , up ( z ) + h ], . . . , [top Ψ ( z ) , top Ψ ( z ) + h ], and [ y + 1 , y + h ]. (7.14)Define cover z ( λ ) = λ + ǫ [ y +1 ,y + h ] − ǫ [ z,z + h ] . If y is undefined or, equivalently, h = 0 then cover z ( λ ) = µ (we interpret [ y + 1 , y + h ] = ∅ in this case). We also define the bounce from λ to cover z ( λ ) bybounce(cover z ( λ ) , λ ) = h · c. (7.15)For the proof of k -Schur straightening II below and the results of the next subsection,we also define integers c ′ and h x as follows: if λ z + h > λ z + h +1 , then c ′ := −
1; otherwise, c ′ := max (cid:8) i ∈ [0 , c − | µ x + h = µ x + h +1 for all x ∈ path Ψ (up i Ψ ( z ) , up Ψ ( z )) (cid:9) . (7.16)And for x ∈ uppath Ψ ( z ), let h x = ( h + 1 if x = up s Ψ ( z ) with s ≤ c ′ ,h otherwise . (7.17) Lemma 7.10.
The following facts clarify Definition 7.9. (i) If y is defined (equivalently, h > ), then µ y > µ y +1 . (ii) cover z ( λ ) ∈ Par kℓ if and only if λ z + h > λ z + h +1 . (iii) If up c Ψ ( z + 1) is defined and z < ℓ , then (7.14) holds with h = 1 (so there doesexist a largest h ∈ [ ℓ − z ] satisfying (7.14) ). Note that we have used Remark 7.5 to handle corner cases in (i) and (ii).
Proof.
Statements (i) and (iii) are straightforward consequences of Proposition 7.3 and(ii) is immediate from (i). (cid:3)
Lemma 7.11.
The intervals in (7.14) are pairwise disjoint.Proof.
It suffices to show that x + h x < down Ψ ( x ) for all x ∈ uppath Ψ ( z ) \ { z } and y + h < top Ψ ( z ). We begin by proving the former (this is stronger than what we need,but we will use this version later). Suppose for a contradiction that this fails; then thereis a largest x ∈ uppath Ψ ( z ) \ { z } such that x + h x ≥ down Ψ ( x ). By definition of h x , wehave µ x = · · · = µ down Ψ ( x ) = · · · = µ x + h x . We thus cannot have x = up Ψ ( z ) since thiswould contradict µ z − > µ z . So x = up a Ψ ( z ) for some a ≥
2. We then have h down Ψ ( x ) ≥ h x ≥ down Ψ ( x ) − x = k + 1 − µ x = k + 1 − µ down Ψ ( x ) = down ( x ) − down Ψ ( x ) , where we have used Proposition 7.3 for the first and third equalities. This contradicts ourchoice of x .Now to prove y + h < top Ψ ( z ), suppose for a contradiction that y + h ≥ top Ψ ( z ). Thenby definition of h , we have µ y +1 = · · · = µ top Ψ ( z ) = · · · = µ y + h . If z = top Ψ ( z ), thiscontradicts µ z − > µ z and we are done, so assume top Ψ ( z ) < z . We have h ≥ top Ψ ( z ) − y = down Ψ ( y + 1) − ( y + 1)= k + 1 − µ y +1 = k + 1 − µ top Ψ ( z ) = down Ψ (top Ψ ( z )) − top Ψ ( z ) , where the the first, second, and fourth equalities follow from Proposition 7.3. This con-tradicts the result of the previous paragraph x + h x < down Ψ ( x ) for x = top Ψ ( z ). (cid:3) Theorem 7.12 ( k -Schur straightening II) . Maintain the notation of Definition 7.9. Then s ( k ) µ = t hc s ( k )cover z ( λ ) = t bounce(cover z ( λ ) ,λ ) s ( k )cover z ( λ ) (7.18) and this is equal to 0 if cover z ( λ ) / ∈ Par kℓ .Proof. We first prove (7.18). This is trivial if h = 0, so assume h > Ψ ( y + 1 , z + 1) = ( b c + 1 , b c − + 1 , . . . , b + 1) (thus b = z and b c = y ). Set µ i = µ + ǫ [ b i +1 ,b i + h ] − ǫ [ z +1 ,z + h ] for i ∈ [ c ] (thus µ c = cover z ( λ )). By c applications of Lemma 7.8, we obtain s ( k ) µ = t h s ( k ) µ = · · · = t hc s ( k ) µ c . The hypotheses of the lemma are satisfied at each step: (7.9)–(7.10) are clear from thedefinition of h , (7.11) follows from the definition of h together with Lemma 7.11, and(7.12) follows from µ + ǫ z ∈ Par kℓ . ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 27 Now suppose ν := cover z ( λ ) / ∈ Par kℓ ; this can only happen if λ z + h = λ z + h +1 byLemma 7.10 (ii). If z + h = ℓ , then λ z = λ z + h = 0, and we have s ( k ) ν = 0 by Propo-sition 4.7 and the fact that H γ = 0 for γ ℓ <
0. So now assume z + h < ℓ . Set ˜Ψ = ∆ k ( ν ).We will apply Lemma 6.4 to show s ( k ) ν = 0.First consider the case h > b i as above. Let c ′ be as in (7.16). Note that˜Ψ and Ψ are identical in rows [ y + h + 1 , z −
1] and have removable roots in these rows.It then follows from Proposition 7.3 and the definitions of h and c ′ thatdown ˜Ψ ( b i + h ) = down Ψ ( b i + h ) = b i − + h anddown ˜Ψ ( b i + h + 1) = down Ψ ( b i + h + 1) = b i − + h + 1 for each i ∈ [ c ′ ] . (7.19)(We need that b i + h + 1 ≤ z −
1, which holds since µ b i = · · · = µ b i + h +1 and µ z − > µ z .)Also by the definitions of h and c ′ , we have µ b c ′ +1 + h > µ b c ′ +1 + h +1 (we need to check that µ b c ′ +1 + h < µ b c ′ +1 + h +1 cannot occur, but this follows from µ z − > µ z and µ b c ′ +1 = · · · = µ b c ′ +1 + h ). This implies ν b c ′ +1 + h > ν b c ′ +1 + h +1 . It then follows from Proposition 7.3 thatdown ˜Ψ ( b c ′ +1 + h ) ≤ down Ψ ( b c ′ +1 + h ) = b c ′ + h and down ˜Ψ ( b c ′ +1 + h + 1) > b c ′ + h + 1.This means that ˜Ψ has a ceiling in columns b c ′ + h, b c ′ + h + 1. Together with (7.19), thisshows that the hypotheses of Lemma 6.4 are satisfied for the indexed root ideal ( ˜Ψ , ν )(with y , z , w of the lemma equal to b c ′ + h , z + h , z + h , respectively) and hence s ( k ) ν = 0.The case h = 0 is similar but easier. Here we define the b i by uppath Ψ ( z ) = ( b , b , . . . , b c − ).The proof in the previous paragraph still works (though some arguments simplify since ν = µ and ˜Ψ = Ψ) except for the proof that ˜Ψ has a ceiling in columns b c ′ + h, b c ′ + h + 1 inthe case c ′ = c −
1; but this follows directly from the fact that up Ψ ( b c ′ + h + 1) is undefined(by Definition 7.9, up c Ψ ( z + 1) is undefined since h = 0, λ z = λ z +1 , and z < ℓ ). (cid:3) Example 7.13.
We illustrate Definition 7.9 and different cases of the proof of Theo-rem 7.12 with several examples, all with k = 4. First, let λ = 222222221, µ = 222221221, z = 6. Then uppath ∆ k ( µ ) ( z ) = (6 , ∆ k ( µ ) ( z + 1) = (7 , , y + 1 = 1, c = 2, h = 2, and cover z ( λ ) = 332221111. We illustrate the two applications of Lemma 7.8 usedin the proof of Theorem 7.12 to obtain s (4)222221221 = t s (4)332221111 : = t = t . For comparison, here is a similar example where s ( k ) µ = 0 : λ = 222222222, µ =222221222, z = 6. Then c = 2, h = 2, c ′ = 0, and cover z ( λ ) = 332221112. The proof of Theorem 7.12 yields = t = 0 . Lastly, an example where s ( k ) µ = 0 and c ′ > λ = 432222222, µ = 432221222, z = 6.Then uppath ∆ k ( µ ) ( z ) = (6 , ∆ k ( µ ) ( z + 1) = (7 , , y + 1 = 2, c = 2, h = 1, c ′ = 1, and cover z ( λ ) = 442221122. The proof of Theorem 7.12 yields = t = 0 . k -Schur straightening and the subset lowering operators. Here we show that k -Schur straightening “commutes” with the subset lowering operators under some mildassumptions. Lemma 7.14.
Let µ ∈ g Par kℓ , Ψ = ∆ k ( µ ) , z ∈ [ ℓ − , and V ⊂ [ ℓ ] . Suppose that (7.4) – (7.6) hold and V contains both or neither of z, z + 1 . Recall that y + 1 := up Ψ ( z + 1) by (7.4) . Then for any d ≥ , L d,V s ( k ) µ = t L d,V s ( k ) µ + ǫ y +1 − ǫ z +1 (7.20) Proof.
Repeat the proof of Lemma 7.4 with (6.11) in place of (5.2), Lemma 6.10 in placeof Lemma 6.2, and Lemma 6.13 in place of Lemma 6.5. (cid:3)
Lemma 7.15.
Let µ ∈ g Par kℓ , Ψ = ∆ k ( µ ) , h ∈ Z ≥ , z ∈ [ ℓ − h ] , and V ⊂ [ ℓ ] . Suppose (7.9) – (7.12) hold and V contains all or none of the interval [ z, z + h ] . Then for any d ≥ , L d,V s ( k ) µ = t h L d,V s ( k ) µ + ǫ [ y +1 ,y + h ] − ǫ [ z +1 ,z + h ] . (7.21) Proof.
Repeat the proof of Lemma 7.8 with Lemma 7.14 in place of Lemma 7.4. (cid:3)
Theorem 7.16.
Maintain the notation of Definition 7.9 and (7.17) . Let V ⊂ [ ℓ ] be suchthat V contains all or none of the interval [ x, x + h x ] for all x ∈ uppath Ψ ( z ) . Then forany d ≥ , L d,V s ( k ) λ − ǫ z = ( t bounce(cover z ( λ ) ,λ ) L d,V s ( k )cover z ( λ ) if cover z ( λ ) ∈ Par kℓ , otherwise. (7.22) Proof.
Repeating the proof of (7.18) with Lemma 7.15 in place of Lemma 7.8 gives L d,V s ( k ) λ − ǫ z = t bounce(cover z ( λ ) ,λ ) L d,V s ( k )cover z ( λ ) (without the restriction cover z ( λ ) ∈ Par kℓ ); here ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 29 we need that V contains all or none of the interval [ x, x + h ] for all x ∈ uppath Ψ ( z ), whichis certainly implied by our assumption on V .If cover z ( λ ) ∈ Par kℓ we are done. Otherwise, repeat the last three paragraphs of theproof of Theorem 7.12 with Lemma 6.12 in place of Lemma 6.4; Lemma 6.12 requires that V contains all or none of [ x + h, x + h + 1] for all x ∈ path ˜Ψ (up c ′ ˜Ψ ( z ) , z ) = path Ψ (up c ′ Ψ ( z ) , z ),where the equality is by (7.19) and c ′ and ˜Ψ are as in the proof of Theorem 7.12; this con-dition on V combined with that of the previous paragraph is equivalent to the assumptionon V in the statement. (cid:3) Example 7.17.
We continue Example 7.13. For the first example, µ = 222221221, wehave h x = h = 2 for all x ∈ uppath ∆ k ( µ ) ( z ) = (6 , V inTheorem 7.16 is that V must contain all or none of the intervals { , , } and { , , } .For the second example, µ = 222221222, we have uppath ∆ k ( µ ) ( z ) = (6 , h = 2, h = 3, h = 2. Thus the intervals [ x, x + h x ] are { , , } and { , , , } . So, for instance, V = [8] would work for the previous example, but not this one. For the third example, µ = 432221222, we have uppath ∆ k ( µ ) ( z ) = (6 , h = 1, h = 2, h = 2. Thus theintervals [ x, x + h x ] are { , , } and { , , } .Theorem 7.16 gives the most general conditions on V for which k -Schur straighteningis possible, however we will only need it for V = [ m −
1] for each m ∈ uppath Ψ ( z ).Theorem 7.16 does indeed apply in this case, as the following result shows. Lemma 7.18.
Let µ , Ψ = ∆ k ( µ ) , and z be as in Definition 7.9, and h x be as in (7.17) .Then for any m ∈ uppath Ψ ( z ) , the set [ m − contains all or none of the interval [ x, x + h x ] for all x ∈ uppath Ψ ( z ) .Proof. This is immediate from the first paragraph of the proof of Lemma 7.11. (cid:3) The bounce graph to core dictionary
Here we connect the combinatorics associated to bounce graphs and k -Schur straight-ening to strong (marked) covers. In particular, we give an explicit description of strong(marked) covers in terms of the corresponding k -bounded partitions (Propositions 8.10and 8.12). To make this connection, we use a description of strong covers in terms of edgesequences and offset sequences; we follow [21] for this background.Throughout this section, fix a positive integer k and let n = k + 1.The edge sequence of a partition κ is the bi-infinite binary word p ( κ ) = p = . . . p − p p . . . obtained by tracing the border of the diagram of κ from southwest to northeast, such thatevery letter 1 (resp. 0) represents a north (resp. east) step. We adopt the convention that the meeting point of the edges p i and p i +1 is the southeast corner of a box in diagonal i , where the diagonal of a box ( r, c ) of a partition diagram is c − r .A partition κ is an n -core if and only if each subsequence . . . p i − n p i − n p i p i + n p i +2 n . . . has the form . . . . . . . Thus an n -core κ is specified by recording, for each i ∈ Z , the This is off by 1 from the convention in [21], but our extended offset sequences agree. integer d i such that p i + n ( d i − = 1 and p i + nd i = 0. The sequence ( d i ) i ∈ Z is the extendedoffset sequence of κ . Note that d i − n = d i + 1 for all i ∈ Z . (8.1)The affine symmetric group b S n can be identified with the set of permutations w of Z such that w ( i + n ) = w ( i )+ n for all i ∈ Z and P ni =1 ( w ( i ) − i ) = 0 (see [28]). For r < s with r s mod n , the reflection t r,s ∈ b S n is defined by t r,s ( r + jn ) = s + jn , t r,s ( s + jn ) = r + jn for all j ∈ Z and t r,s ( i ) = i for all i ∈ Z such that r − i, s − i / ∈ n Z . There is a naturalaction of b S n on edge sequences and extended offset sequences: the reflection t r,s acts on anedge sequence p by exchanging the bits p r + in and p s + in for all i ∈ Z , and t r,s d is obtainedfrom d in the same way. This gives an b S n action on n -cores.It is worth pointing (though we will only make use of this implicitly through citations)that there is a bijection between minimal coset representatives b S n / S n and n -cores, com-patible with the b S n actions, given by w S n w · ∅ , where ∅ denotes the empty partition.Moreover, this bijection matches strong Bruhat order with the inclusion partial order on n -cores (see Propositions 8.7, 8.8, 8.9, and 9.3 of [21]).We also need one new definition, not given in the reference [21]. For an n -core κ , the row map of κ is the function f : Z ≥ → Z , given by f ( z ) = κ z − z + 1 . (8.2)With this definition, f ( z ) − z, κ z ) on the eastern border of κ . By our convention above for the sequence p , this means that p f ( z ) corresponds to thenorth step in row z ; in other words, f ( z ) is equal to the index i such that p i = 1 and p i p i +1 · · · contains z Example 8.1.
Let k = 4, n = 5. Below is the diagram of the n -core κ = 665443221 withits edge sequence p labeled, where the • separates p and p : κ = •
11 01 011 01 0 i − − − − − − − − − p i d i − − − − − ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 31 It is convenient to depict the edge sequence in an ∞× n array with the r -th row equal tothe sequence p nr p nr . . . p n + nr , where the horizontal line divides rows 0 and −
1. Thenthe entries d d · · · d n record the heights the 1’s attain in columns 1 , , . . . , n . ... ... ... ... ...0 0 0 0 01 0 0 0 01 0 1 0 11 0 1 0 11 0 1 0 11 1 1 1 1... ... ... ... ... The row map is given by f (1) , f (2) , · · · = 6 , , , , , − , − , − , − , − , − , − , . . . .Our first task in this section is to give the translation between bounce paths andextended offset sequences. We write c for the inverse of the bijection p , which is a mapfrom Par k to k + 1-cores. Proposition 8.2.
Let λ ∈ Par kℓ and Φ = ∆ k ( λ ) . Let κ = c ( λ ) have edge sequence p ,extended offset sequence d , and row map f . Let r ∈ Z ≥ and z ∈ [ ℓ ] . Then (a) f ( r ) = f ( r + n − λ r ) + n ; (b) f (up Φ ( z )) = f ( z ) + n whenever up Φ ( z ) is defined; (c) d f ( z ) = d f (up Φ ( z )) + 1 whenever up Φ ( z ) is defined; (d) d f ( z ) > ⇐⇒ up Φ ( z ) is defined; (e) d f ( z ) = | uppath Φ ( z ) | ; (f) n − λ z = p f ( z ) − n +1 + p f ( z ) − n +2 + · · · + p f ( z ) ; (g) λ z = λ z +1 ⇐⇒ there are no 0’s in d f ( z +1) d f ( z +1)+1 . . . d f ( z ) .Proof. We first prove (a). First suppose κ r > λ r and let ( r, c ) = ( r, κ r − λ r ) be theeasternmost box in the r -th row of κ having hook length > n . For this box to havehook length > n , the box ( r + n − λ r , c ) must lie in κ . By definition of the map p (seeSection 2), the box ( r, c + 1) has hook length < n in κ , and thus ( r + n − λ r , c + 1) / ∈ κ .Hence ( r + n − λ r , c ) lies on the eastern border of κ and we have f ( r ) = κ r − r + 1 = c + λ r − r + 1 = c − ( r + n − λ r ) + 1 + n = f ( r + n − λ r ) + n. This argument also works in the case κ r = λ r if we consider the column { ( i, | i ∈ Z ≥ } to be part of κ .Statement (b) follows from (a) by setting r = up Φ ( z ) and using that z = down Φ ( r ) = r + n − λ r . Statement (c) follows from (b) and (8.1), (e) follows from (c) and (d), (f)follows from (a), and (g) follows from (f).It remains to prove (d). The ⇐ direction is immediate from (c). For the ⇒ direction,suppose d f ( z ) >
1. Then p f ( z )+ n = 1 and f ( r ) = f ( z ) + n for some r < z ; hence z = r + n − λ r by (a). Note that down Φ ( r ) is defined exactly when r + n − λ r ≤ ℓ (byProposition 7.3); since z ≤ ℓ we then have z = down Φ ( r ) and up Φ ( z ) = r . (cid:3) Recall that a strong cover τ ⇒ κ is a pair of n -cores such that τ ⊂ κ and | p ( τ ) | + 1 = | p ( κ ) | . Lemma 8.3.
Let κ be an n -core with extended offset sequence d and t r,s ∈ b S n a reflection. (i) κ ⇒ t r,s κ if and only if d r > d s and for all r < i < s , d i / ∈ [ d s , d r ] . (ii) t r,s κ ⇒ κ if and only if s − n < r , d r < d s , and for all r < i < s , d i / ∈ [ d r , d s ] . (iii) If t r,s κ ⇒ κ , then the skew shape κ/ ( t r,s κ ) has components R d r , . . . , R d s − ,where R j is the ribbon with s − r boxes in diagonals r + nj, r + 1 + nj, . . . , s − nj . (iv) For each j ∈ [ d r , d s − , let z j be the smallest row index of the ribbon R j from(iii). Then f ( z j ) = s + nj and uppath ∆ k ( p ( κ )) ( z d r ) = ( z d r , z d r +1 , . . . , z d s − ) .Proof. Statement (i) is [21, Lemma 9.4 (2)]. Statement (ii) follows from (i) and (8.1).Statement (iii) is essentially [21, Proposition 9.5]; it follows from the interpretation of t r,s κ ⇒ κ in terms of edge sequences. We now prove (iv). By (iii), the northeastmostbox of the ribbon R j is the box in diagonal s − nj on the eastern border of κ . Therow z j containing this box satisfies f ( z j ) = s + nj by the discussion following (8.2). Thestatement about the uppath then follows from Proposition 8.2 (b) and (e). (cid:3) The next result gives a description of strong covers with the focus on the row indices ofthe shape κ rather than diagonals. This result as well as Lemmas 8.7 and 8.9 prepare usto prove the main results of this section, which connect strong covers to cover z ( λ ) from k -Schur straightening. Lemma 8.4.
Let κ be an n -core with extended offset sequence d and row map f . Let z ∈ [ ℓ ( κ )] and set s = f ( z ) . Let r < s be as small as possible such that (cid:0) d i > d s or d i < (cid:1) for all i ∈ [ r + 1 , s − . (8.3) There exists a strong cover τ ⇒ κ such that the southwestmost component of κ/τ hassmallest row index z if and only if ( d r = 0 and s − n < r ) . Moreover, this cover is uniqueif it exists and τ = t r,s κ . Definition 8.5.
We define cover z ( κ ) = τ if the strong cover in Lemma 8.4 exists, andotherwise we say that cover z ( κ ) does not exist. Proof.
For the “only if” direction, suppose τ ⇒ κ is a strong cover such that the south-westmost component of κ/τ has smallest row index z . We can write τ = t r ′ ,s ′ κ with r ′ < s ′ determined uniquely by requiring d r ′ = 0. Then s = f ( z ) = s ′ + nd r ′ = s ′ byLemma 8.3 (iii)–(iv). By Lemma 8.3 (ii), r = r ′ , and hence d r = 0 and s − n < r . Thisalso establishes uniqueness.For the “if” direction, suppose d r = 0 and s − n < r . Note that d s > s ∈ Image( f ). Then by Lemma 8.3 (ii), t r,s κ ⇒ κ . By Lemma 8.3 (iii)–(iv), the southwestmostcomponent of κ/τ has smallest row index z . (cid:3) Example 8.6.
For κ = 665443221 and z = 6, we have cover z ( κ ) = 663331111. Thestrong cover cover z ( κ ) ⇒ κ is the same as that of Example 2.11. The key relevantquantities from Lemma 8.4 are s = − r = −
6, and d r d r +1 · · · d s = 0 3 3 − ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 33 Lemma 8.7.
Let λ ∈ Par kℓ and set Φ = ∆ k ( λ ) . Let j ∈ [ ℓ ] and i ∈ uppath Φ ( j ) . Then | path Φ ( i, j ) | ≤ | uppath Φ ( j + 1) | if and only if λ x = λ x +1 for all x ∈ path Φ ( i, j ) \ { j } ; ifthese conditions hold, then path Φ ( i + 1 , j + 1) = { x + 1 | x ∈ path Φ ( i, j ) } .Proof. This is a direct consequence Proposition 7.3. (cid:3)
Remark 8.8.
Let λ ∈ Par kℓ and µ = λ − ǫ z ( z ∈ [ ℓ ]), and set Φ = ∆ k ( λ ) and Ψ = ∆ k ( µ ).We have uppath Φ ( z ) = uppath Ψ ( z ) and if | uppath Ψ ( z + 1) | >
1, then uppath Φ ( z + 1) =uppath Ψ ( z + 1). Lemma 8.9.
Maintain the notation of Definition 7.9 and set
Φ = ∆ k ( λ ) . Then h = min( k − λ z , h ′ ) , where h ′ is the largest element of [0 , ℓ − z ] such that | uppath Φ ( z ) | < | uppath Φ ( z + i ) | and λ z = λ z + i for all i ∈ [ h ′ ] . (8.4) Proof.
One checks directly that h = 0 if and only if h ′ = 0. So now assume h > h ′ >
0. One checks using Remark 8.8 and Lemma 8.7 with j = z + 1, then j = z + 2, . . . , j = z + h ′ − i = up c Φ ( j ) that h ′ is the largest element of [ z − ℓ ] such that λ is constant on each of the intervals [ z + 1 , z + h ′ ], [up Ψ ( z ) , up Ψ ( z ) + h ′ ],[up ( z ) , up ( z ) + h ′ ], . . . , [top Ψ ( z ) , top Ψ ( z ) + h ′ ], and [ y + 1 , y + h ′ ].This is the same as the definition of h except with λ in place of µ = λ − ǫ z . Hence showing h = min( k − λ z , h ′ ) amounts to checkingmin (cid:0) k − λ z , g ( λ ) (cid:1) = min (cid:0) z − up Ψ ( z + 1) , g ( λ ) (cid:1) = g ( µ ) , (8.5)where g ( ν ) := max (cid:8) i | ν is constant on [up Ψ ( z + 1) , up Ψ ( z + 1) + i − (cid:9) for ν ∈ Z ℓ (the second equality of (8.5) holds by µ z − > µ z = ⇒ g ( µ ) ≤ z − up Ψ ( z + 1)). We have k − λ z ≥ k − λ up Ψ ( z +1) = k − µ up Ψ ( z +1) = z − up Ψ ( z + 1) , with equality if and only if g ( λ ) > z − up Ψ ( z + 1) (the last equality is by Proposition 7.3).This proves (8.5). (cid:3) The next three results establish a dictionary between constructions on the core side(strong covers, spin) and constructions on bounce graph side (cover z ( λ ), bounce). Proposition 8.10.
Let λ ∈ Par kℓ , κ = c ( λ ) , and z ∈ [ ℓ ] . Then (i) cover z ( κ ) exists if and only if cover z ( λ ) ∈ Par kℓ ; (ii) if cover z ( λ ) ∈ Par kℓ , then cover z ( λ ) = p ( cover z ( κ )) .Proof. If λ z = 0, then both conditions in (i) fail. So now assume λ z >
0. Maintain thenotation of Lemma 8.4 so that s = f ( z ) and r is defined by (8.3). The edge sequence,extended offset sequence, and row map of κ are denoted p , d , and f , respectively.We first prove (i). For an index r ′ < s , define ˜ h ( r ′ ) = p r ′ +1 + p r ′ +2 + · · · + p s − = |{ i ∈ [ r ′ + 1 , s −
1] : d i > }| . One checks using Proposition 8.2 (e) and (g) that ˜ h ( r ) isequal to the h ′ defined in Lemma 8.9 and d r = 0 ⇐⇒ λ z + h ′ > λ z + h ′ +1 . (8.6) By Lemma 8.4 and Lemma 7.10 (ii), statement (i) amounts to showing (cid:0) d r = 0 and s − n < r (cid:1) ⇐⇒ λ z + h > λ z + h +1 , (8.7)where h is as in Definition 7.9. We treat the cases s − n < r and s − n ≥ r sepa-rately. By Proposition 8.2 (f), k − λ z = ˜ h ( s − n ); this together with Lemma 8.9 yields s − n < r = ⇒ h = h ′ , and s − n ≥ r = ⇒ h = k − λ z . Hence in the case s − n < r , (8.6)implies (8.7). In the case s − n ≥ r , using that d s − n = d s +1 > h = k − λ z = ˜ h ( s − n )we have f ( z + h + 1) = s − n ; by definition of r , there are no 0’s in d f ( z + h +1) · · · d f ( z + h ) ,and thus λ z + h = λ z + h +1 by Proposition 8.2 (g).We now prove (ii). By (i), τ := cover z ( κ ) exists. The border sequence of τ is q := t r,s p .Define the sequence ˜ p by ˜ p i = n − P j ∈ [ i − n +1 ,i ] p i , and define ˜ q similarly in terms of q .Since q = p + P i ∈ [0 ,d s − ( ǫ r + in − ǫ s + in ), we have˜ q = ˜ p + ǫ [ r + d s n,s + d s n − − ǫ [ r,s − . (8.8)By Proposition 8.2 (f), λ (resp. p ( τ )) is the subsequence of ˜ p (resp. ˜ q ) over those indices j such that p j = 1 (resp. q j = 1). By the definition of r , the sequence ˜ p is constant oneach interval [ r + in, s + in ] for i ∈ [0 , d s − q = t r,s p , one checksthat p ( τ ) = λ + ǫ I − ǫ J , where I = f − ([ r + d s n, s + d s n − J = f − ([ r, s ]). Let y be as in Definition 7.9. We have I = [ y + 1 , y + h ] and J = [ z, z + h ] by h = ˜ h ( r ) (fromthe proof of (i)) and Proposition 8.2 (b) and (e). Hence p ( τ ) = cover z ( λ ) as desired. (cid:3) Example 8.11.
Let λ = 222222221 and κ = 665443221. By Examples 7.13 and 8.6,cover z ( λ ) = 332221111 = p (663331111) = p ( cover z ( κ )) , in agreement with Proposition 8.10 (ii). Proposition 8.12.
Fix λ ∈ Par kℓ and set Φ = ∆ k ( λ ) and κ = c ( λ ) . There is a bijection (cid:8) ( ν, z, m ) ∈ Par kℓ × [ ℓ ] × [ ℓ ] | z ∈ downpath Φ ( m ) , ν = cover z ( λ ) (cid:9) ∼ = −→ VSMT k (1) ( λ ) given by ( ν, z, m ) ( c ( ν ) m == ⇒ κ ) .Proof. The set VSMT k (1) ( λ ) is just another notation for the set of all strong marked covers τ m == ⇒ κ , which, by Lemma 8.4, can be written as the union of { ( cover z ( κ ) m == ⇒ κ ) } overall z, m ∈ [ ℓ ] such that cover z ( κ ) exists and m is a possible marking of cover z ( κ ) ⇒ κ .By Lemma 8.3 (iii)–(iv), the possible markings of cover z ( κ ) ⇒ κ are exactly the elementsof uppath Φ ( z ). The result then follows from Proposition 8.10. (cid:3) Recall the definitions of spin from Section 2 and bounce from Definition 7.9.
Proposition 8.13.
The bijection of Proposition 8.12 takes bounce to spin: for (cover z ( λ ) , z, m ) T , we have bounce(cover z ( λ ) , λ ) + B Φ ( m, z ) = spin( T ) . Proof.
We have T = ( c (cover z ( λ )) m == ⇒ κ ) = ( cover z ( κ ) m == ⇒ κ ) (by Proposition 8.10).Let c and h be as in Definition 7.9. By Lemma 8.3 (iv), the number of components of κ/ cover z ( κ ) is | uppath Φ ( z ) | = c. By Lemma 8.3 (iii), the height of each componentis the number of 1’s (north steps) in p r p r +1 · · · p s − p s , where p is the edge sequence of ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 35 κ and cover z ( κ ) = t r,s κ as in Lemma 8.4. This is equal to ˜ h ( r ) + 1 = h + 1 bythe proof of Proposition 8.10 (i). So bounce(cover z ( λ ) , λ ) is equal to the number ofcomponents of κ/ cover z ( κ ) times one less than the height of each component. Finally,by Lemma 8.3 (iv), the number of components of κ/ cover z ( κ ) entirely contained in rows > m is equal to | path Φ (down Φ ( m ) , z ) | = B Φ ( m, z ). (cid:3) Example 8.14.
In Example 2.11 we computed the spins of the possible markings of thestrong cover 663331111 = τ ⇒ κ = 665443221 (reproduced on the right in (8.9)–(8.10)).We have τ = cover z ( κ ) for z = 6 (see Example 8.6) and λ = p ( κ ) = 222222221. Weverify Proposition 8.13 for each m ∈ uppath ∆ k ( λ ) ( z ) = { , } :bounce(cover z ( λ ) , λ ) + B ∆ k ( λ ) (6 , z ) = 4 + 0 = 4 = spin( τ = ⇒ κ ) , (8.9)bounce(cover z ( λ ) , λ ) + B ∆ k ( λ ) (3 , z ) = 4 + 1 = 5 = spin( τ = ⇒ κ ) , (8.10)where bounce(cover z ( λ ) , λ ) = h · c = 2 · The dual Pieri rules
To better understand the dual Pieri rules for k -Schur functions, let us first considerthe dual Pieri rule for ordinary Schur functions for d = 1. This rule can be stated in thefollowing elegant way: e ⊥ s λ = ℓ X z =1 s λ − ǫ z . This is the same as the sum of Schur functions indexed by partitions obtained by removinga corner from λ since s λ − ǫ z is nonzero if and only if ( z, λ z ) is a removable corner. We havethe following analogous formula for the s ( k ) λ (by summing (9.3) below over m ∈ [ ℓ ]): e ⊥ s ( k ) λ = X m ∈ [ ℓ ] X z ∈ downpath Φ ( m ) t B Φ ( m,z ) s ( k ) λ − ǫ z ; (9.1)the (vertical) dual Pieri rule for d = 1 is then obtained by evaluating the right side using k -Schur straightening II.To prove the vertical dual Pieri rule for general d , we need a more refined version toinduct on d . This refined statement involves the subset lowering operators, recalled below.Somewhat miraculously, the combinatorics of bounce graphs and k -Schur straighteningmatches exactly the combinatorics of strong covers—we obtain an algebraic meaning forthe sum over strong covers with a fixed marking; see Theorem 9.2. This is the refinedstatement we need for d = 1. The general case is then handled in Theorem 9.3. (Theformer is a special case of the latter so is unnecessary but we include it as an instructivewarmup.) We will need the following variants of the notation from Definition 6.7 for the subsetlowering operators: for d ≥ m ∈ [ ℓ ], and an indexed root ideal (Φ , λ ) of length ℓ , define L d,m H (Φ; λ ) := L d, [ m ] H (Φ; λ ) = X S ⊂ [ m ] , | S | = d H (Φ; λ − ǫ S ) ;˜ L d,m H (Φ; λ ) := L d,m H (Φ; λ ) − L d,m − H (Φ; λ ) = X S ⊂ [ ℓ ] , | S | = d, max( S )= m H (Φ; λ − ǫ S ) . (9.2)By Lemma 4.11, L d,ℓ H (Φ; λ ) = e ⊥ d H (Φ; λ ) for any d ≥
0, and e ⊥ d H (Φ; λ ) = 0 when d > ℓ .Note that L d,m H (Φ; λ ) = ˜ L d,m H (Φ; λ ) = 0 when d > m , the natural generalization of thislatter fact. Proposition 9.1.
Let λ ∈ Par kℓ , Φ = ∆ k ( λ ) , and m ∈ [ ℓ ] . Then ˜ L ,m s ( k ) λ = H (Φ; λ − ǫ m ) = X z ∈ downpath Φ ( m ) t B Φ ( m,z ) s ( k ) λ − ǫ z . (9.3) Proof.
We have H (Φ; λ − ǫ m ) = X z ∈ downpath Φ ( m ) t B Φ ( m,z ) H (Ψ z ; λ − ǫ z ) = X z ∈ downpath Φ ( m ) t B Φ ( m,z ) s ( k ) λ − ǫ z , where Ψ z := Φ \ { ( z, down Φ ( z )) } for z = bot Φ ( m ) and Ψ z := Φ for z = bot Φ ( m ). Thefirst equality is by Corollary 5.7 and the second is by ∆ k ( λ − ǫ z ) = Ψ z . (cid:3) For a (vertical) strong marked tableau T = ( κ (0) r == ⇒ κ (1) r == ⇒ · · · r m == ⇒ κ ( m ) ), definemarks( T ) to be the set { r , . . . , r m } of markings that appear in T . Theorem 9.2.
For any λ ∈ Par kℓ and m ∈ [ ℓ ] , ˜ L ,m s ( k ) λ = X T ∈ VSMT k (1) ( λ ) , marks( T )= { m } t spin( T ) s ( k )inside( T ) . (9.4) Proof.
Set Φ = ∆ k ( λ ). Beginning with Proposition 9.1, we obtain˜ L ,m s ( k ) λ = X z ∈ downpath Φ ( m ) t B Φ ( m,z ) s ( k ) λ − ǫ z (9.5)= X { z ∈ downpath Φ ( m ) | cover z ( λ ) ∈ Par kℓ } t B Φ ( m,z )+bounce(cover z ( λ ) ,λ ) s ( k )cover z ( λ ) (9.6)= X T ∈ VSMT k (1) ( λ ) , marks( T )= { m } t spin( T ) s ( k )inside( T ) , (9.7)where the second equality is by Theorem 7.12 and the third equality is by Propositions8.12 and 8.13. (cid:3) ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 37 The vertical dual Pieri rule.Theorem 9.3.
For any λ ∈ Par kℓ and integers d ≥ , m ∈ [ ℓ ] , we have L d,m s ( k ) λ = X T ∈ VSMT k ( d ) ( λ ) , marks( T ) ⊂ [ m ] t spin( T ) s ( k )inside( T ) ; (9.8)˜ L d,m s ( k ) λ = X T ∈ VSMT k ( d ) ( λ ) , max(marks( T ))= m t spin( T ) s ( k )inside( T ) . (9.9)In the special case m = ℓ , the condition marks( T ) ⊂ [ m ] in (9.8) is no restriction at all.Hence this proves Property (2.6). Proof.
Since ˜ L d,m s ( k ) λ = L d,m s ( k ) λ − L d,m − s ( k ) λ , statements (9.8) and (9.9) are equivalent.We will prove them simultaneously by induction on d . Specifically, we will prove (9.9)using (9.8) for d − , m −
1. The base cases d = 0 and m < d are trivial. So now assume0 < d ≤ m .Set Φ = ∆ k ( λ ). By definition (9.2), ˜ L d,m s ( k ) λ = P S ⊂ [ ℓ ] , | S | = d, max( S )= m H (Φ; λ − ǫ S ). Foreach term H (Φ; λ − ǫ S ) in this sum, expand on downpath Φ ( m ) using Corollary 5.7 toobtain H (Φ; λ − ǫ S ) = X z ∈ downpath Φ ( m ) t B Φ ( m,z ) H (Ψ z ; λ − ǫ z − ǫ S ′ ) , (9.10)where Ψ z := Φ \ { ( z, down Φ ( z )) } for z = bot Φ ( m ) and Ψ z := Φ for z = bot Φ ( m ), and S ′ := S \ { m } .Summing (9.10) over all S ⊂ [ ℓ ] such that | S | = d and max( S ) = m , we obtain˜ L d,m s ( k ) λ = X S ′ ⊂ [ m − , | S ′ | = d − X z ∈ downpath Φ ( m ) t B Φ ( m,z ) H (Ψ z ; λ − ǫ z − ǫ S ′ )= X z ∈ downpath Φ ( m ) t B Φ ( m,z ) L d − ,m − H (Ψ z ; λ − ǫ z )= X { z ∈ downpath Φ ( m ) | cover z ( λ ) ∈ Par kℓ } t B Φ ( m,z )+bounce(cover z ( λ ) ,λ ) L d − ,m − s ( k )cover z ( λ ) = X V ∈ VSMT k (1) ( λ ) , marks( V )= { m } t spin( V ) L d − ,m − s ( k )inside( V ) = X V ∈ VSMT k (1) ( λ )marks( V )= { m } X U ∈ VSMT k ( d − (inside( V ))marks( U ) ⊂ [ m − t spin( V )+spin( U ) s ( k )inside( U ) = X T ∈ VSMT k ( d ) ( λ ) , max(marks( T ))= m t spin( T ) s ( k )inside( T ) . We need to justify the last four equalities. The third equality is the delicate step where weapply Theorem 7.16 to each term L d − ,m − H (Ψ z ; λ − ǫ z ) = L d − ,m − s ( k ) λ − ǫ z ; the hypothesesof the theorem are satisfied by Lemma 7.18 and Remark 8.8. The fourth equality is by Propositions 8.12 and 8.13 (just as in the proof of Theorem 9.2). The fifth equality is bythe inductive hypothesis, and the last equality holds since a vertical strong marked tableau T of weight ( d ) with max(marks( T )) = m is the same as a vertical strong marked tableau V of weight (1) with marks( V ) = { m } followed by a vertical strong marked tableau U ofweight ( d −
1) such that marks( U ) ⊂ [ m −
1] and inside( V ) = outside( U ). (cid:3) Example 9.4.
Let k = 3, d = 2, and m = 4. According to Theorem 9.3, L d,m s ( k )222222 equals the sum of t spin( T ) s ( k )inside( T ) over strong marked tableaux T ∈ VSMT k ( d ) (222222) suchthat marks( T ) ⊂ [ m ], as shown: T ⋆ ⋆
12 1 ⋆ ⋆
12 1 ⋆ ⋆ ⋆
211 1 2 ⋆
12 1 ⋆
21 2 ⋆ k -skew(inside( T ))spin( T ) 2 3 4 4 3inside( T ) 222211 222211 222211 322111 222220 L , s (3)222222 = t s (3)222211 + t s (3)222211 + t s (3)222211 + t s (3)322111 + t s (3)222220 . Remark 9.5.
Theorem 9.3 is somewhat delicate. By definition, L d,m s ( k ) λ = X S ⊂ V, | S | = d H (∆ k ( λ ); λ − ǫ S ) . However, each Catalan function H (∆ k ( λ ); λ − ǫ S ) in the sum is not necessarily equal to P T ∈ VSMT k ( d ) ( λ ) , marks( T )= S t spin( T ) s ( k )inside( T ) . In fact, the Catalan functions H (∆ k ( λ ); λ − ǫ S )are not always Schur positive, as can be seen from a more detailed study of the previousexample. Setting Φ = ∆ (222222), we have L , s (3)222222 = H (Φ; 112222) + H (Φ; 121222) + H (Φ; 122122)+ H (Φ; 211222) + H (Φ; 212122) + H (Φ; 221122) , and these terms have the following expansions into 3-Schur functions: H (Φ; 112222) = t s (3)222211 , H (Φ; 121222) = − t s (3)222211 , H (Φ; 122122) = t s (3)222211 ,H (Φ; 211222) = t s (3)222211 , H (Φ; 212122) = t s (3)322111 + t s (3)222220 , H (Φ; 221122) = t s (3)222211 . Summing these, we recover the positive expression in Example 9.4. The proof of Theo-rem 9.3 implicitly handles cancellation of this form to produce the positive sum in (9.8).
ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 39 The horizontal dual Pieri rule.
We prove this rule (Property (2.5)) using the ver-tical dual Pieri rule and a trick involving symmetric functions in noncommuting variables,inspired by [9, 2].Fix positive integers ℓ and k . Recall that by Theorem 2.8, { s ( k ) µ | µ ∈ Par kℓ } is a basisfor Λ kℓ ⊂ Λ. We define the strong marked cover operators u , . . . , u ℓ ∈ End(Λ kℓ ) by s ( k ) µ · u r = X T ∈ SMT k (1) ( µ ) , marks( T )= { r } t spin( T ) s ( k )inside( T ) . (9.11)We have found it more natural here to work with right operators, so for this subsectionwe consider all operators to act on the right. Note that the set of T in the sum is justanother notation for the set of strong marked covers τ r = ⇒ κ with p ( κ ) = µ .For d ≥
0, define the following versions of the elementary and homogeneous symmetricfunctions in the (noncommuting) operators u i :˜ e d = X ℓ ≥ i >i > ··· >i d ≥ u i u i · · · u i d , (9.12)˜ h d = X ≤ i ≤ i ≤···≤ i d ≤ ℓ u i u i · · · u i d ; (9.13)by convention, ˜ e = ˜ h = 1 and ˜ e d = 0 for d > ℓ . Proof of Property (2.5) . First, we have the basic identity h ⊥ m − h ⊥ m − e ⊥ + h ⊥ m − e ⊥ − · · · + ( − m e ⊥ m = 0 for m > . (9.14)This follows directly from the well-known identity P mi =0 ( − i e i h m − i = 0 [36, Equation 7.13].Hence the operators h ⊥ , h ⊥ , . . . can be recursively expressed in terms of e ⊥ , e ⊥ , . . . .The operators ˜ h d and ˜ e d were cooked up so that s ( k ) µ · ˜ h d (resp. s ( k ) µ · ˜ e d ) equals the rightside of (2.5) (resp. (2.6)). Hence by Property (2.6), e ⊥ d restricts to an operator on Λ kℓ which agrees with ˜ e d . Then by the previous paragraph, h ⊥ d also restricts to an operatoron Λ kℓ and Property (2.5) is equivalent to the identity h ⊥ d = ˜ h d of operators on Λ kℓ . It nowsuffices to show that (9.14) holds with ˜ e d and ˜ h d in place of e ⊥ d and h ⊥ d .Let y be a formal variable that commutes with u , . . . , u ℓ . The elements ˜ e d and ˜ h d canbe packaged into the generating functions˜ E ( y ) = ℓ X d =0 y d ˜ e d = (1 + yu ℓ ) · · · (1 + yu ) ∈ End(Λ kℓ )[ y ] , ˜ H ( y ) = ∞ X d =0 y d ˜ h d = (1 − yu ) − · · · (1 − yu ℓ ) − ∈ End(Λ kℓ )[[ y ]] . The identity ˜ H ( y ) ˜ E ( − y ) = 1 implies that for any m > h m − ˜ h m − ˜ e + ˜ h m − ˜ e − · · · + ( − m ˜ e m = 0 , (9.15)as desired. (cid:3) The Chen-Haiman root ideal for skew-linking diagrams
Consider a skew partition κ/η and denote its rows by λ = ( κ − η , . . . , κ ℓ − η ℓ ) andthe rows of its transpose ( κ/η ) ′ by µ . If λ and µ are both partitions, we say κ/η is a skew-linking diagram and write λ κ/η −−→ µ . Any k -skew diagram (defined in § k + 1-core is again a k + 1-core and the existence of the bijection p . Definition 10.1 ([7, § . The root ideal associated to a skew-linking diagram λ κ/η −−→ µ is given by Φ( κ/η ) = { ( i, j ) ∈ ∆ + ℓ ( κ ) | i ≤ ℓ ( η ) and j ≥ µ η i + i } . Equivalently, Φ( κ/η ) is the root ideal with removable roots { ( i, µ η i + i ) | i ≤ ℓ ( η ) } .See Example 10.5.Chen and Haiman conjectured that the Catalan functions include the k -Schur functionsas a subclass but used the root ideal Φ( k -skew( λ )) rather than our ∆ k ( λ ). We reconcilethis difference in Theorem 10.4 below. This conjecture fits in the broader context ofa study of Catalan functions associated to skew-linking diagrams. We briefly mentiontwo conjectures of Chen-Haiman arising in this study: for any skew-linking diagram λ κ/η −−→ µ , the Catalan function H (Φ( κ, η ); λ ) (1) is the graded character of an explicitlyconstructed module for the symmetric group (see [7, Corollary 5.4.6]), and (2) is equalto the Schur positive sum P t charge( T ) s shape( T ) over semistandard tableaux T satisfyingcertain katabolizability conditions [7, Conjecture 5.4.3]. Lemma 10.2.
Let λ κ/η −−→ µ be a skew-linking diagram. If κ y = κ y +1 , then η y = η y +1 and λ y = λ y +1 . If η y = η y +1 > , then κ µ ηy + y = κ µ ηy + y +1 .Proof. These facts are direct consequences of the definition of a skew-linking diagram.The first uses that λ is a partition and the second that µ is. (cid:3) Lemma 10.3.
Let λ κ/η −−→ µ be a skew-linking diagram and Ψ a root ideal which agreeswith Φ( κ/η ) in rows ≥ y . Suppose that Ψ has a removable root of the form α = ( a, y ) , Ψ has a ceiling in columns y, y + 1 , and κ y = κ y +1 . Then H (Ψ; λ ) = H (Ψ \ α ; λ ) .Proof. Apply Lemma 6.5 with indexed root ideal (Ψ , λ ) and w = bot( y ); we need to verifythe hypotheses (6.4)–(6.6), (6.9), and w < ℓ ( κ ). The first, (6.4), holds by assumption.Next, by Lemma 10.2, κ y = κ y +1 implies λ y = λ y +1 and η y = η y +1 . By definitionof Φ( κ/η ), if η y >
0, then down Ψ ( y ) = µ η y + y and thus κ down Ψ ( y ) = κ down Ψ ( y )+1 byLemma 10.2. Iterating this argument gives κ x = κ x +1 , λ x = λ x +1 , and η x = η x +1 for all x ∈ downpath Ψ ( y ), which verifies (6.5), (6.9), and also w < ℓ ( κ ) (since κ w = κ w +1 > i for z ≤ i ≤ ℓ ( η ) and no roots in rows > ℓ ( η ),we have w > ℓ ( η ) and Ψ has a wall in rows w, w + 1, i.e., (6.6) holds. (cid:3) Theorem 10.4.
Let λ ∈ Par k and κ/η be the k -skew diagram of λ . Then s ( k ) λ = H (∆ k ( λ ); λ ) = H (Φ( κ/η ); λ ) . ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 41 It is most natural to regard (∆ k ( λ ) , λ ) as an indexed root ideal of length ℓ ( λ ) = ℓ ( κ ), andwe do this in the proof below, however this is not strictly necessary by Proposition 4.12. Proof.
Let µ denote the rows of ( κ/η ) ′ . LetΨ s = { ( i, j ) ∈ ∆ k ( λ ) | i < s } ⊔ { ( i, j ) ∈ Φ( κ/η ) | i ≥ s } . We have Ψ = Φ( κ/η ) and Ψ ℓ ( η )+1 = ∆ k ( λ ) since both Ψ and ∆ k ( λ ) have no roots ( i, j )for i > ℓ ( η ); namely, when i > ℓ ( η ), κ/η is a k -skew diagram implies that k + 1 − λ i + i ≥ µ + i > ℓ ( η ) + µ = ℓ ( κ ). It thus suffices to show H (Ψ s +1 ; λ ) = H (Ψ s ; λ ) for all s ∈ [ ℓ ( η )].Let s ∈ [ ℓ ( η )]. Since κ/η is a k -skew diagram, λ s + µ η s > k and thusΨ s +1 = Ψ s ⊔ { ( s, j ) | j ∈ J } , where J = [ k + 1 − λ s + s, µ η s + s − . (10.1)We have H (Ψ s +1 ; λ ) = H (Ψ s +1 \ ( s, k + 1 − λ s + s ); λ ) = · · · = H (Ψ s ; λ )by repeated application of Lemma 10.3. The hypotheses hold at each step by (10.1) andthe facts (1) Ψ s +1 has a ceiling in columns j, j + 1 for j ∈ J , and (2) κ j = κ j +1 for j ∈ J .Fact (1) holds by (10.1) and down Ψ s +1 ( s + 1) = µ η s +1 + s + 1 > µ η s + s . For (2), note thatthe box ( s, η s + 1) has hook length ≤ k in κ , and hence the box ( k + 1 − λ s + s, η s + 1) / ∈ κ .Together with ( µ η s + s, η s ) ∈ κ , this implies κ k +1 − λ s + s = η s = κ µ ηs + s . (cid:3) Example 10.5.
For k = 7 and λ = 6643321111, here are the k -skew diagram κ/η of λ and the two associated Catalan functions from Theorem 10.4: κ/η H (Φ( κ/η ); λ ) H (∆ k ( λ ); λ ) . Acknowledgments.
We thank Elaine So for help typing and typesetting figures.
References [1] Sami H. Assaf and Sara C. Billey. Affine dual equivalence and k -Schur functions. J. Comb. , 3(3):343–399, 2012.[2] Jonah Blasiak and Sergey Fomin. Noncommutative Schur functions, switchboards, and Schur posi-tivity.
Selecta Math. (N.S.) , 23(1):727–766, 2017.[3] Abraham Broer. A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles.
J.Reine Angew. Math. , 493:153–169, 1997.[4] Bram Broer. Line bundles on the cotangent bundle of the flag variety.
Invent. Math. , 113(1):1–20,1993.[5] Bram Broer. Normality of some nilpotent varieties and cohomology of line bundles on the cotangentbundle of the flag variety. In
Lie theory and geometry , volume 123 of
Progr. Math. , pages 1–19.Birkh¨auser Boston, Boston, MA, 1994.[6] Erik Carlsson and Anton Mellit. A proof of the shuffle conjecture.
J. Amer. Math. Soc. , 2017. [7] Li-Chung Chen.
Skew-Linked Partitions and a Representation-Theoretic Model for k -Schur Func-tions . PhD thesis, UC Berkeley, 2010.[8] J. D´esarm´enien, B. Leclerc, and J.-Y. Thibon. Hall-Littlewood functions and Kostka-Foulkes poly-nomials in representation theory. S´em. Lothar. Combin. , 32:Art. B32c, approx. 38, 1994.[9] Sergey Fomin and Curtis Greene. Noncommutative Schur functions and their applications.
DiscreteMath. , 193(1-3):179–200, 1998. Selected papers in honor of Adriano Garsia (Taormina, 1994).[10] Adriano M. Garsia. Orthogonality of Milne’s polynomials and raising operators.
Discrete Math. ,99(1-3):247–264, 1992.[11] Adriano M. Garsia and Claudio Procesi. On certain graded S n -modules and the q -Kostka polynomi-als. Adv. Math. , 94(1):82–138, 1992.[12] J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov. A combinatorial formula for thecharacter of the diagonal coinvariants.
Duke Math. J. , 126(2):195–232, 2005.[13] Mark Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture.
J. Amer. Math.Soc. , 14(4):941–1006 (electronic), 2001.[14] Mark Haiman. Combinatorics, symmetric functions, and Hilbert schemes. In
Current developmentsin mathematics, 2002 , pages 39–111. Int. Press, Somerville, MA, 2003.[15] Wim H. Hesselink. Cohomology and the resolution of the nilpotent variety.
Math. Ann. , 223(3):249–252, 1976.[16] Nai Huan Jing. Vertex operators and Hall-Littlewood symmetric functions.
Adv. Math. , 87(2):226–248, 1991.[17] Joel Kamnitzer. Geometric constructions of the irreducible representations of GL n . In Geometricrepresentation theory and extended affine Lie algebras , volume 59 of
Fields Inst. Commun. , pages1–18. Amer. Math. Soc., Providence, RI, 2011.[18] Anatol N. Kirillov, Anne Schilling, and Mark Shimozono. A bijection between Littlewood-Richardsontableaux and rigged configurations.
Selecta Math. (N.S.) , 8(1):67–135, 2002.[19] Thomas Lam. Schubert polynomials for the affine Grassmannian.
J. Amer. Math. Soc. , 21(1):259–281, 2008.[20] Thomas Lam. Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras.
Bull. Lond.Math. Soc. , 43(2):328–334, 2011.[21] Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono. Affine insertion and Pieri rulesfor the affine Grassmannian.
Mem. Amer. Math. Soc. , 208(977):xii+82, 2010.[22] Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono. The poset of k -shapes andbranching rules for k -Schur functions. Mem. Amer. Math. Soc. , 223(1050):vi+101, 2013.[23] L. Lapointe, A. Lascoux, and J. Morse. Tableau atoms and a new Macdonald positivity conjecture.
Duke Math. J. , 116(1):103–146, 2003.[24] L. Lapointe and J. Morse. Schur function analogs for a filtration of the symmetric function space.
J.Combin. Theory Ser. A , 101(2):191–224, 2003.[25] Luc Lapointe and Jennifer Morse. Tableaux on k + 1-cores, reduced words for affine permutations,and k -Schur expansions. J. Combin. Theory Ser. A , 112(1):44–81, 2005.[26] Luc Lapointe and Jennifer Morse. A k -tableau characterization of k -Schur functions. Adv. Math. ,213(1):183–204, 2007.[27] Luc Lapointe and Jennifer Morse. Quantum cohomology and the k -Schur basis. Trans. Amer. Math.Soc. , 360(4):2021–2040, 2008.[28] George Lusztig. Some examples of square integrable representations of semisimple p -adic groups. Trans. Amer. Math. Soc. , 277(2):623–653, 1983.[29] I. G. Macdonald.
Symmetric functions and Hall polynomials . Oxford Mathematical Monographs.The Clarendon Press, Oxford University Press, New York, second edition, 1995.[30] Dmitri I. Panyushev. Generalised Kostka-Foulkes polynomials and cohomology of line bundles onhomogeneous vector bundles.
Selecta Math. (N.S.) , 16(2):315–342, 2010.[31] Anne Schilling and S. Ole Warnaar. Inhomogeneous lattice paths, generalized Kostka polynomialsand A n − supernomials. Comm. Math. Phys. , 202(2):359–401, 1999.
ATALAN FUNCTIONS AND k -SCHUR POSITIVITY 43 [32] Mark Shimozono. A cyclage poset structure for Littlewood-Richardson tableaux. European J. Com-bin. , 22(3):365–393, 2001.[33] Mark Shimozono. Affine type A crystal structure on tensor products of rectangles, Demazure char-acters, and nilpotent varieties.
J. Algebraic Combin. , 15(2):151–187, 2002.[34] Mark Shimozono and Jerzy Weyman. Graded characters of modules supported in the closure of anilpotent conjugacy class.
European J. Combin. , 21(2):257–288, 2000.[35] Mark Shimozono and Mike Zabrocki. Hall-Littlewood vertex operators and generalized Kostka poly-nomials.
Adv. Math. , 158(1):66–85, 2001.[36] Richard P. Stanley.
Enumerative combinatorics. Vol. 2 , volume 62 of
Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 1999.[37] Michael Zabrocki.
On The Action of the Hall-Littlewood Vertex Operator . PhD thesis, UC San Diego,1998.
Department of Mathematics, Drexel University, Philadelphia, PA 19104
E-mail address : [email protected] Department of Math, University of Virginia, Charlottesville, VA 22904
E-mail address : [email protected] Department of Mathematics, Drexel University, Philadelphia, PA 19104
E-mail address : [email protected] Department of Mathematics, Drexel University, Philadelphia, PA 19104
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