Demazure crystals and the Schur positivity of Catalan functions

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_q(\widehat{\mathfrak{sl}}_\ell)$-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

A broader framework has emerged over the last decades. Broer [13] and Shimozono-Weyman [81], in their study of nilpotent conjugacy class closures, replaced the set of all positive roots ∆ + by a parabolic subset-the roots ∆(η) ⊂ ∆ + above a block diagonal matrix. Panyushev [73] and Chen-Haiman [15] went further, taking any one of Catalan many upper order ideals Ψ ⊂ ∆ + . The associated symmetric Catalan functions, H(Ψ; µ)(x; q) = λ K Ψ λµ (q) s λ (x), indexed by Ψ and partition µ, are graded Euler characteristics of vector bundles on the flag variety.
The broader scope deepened ties to Kazhdan-Lusztig theory, advanced by the discovery of LLT polynomials [56,59,40,23], and inspired a generalization of Jing's Hall-Littlewood Demazure-like subsets of tensor products of KR crystals. Explicit katabolism combinatorics arises naturally by unraveling the F w i operators on the DARK side.
(4) A katabolism tableau formula for Catalan functions. In the parabolic case, it agrees with and settles the Shimozono-Weyman conjecture.

Main results
The basic approach of [11] is to open the door to powerful inductive techniques by realizing k-Schur functions as a subclass of (symmetric) Catalan functions. In a similar spirit, our inductive approach here depends crucially on viewing the Catalan functions as a subclass of a larger family of nonsymmetric Catalan functions. Nonsymmetric More generally, for any w ∈ H ℓ , let w = s i 1 s i 2 · · · s im and define the associated Demazure operator by π w := π i 1 π i 2 · · · π im ; this is well defined as the π i satisfy the 0-Hecke relations. A root ideal is an upper order ideal of the poset ∆ + = ∆ + ℓ := {(i, j) | 1 ≤ i < j ≤ ℓ with partial order given by (a, b) ≤ (c, d) when a ≥ c and b ≤ d. A labeled root ideal of length ℓ is a triple (Ψ, γ, w) consisting of a root ideal Ψ ⊂ ∆ + ℓ , a weight γ ∈ Z ℓ , and w ∈ H ℓ . Definition 2.1. The nonsymmetric Catalan function associated to the labeled root ideal (Ψ, γ, w) of length ℓ is H(Ψ; γ; w)(x; q) := π w poly (i,j)∈Ψ where poly denotes the polynomial truncation operator, defined by its action on key polynomials: poly(κ α ) = κ α for α ∈ Z ℓ ≥0 and poly(κ α ) = 0 for α ∈ Z ℓ \ Z ℓ ≥0 (see §5). In the case w = w 0 , the longest element in H ℓ , we recover the (symmetric) Catalan functions studied in [10,11,15,73].
Its proof requires an in-depth understanding of polynomial truncation and is given in Section 5. The operator Φ arises in a recurrence for nonsymmetric Macdonald polynomials, and we will see in Section 8 that its appearance here is no coincidence.

2.2.
Affine generalized Demazure crystals and key positivity. Theorem 2.3 allows us to connect tame nonsymmetric Catalan functions with affine Demazure crystals. We describe this connection here, but defer a thorough treatment of crystals to Section 4.
Let U q (g) be the quantized enveloping algebra of a symmetrizable Kac-Moody Lie algebra g (as in [37]). Among the data specifying a U q (g)-crystal B are mapsf i : B⊔{0} → B ⊔ {0} for i ranging over the Dynkin node set I. For a subset S of B and i ∈ I, define For a dominant integral weight Λ ∈ P + , let B(Λ) denote the highest weight U q (g)-crystal of highest weight Λ and u Λ its highest weight element.
A U q ( sl ℓ )-generalized Demazure crystal is a subset of a tensor product of highest weight crystals of the form F w 1 F w 2 · · · F w p−1 F wp {u Λ p } ⊗ u Λ p−1 · · · ⊗ u Λ 2 ⊗ u Λ 1 for some Λ 1 , . . . , Λ p ∈ P + and w 1 , . . . , w p ∈ H ℓ . Theorem 2.6 and the well-definedness of F w on D( sl ℓ ) show that these are well-defined and yield the following corollary (this argument is essentially due to [45], with the extended affine setup treated carefully in [70]).
It can further be shown that the U q (sl ℓ )-restriction of a U q ( sl ℓ )-Demazure crystal is isomorphic to a disjoint union of U q (sl ℓ )-Demazure crystals (Theorem 4.1). Combining this with Corollary 2.7 and Theorem 2.8 proves that Corollary 2.9. The tame nonsymmetric Catalan functions are key positive.
More detailed versions of Theorem 2.8 and Corollary 2.9-Theorem 7.5 and Corollary 7.13-are stated and proved in Section 7. They include explicit positive formulas for the key expansions.
2.3. DARK crystals. To extract key positive formulas from Theorem 2.8, we use a technique of Naoi [70] to match generalized Demazure crystals with subsets of tensor products of KR crystals, termed DARK crystals; the latter appears to have simpler combinatorics and, remarkably, exactly matches the katabolism combinatorics conjectured in [81].
The following modification of [70,Proposition 5.16] allows us to port results in crystal theory from AGD to DARK crystals. Theorem 2.11 ([8,Corollary 3.11]). Let w, µ, µ i be as in (2.6). There is a strict embedding of U ′ q ( sl ℓ )-seminormal crystals (see §4.1) it is an isomorphism from the domain onto a disjoint union of connected components of the codomain. And under this map, Θ µ (B µ;w ⊗ u µ 1 Λ 0 ) = AGD(µ; w). Here, the B(sΛ i ) are regarded as U ′ q ( sl ℓ )-seminormal crystals by restriction-see §4.4.
Remark 2.12. This article makes important use of the U ′ q ( sl ℓ )-crystal structures of B µ ⊗ B(µ 1 Λ 0 ) and B(µ p Λ p ) ⊗ · · · ⊗ B(µ 1 Λ 1 ), but not of B µ -it does not seem to be the right object for the combinatorics of interest here. However, the U q (sl ℓ )-restriction of B µ , being isomorphic to that of B µ ⊗ B(µ 1 Λ 0 ), is of interest and will be frequently used.
2.4. Katabolism and Schur positive formulas. We establish the Schur positivity of Catalan functions in the strongest possible terms with a streamlined tableau formula. It arises naturally from DARK crystals by unraveling the F w i , F τ , and tensor operations in their construction (in the spirit of [45,46]).
Given a weak composition α = (α 1 , . . . , α ℓ ) ∈ Z ℓ ≥0 , the diagram of α consists of a left justified array of boxes with α i boxes in row i (rows are allowed to be empty). A tabloid T of shape α is a filling of the diagram of α with weakly increasing rows, drawn in English notation with rows labeled 1, 2, . . . , ℓ from the top down. Set shape(T ) = α. The content of T is the vector (c 1 , . . . , c p ), where c i is the number of times letter i appears in T .
The elements of B µ are naturally labeled by biwords whose top word is p µp · · · 2 µ 2 1 µ 1 and whose bottom word is weakly increasing on the intervals with constant top word. Define the bijection inv : B µ ∼ = − → Tabloids ℓ (µ) as follows: for all i, the i-th row of inv(b) is obtained by sorting the letters above the i's in the bottom word of b ∈ B µ . (This is essentially the well-known inverse map on biwords generalizing the inverse of a permutation-see §6.4.) Katabolism exactly characterizes the image of DARK crystals under inv. Theorem 2.17. For a partition µ and root ideal Ψ, the map inv gives a bijection which takes content to shape. Here, P (T ) denotes the insertion tableau of the row reading word T ℓ · · · T 1 of T .
We settle Conjecture 1.1 with a manifestly positive formula.
This gives the first proof of positivity for the Catalan functions and generalized Kostka polynomials in the parabolic case.
Remark 2.19. Shimozono [78] and Schilling-Warnaar [77] give a positive formula for the dominant rectangle Catalan functions H(∆(η); µ; w 0 ) (i.e., µ = a η 1 1 · · · a ηr r , a 1 ≥ · · · ≥ a r ) using tensor products of arbitrary KR crystals in type A. Included in Theorem 2.18 is a different formula addressing this case, using subsets of tensor products of single row KR crystals. Conjecture 10 of [40] proposes a map to reconcile these two different formulas.
We further obtain a positive combinatorial formula for the key expansion of any tame nonsymmetric Catalan function of partition weight by similar methods (Corollary 7.13).

2.5.
Consequences for t = 0 nonsymmetric Macdonald polynomials. A deep theory of nonsymmetric Macdonald polynomials has developed over the last 30 years, beginning with the work of Opdam-Heckman [71], Macdonald [67], and Cherednik [16]. Our results apply to the type A nonsymmetric Macdonald polynomials at t = 0, E α (x; q, 0), a nonsymmetric generalization of the modified Hall-Littlewood polynomials. They were connected to affine Demazure characters by Sanderson [76] and the subject of recent results and conjectures on key positivity [1,2,4,5,6]. The t = 0 nonsymmetric Macdonald polynomials in other types have also received considerable attention [30,61,62,72]. Our results yield the following.
Proof. Statement (1) is due to Sanderson [76], and we also recover it as a special case of our character formula ( The formula (2.13) generalizes Lascoux's formula for cocharge Kostka-Foulkes polynomials [53], answering a call put out in [2,Conjecture 15], [55, p. 267-268] for a description of the key coefficients of E α (x; q, 0) in this style. Assaf-Gonzalez [5,6] studied the problem from a different point of view and realized the coefficients in terms of crystals on nonattacking fillings with no coinversion triples (objects defined in [24]). See also Remark 8.6.
A combinatorial formula for the Schur expansion of s (k) µ was given in [11] in terms of chains in Bruhat order on the affine symmetric group S k+1 and the spin statistic. Theorem 2.18 yields a very different formula: µ has the following Schur positive expansion: (2.14) Namely, T occurs in the sum as follows: remove the µ 1 1's from the first row of T and column insert the remainder of row 1 into rows larger than min{k − µ 1 , ℓ − 1}; remove µ 2 2's from first row and column insert its remainder into rows larger than min{k −µ 2 , ℓ−2}; continue until reaching an i such that there are not µ i i's in the first row; T survives if no such i occurs.
Example 7.14 illustrates (2.14) for s 22211 . This formula has the same spirit as the original definition of k-Schur functions [49], which expressed them in terms of sets of     Similar examples are also given in Figure 5 on the last page. Figure 1 (right) depicts the DARK crystal B µ;w for ℓ = 3, µ = (2, 1, 1), w = (id, s 2 s 1 , s 2 s 1 ); it can be constructed step by step using the F i , F τ , and tensor operations as illustrated.
The first two lines give two different names for each DARK crystal. The connected components of solid edges decompose them into U q (sl ℓ )-Demazure crystals, each of which has character equal to a key polynomial; the key expansions of their charge weighted characters (see §7) are given in the third to last line, written so that reading left to right gives the components top to bottom, e.g., {3211} has character κ 211 = x 2 1 x 2 x 3 . By Corollary 7.13, these characters are tame nonsymmetric Catalan functions (second to last line), though this requires rewriting the DARK crystals appropriately (last line), e.g., Here, b s denotes the element of B 1,s labeled by 1 s , with b 0 the empty word (see §6.1).
The dashed arrows are thef 0 -edges of B µ;w ⊗ u 2Λ 0 (technically this is just a subset of the U ′ q ( sl ℓ )-seminormal crystal B µ ⊗ B(2Λ 0 ) but we often think of it as coming with the edgesf i ,ẽ i (i ∈ I) which have both ends in the subset). By Theorem 2.11, AGD(µ; w) = Θ µ (B µ;w ⊗ u 2Λ 0 ), which is isomorphic to a disjoint union of U q ( sl ℓ )-Demazure crystals; the corresponding decomposition of B µ;w is given by the components of dashed and solid edges (in the rightmost crystal). Here there are two such components, so AGD(µ; w) is not a single U q ( sl ℓ )-Demazure crystal; this demonstrates a fundamental difference between this work and earlier work [44,76,79] relating generalizations of Kostka-Foulkes polynomials to Demazure crystals, where only single U q ( sl ℓ )-Demazure crystals were used. Figure 2 depicts the tabloids obtained by applying inv to the rightmost two DARK crystals in Figure 1. By Theorem 6.20 (the full version of Theorem 2.17), the tabloids on the right are also the T ∈ Tabloids ℓ (211) which are w-katabolizable in the sense of Definition 6.14; the ones on the left are the T ∈ Tabloids ℓ (11) which are (s 2 s 1 , s 2 s 1 )-katabolizable. The bold tabloids, by reading off their shapes and charges, give the rightmost two key expansions in Figure 1; this will be explained in Corollary 7.13.

Higher cohomology vanishing and nonsymmetric Catalan functions
This section uses notation in §1, (2.1)-(2.2), and Definition 5.1, but is otherwise notationally independent from the remainder of the paper.
Let G = GL ℓ (C) and B ⊂ G the standard upper triangular Borel subgroup. For w ∈ S ℓ , let X w = B · wB ⊂ G/B denote the Schubert variety. 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 (N) denote the locally free O G/B -module of its sections. We also denote by L (N) = L (N)| Xw the restriction of L (N) to X w .
Consider the adjoint action of B on the Lie algebra u of strictly upper triangular matrices. The B-stable (or "ad-nilpotent") ideals of u are in bijection with root ideals via the map sending the root ideal Ψ to the B-submodule, call it u Ψ , of u with weights The character of a B-module N is char For γ ∈ Z ℓ , let C γ denote the one-dimensional B-module of weight γ. We need the following result of Demazure [17, §5.5] (this assumes G is semisimple; see also [31,II.14.18 (a)] where reductive G are allowed). Theorem 3.1. For any weight γ ∈ Z ℓ and w ∈ S ℓ , Nonsymmetric Catalan functions appear naturally as graded Euler characteristics, extending a description of the Catalan functions in [73,15]: For any labeled root ideal (Ψ, γ, w), where S j u * Ψ denotes the j-th symmetric power of the B-module u * Ψ .
Proof. The series d (i,j)∈Ψ 1 − qx i /x j −1 x γ gives the character of j S j u * Ψ ⊗ C * γ where q keeps track of the grading. Each homogeneous component S j u * Ψ ⊗ C * γ has a B-module filtration into one-dimensional weight spaces. Then by the additivity of the Euler characteristic and Theorem 3.1, Applying poly • d to both sides, the right side becomes the nonsymmetric Catalan function H(Ψ; γ; w) from Definition 2.1 after using poly • π w = π w • poly (Proposition 5.5 (i)). For ν = (ν 1 ≥ · · · ≥ ν ℓ ) ∈ Z ℓ , let V (ν) be the irreducible G-module of highest weight ν. Let α ∈ Z ℓ and α + be the weakly decreasing rearrangement of α. The Demazure module D(α) ⊂ V (α + ) is the B-module B u α , where u α is an element of the (onedimensional) α-weight space of V (α + ). The Demazure atom moduleD(α) is the quotient of D(α) by the sum of all Demazure modules properly contained in D(α). The characters κ α (x) = char(D(α)) andκ α (x) = char(D(α)) are the key polynomial and Demazure atom, respectively which will be discussed further in §4.8 and §5.2.
As in [84, §2.3], say a B-module N has an excellent filtration (resp. relative Schubert filtration) if its dual N * has a B-module filtration whose subquotients are isomorphic to Demazure modules (resp. Demazure atom modules).
In this paragraph we discuss the w = w 0 (X w = G/B) case of Conjecture 3.4. First note that the cohomology groups are G-modules, so (iii)-(iv) hold and (ii) implies (i). Conjecture (ii) was posed by Chen-Haiman [15,Conjecture 5.4.3]; this generalized a conjecture of Broer for parabolic Ψ, which he settled in the dominant rectangle case [13,Theorem 2.2]. Hague [25,Theorems 4.15 and 4.23] extended this result to some other classes of weights (still parabolic Ψ). Panyushev proved that (ii) holds when the weight µ − ρ + (i,j)∈∆ + \Ψ ǫ i − ǫ j is weakly decreasing, where ρ = (ℓ − 1, ℓ − 2, . . . , 0). Frobenius splitting methods [43] give another proof of a subcase of Broer's result; this method has the advantage of applying to G over algebraically closed fields of prime characteristic.

Background on crystals
We begin by reviewing crystals for any symmetrizable Kac-Moody Lie algebra g and prove that restrictions of Demazure crystals are disjoint unions of Demazure crystals. We then fix notation and conventions for g = sl ℓ following Naoi [70] and Kac [33]; note that the notation I, P, P + , α i , α ∨ i is for general g in §4.1-4.2 and for sl ℓ from §4.3 through the remainder of the paper. 4.1. U q (g)-(seminormal) crystals. The quantized enveloping algebra U q (g) is specified by a Dynkin node set I, coweight lattice P * , weight lattice P = Hom Z (P * , Z), coroots {α ∨ i } i∈I ⊂ P * , roots {α i } i∈I ⊂ P , and a symmetric bilinear form (·, ·) : P ×P → Q subject to several conditions (see [37, §2.1]). This data given, a U q (g)-seminormal crystal is a set B equipped with a weight function wt : B → P and crystal operatorsẽ i ,f i : B⊔{0} → B⊔{0} (i ∈ I) such that for all i ∈ I and b ∈ B, there holdsẽ i (0) =f i (0) = 0 and This agrees with the notion of a seminormal crystal in [37, §7], the notion of a crystal in [70], and the notion of a P -weighted I-crystal in [79].
A strict embedding of U q (g)-seminormal crystals B, B ′ is an injective map Ψ : B ⊔{0} → B ′ ⊔ {0} such that Ψ(0) = 0 and Ψ commutes with wt, ε i , φ i ,ẽ i , andf i for all i ∈ I. It is necessarily an isomorphism from B onto a disjoint union of connected components of B ′ .
For U q (g)-seminormal crystals B 1 and B 2 , their tensor product and crystal operators (we use the convention opposite Kashiwara's) Assume for this paragraph that the roots and coroots are linearly independent. Let O int (g) denote the category whose objects are the U q (g)-modules isomorphic to a direct sum of integrable highest weight U q (g)-modules (see, e.g., [37, §2.4]). Any M in O int (g) has a unique local crystal basis (L, B) up to isomorphism [34], and extracting the associated combinatorial data yields a U q (g)-seminormal crystal (see [37, §4.2, §7.5]). We define a U q (g)-crystal to be a U q (g)-seminormal crystal arising in this way.
is the U q (g)-crystal arising from the local crystal basis of the irreducible highest weight module V (Λ) in O int (g). So with this notation, any U q (g)-crystal is a disjoint union of highest weight U q (g)-crystals by [34].
be the subalgebra generated by e i , f i , i ∈ J, and q h , h ∈P * ; it is a quantized enveloping algebra and its defining data includes J, {α ∨ i } i∈J ⊂P * , {z(α i )} i∈J ⊂P . It is straightforward to verify that for any M in O int (g), the local crystal basis (L, B) of M is also a local crystal basis of the U q (g J )-restriction of M and so is isomorphic to the direct sum of local crystal bases of highest weight U q (g J )-modules by [34] (see [37, §4.6] for a similar result). Moreover, the associated U q (g)-crystal B of (L, B) and U q (g J )-crystal B of Res Uq(g J ) (L, B) are related as follows:B is obtained from B by replacing its weight function with z • wt : B →P and taking only the crystal operatorsẽ i ,f i for i ∈ J. We sayB is the U q (g J )-restriction of B and denote it Res J B or similar-see §4.4.
The following crystal restriction theorem will be important for obtaining key positivity results. Its proof was communicated to us by Peter Littelmann, and we are also grateful to Wilberd van der Kallen who pointed us to his module-theoretic version [84, Theorem 6.3.1]. A more general module-theoretic version was recently given in [6, Appendix A]. Definition 2.4) and Res J S denotes the set S regarded as a subset of Res J B(Λ), which is isomorphic to a disjoint union of highest weight U q (g J )-crystals by the discussion above.
Remark 4.2. Let U q (g J ) ⊂ U q (g) be as above and assume J = I. Then a subset S of a U q (g)-crystal B is isomorphic to a disjoint union of U q (g)-Demazure crystals if and only if Res J S is isomorphic to a disjoint union of U q (g J )-Demazure crystals. This is immediate from the definitions since B and Res J B have the samef i -edges for all i ∈ J = I.
The convention (4.4) is implicit in [70] and ensures the extended affine Weyl group acts nicely on α i and Λ i , which will be important in §4.6. The {Λ i | i ∈ I} together with the null root δ = i∈I α i form a basis for h * ; note that α ∨ i , δ = 0 for i ∈ I and d, δ = 1. Let P = i∈I ZΛ i ⊕ Z δ 2ℓ ⊂ h * be the weight lattice and P + = i∈I Z ≥0 Λ i + Z δ 2ℓ the dominant weights. Let cl : h * → h * /Cδ be the canonical projection, and set P cl = cl(P ) = i∈I Z cl(Λ i ). Let aff : h * /Cδ → h * be the section of cl satisfying d, aff(λ) = 0 for all Let sl ℓ ⊂ sl ℓ be the simple Lie subalgebra with Dynkin nodes

Type A crystals.
Let U q ( sl ℓ ) be the quantized enveloping algebra specified by the data I, P * = Hom Z (P, Z), P, {α ∨ i } i∈I , {α i } i∈I above and the symmetric bilinear form (·, ·) : Zα ∨ i , and weight latticeP . Let U q (gl ℓ ) be as in [37, §5]; data includes Dynkin nodes [ℓ − 1], weight lattice Z ℓ , and roots be the subalgebra generated by e i , f i , i ∈ I, and q h , h ∈ P * cl = i∈I Zα ∨ i ; it can be considered a quantized enveloping algebra with data I, {α ∨ i } i∈I ⊂ P * cl , {cl(α i )} i∈I ⊂ P cl (it fits the form in [37, Definition 2.1]), but note that the roots are not linearly independent. For U ′ q ( sl ℓ ), we work with U ′ q ( sl ℓ )-seminormal crystals so that we can work with both KR crystals and restrictions of U q ( sl ℓ )-crystals and treat them uniformly, while for g = sl ℓ , gl ℓ , or sl ℓ we only need U q (g)-crystals.
We fix some notation for restricting crystals and specify the projection z of weight lattices (as in (4.2)) for each case.
, and z is the canonical projection P cl →P (this does not fit the form in §4.2 and it need not yield a U q (sl ℓ )crystal, but it does so for all U ′ q ( sl ℓ )-seminormal crystals considered in this paper). For a U q ( sl ℓ )-crystal B, its U ′ q ( sl ℓ )-restriction has the same edges as B and z is cl : P → P cl (it is easily verified that this always yields a U ′ q ( sl ℓ )-seminormal crystal).

4.5.
The affine symmetric group and 0-Hecke monoid. The extended affine symmetric group S ℓ is the group generated by τ and s i (i ∈ I) with relations Here, i, j denote arbitrary elements of I = Z/ℓZ. The affine symmetric group S ℓ is the subgroup of S ℓ generated by the s i for i ∈ I, and the symmetric group S ℓ is the subgroup generated by Following the conventions of [70], S ℓ is also naturally realized as a subgroup of GL(h * ): The 0-Hecke monoid H ℓ of S ℓ is the monoid generated by τ and s i (i ∈ I) with relations (4.6)-(4.9) (with s i 's in place of s i 's) together with for i ∈ I. The 0-Hecke monoid H ℓ of S ℓ is the submonoid of H ℓ generated by The length of w ∈ S ℓ , denoted length(w), is the minimum m such that w = s i 1 s i 2 · · · s im for some i j ∈ I. For w ∈ S ℓ , we can write w = τ i v, v ∈ S ℓ ; define length(w) = length(v). An expression for w ∈ S ℓ as a product of τ 's and s i 's is reduced if it uses length(w) s i 's. Length and reduced expressions for elements of H ℓ are defined similarly. 4.6. Dynkin diagram automorphisms and crystals. Any σ ∈ Σ, viewed as an element of GL(h * ), satisfies σ(P ) = P and since σ(δ) = δ, it also yields an element of GL(h * /Cδ) which satisfies σ(P cl ) = P cl ; hence σ yields automorphisms of P and P cl . For , which follows from σ(α i ) = α σ(i) and the uniqueness of local crystal bases of highest weight modules [34].
It is easily verified that if θ 1 : Thus the tensor product of maps is the natural choice of σ-twist from any tensor product B(Λ 1 ) ⊗ · · · ⊗ B(Λ p ) of highest weight U q ( sl ℓ )-crystals, Λ 1 , . . . , Λ p ∈ P + . We let F τ denote the operator on D( sl ℓ ) (see This agrees with and explains the definition of F τ in §2.2. Similarly, there is a unique τ -twist of Recall that for a subset S of a seminormal crystal B and i ∈ I, Proposition 4.3. The operators F i (i ∈ I) and F τ take U q ( sl ℓ )-Demazure crystals to U q ( sl ℓ )-Demazure crystals. Hence they can be regarded as operators on D( sl ℓ ) and as such they satisfy the 0-Hecke relations (4.6)-(4.10) of H ℓ .
Thus for any w ∈ H ℓ , we can define F w : where w = c 1 · · · c m with each c j ∈ {s i | i ∈ I}⊔{τ } and F s i := F i , and this is independent of the chosen expression for w. Recall that for Λ ∈ P + and w ∈ H ℓ , B w (Λ) : for any Λ ∈ P + and w, w ′ ∈ H ℓ . 4.8. U q (gl ℓ )-Demazure crystals and key polynomials. The symmetric group S ℓ acts on Z ℓ by permuting coordinates. It is also convenient to define an action of H ℓ on Z ℓ by Let B gl (ν) denote the highest weight U q (gl ℓ )-crystal and u ν its highest weight element, parameterized by ν ∈ {λ ∈ Z ℓ | λ 1 ≥ · · · ≥ λ ℓ }, the dominant integral weights for U q (gl ℓ ). Definition 2.4 defines U q (gl ℓ )-Demazure crystals but let us make this more explicit. They are indexed by elements of Z ℓ . Let α ∈ Z ℓ . Denote by α + the weakly decreasing rearrangement of α and p(α) ∈ H ℓ the shortest element such that Remark 4.4. Analogous results to §4.7 hold for U q (gl ℓ )-Demazure crystals. In particular, the F i (i ∈ [ℓ − 1]) can be regarded as operators on the set of U q (gl ℓ )-Demazure crystals and as such satisfy the 0-Hecke relations (4.6), (4.7), (4.10) of H ℓ .
Consider the group ring of the gl ℓ -weight lattice , but we will also regard them as operators on ( for a ground ring A, given by the same formula. They satisfy the 0-Hecke relations (4.6), (4.7), (4.10) of H ℓ (see e.g. [74]). Thus, just as we discussed for F w in §4.7, π w makes sense for any w ∈ H ℓ and π w π w ′ = π ww ′ for all w, w ′ ∈ H ℓ . Definition 4.5. For α ∈ Z ℓ , define the key polynomial or Demazure character by (4.12) If α ∈ Z ℓ is weakly decreasing, then κ α is simply the monomial x α , while if α is weakly increasing, then κ α is the Schur function s α + (x) = s α + (x 1 , x 2 , . . . , x ℓ ).
We record several facts about key polynomials for later use. First, it follows from π s i π w ′ = π s i w ′ for all w ′ ∈ H ℓ , that where s i α is as in (4.11). (4.14) It is immediate from Definition 4.5 and (4.14) that Proof. The first holds by [74,Corollary 7], and the second then follows from (4.15).
The character of a subset S of a U q (gl ℓ )-crystal is char Proposition 4.8. The characters of U q (gl ℓ )-Demazure crystals are key polynomials: for any α ∈ Z ℓ , Proof. This is a consequence of [35]. Note that the setup of [35] encompasses the gl ℓ case with weight lattice Z ℓ (see [37, §5]), and the Demazure operators defined therein match the π i in the definition of key polynomials.

The rotation theorem for tame nonsymmetric Catalan functions
We give the proof of the rotation Theorem 2.3, which requires Demazure operator identities and an in-depth study of polynomial truncation. Interestingly, the expression it gives for tame nonsymmetric Catalan functions is automatically polynomially truncated, whereas we had to explicitly add the truncation in our definition of these functions.
We extend this in the natural way to a linear operator on

Root expansion.
A straightforward yet surprisingly powerful recursion played an important role for the Catalan functions in [11]. This is easily generalized to the nonsymmetric setting. For a root ideal Ψ, we say α ∈ Ψ is a removable root of Proposition 5.2. Let (Ψ, γ, w) be a labeled root ideal. For any removable root α of Ψ, Proof. Apply the linear operator π w • poly to the following identity of series:

Polynomial truncation.
Polynomial truncation is better understood using the following symmetric bilinear form which comes from Macdonald theory and was given a self-contained treatment by Fu and Lascoux [21].
Proof. Statement (i) is immediate from the definition of polynomial truncation and (4.13).
. , x ℓ ] (Proposition 4.6). Since poly acts as the identity on the latter basis by definition, (ii) follows.
To prove (iii), by (5.3), it suffices to show that for any term cx ζ in the monomial Indeed, for such a term we must have ℓ a=k ζ a ≤ ℓ a=k γ a < 0 as needed.

Identities for Demazure operators and polynomial truncation. Recall from (2.4) that Φ is the operator on Z[q][x]
given by Φ(f ) = f (x 2 , . . . , x ℓ , qx 1 ); here we will regard it as an operator on Z[q, Thus, recalling that τ s i τ −1 = s i+1 , we have Proof. This is a direct computation from the definition of the Demazure operator π i : and a ≥ 0, poly(x a 1 Φ(f )) = x a 1 Φ(poly(f )). Proof. Since poly and Φ are linear operators, it is enough to prove this for f ranging over the Z-basis . In light of Remark 4.7, computing poly(κ ζ ) is nontrivial as we have defined polynomial truncation with respect to the basis . However, we can use Demazure operators: write κ ζ = π v x (µ,0) with µ = ζ + ∈ Z ℓ−1 and v = p(ζ) ∈ H ℓ−1 as in Definition 4.5 but for ℓ − 1 in place of ℓ. Then x a 1 Φ(poly(π v x (µ,0) )) = π τ vτ −1 x a 1 Φ(poly(x (µ,0) )) = where the first equality is by Propositions 5.5 (i) and 5.7 and then (4.14); the second equality uses Proposition 5.5 (ii) for the top line and Proposition 5.5 (iii) for the bottom line (µ weakly decreasing implies µ ℓ−1 < 0 if µ / ∈ Z ℓ−1 ≥0 ). On the other hand, there holds The justification is just as in the previous paragraph (the last equality uses a ≥ 0).
Lascoux [55, §4.1] gives a partial description of a Monk's rule for key polynomials, i.e. x i κ α expanded in key polynomials. The computations therein are similar to the next three lemmas, which we need for polynomial part computations. Recall thatπ i = π i − 1.
. Proof. Write κ α = π v x µ with µ = α + and v = p(α) as in Definition 4.5. The proof is by induction on length(v). For the base case v = id, let z be the index such that Choose a length additive factorization v = s j u. Using (5.5) and (5.6) we obtain By the inductive hypothesis, x −1 i−1 π u x µ , x −1 i π u x µ , and x −1 i+1 π u x µ belong to Z κ β | β + = µ − ǫ j for some j ∈ [ℓ] . Hence the result follows from (4.13).

5.4.
Proof of Theorem 2.3. The next theorem shows how to express a tame nonsymmetric Catalan function H(Ψ; γ; w a+1 ) in terms of a smaller one H(R(Ψ); R(γ); w a ) by peeling off its first row, which we can then iterate to unravel it one row at a time and obtain the desired expression involving π i 's and Φ's.

DARK crystals and katabolism
We show that for any DARK crystal B µ;w , katabolism is exactly the condition on Tabloids ℓ (µ) which detects membership in inv(B µ;w ). A connection between KR crystals and Catalan functions in the dominant rectangle case has been well established (see Remark 2.19). One of our key insights is that to go beyond this case, DARK crystals are needed rather than full tensor products of KR crystals. 6.1. Single row Kirillov-Reshetikhin crystals. We will only need an explicit description of the KR crystals B 1,s in type A. For any positive integer s, the U ′ q ( sl ℓ )-seminormal crystal B 1,s consists of all weakly increasing words of length s in the alphabet [ℓ], with weight function wt : B 1,s → P cl given by We also define B 1,0 = {b 0 } to be the trivial U ′ q ( sl ℓ )-seminormal crystal, i.e., wt(b 0 ) = 0 andẽ i (b 0 ) =f i (b 0 ) = 0 for all i ∈ I, and view b 0 as the empty word. 6.2. Products of KR crystals. We now describe in detail the crystals B µ which were briefly introduced in §2.3. Recall that for a partition µ = (µ 1 ≥ · · · ≥ µ p ≥ 0), we let B µ = B 1,µp ⊗ · · · ⊗ B 1,µ 1 , a U ′ q ( sl ℓ )-seminormal crystal. We identify its elements with the biwords whose bottom word has letters in [ℓ] and whose top word is p µp · · · 2 µ 2 1 µ 1 (see Example 6.6); we use a biword b interchangeably with its bottom word when the crystal B µ it belongs to is clear.

Remark 6.2.
We can also regard B µ as a U q (gl ℓ )-crystal (temporarily denote it B µ gl ) with weight function B µ gl → Z ℓ , b → content(b) and the same edges as Res sl ℓ B µ (the restriction from U ′ q ( sl ℓ ) to U q (sl ℓ )); moreover, Res sl ℓ B µ gl = Res sl ℓ B µ by (6.1). From now on we write B µ for both the U ′ q ( sl ℓ )-seminormal and U q (gl ℓ )-crystal, and will clarify when necessary.
The crystal operatorsẽ i andf i on B µ are determined by the above description ofẽ i and f i on B 1,s and the tensor product rule (4.1)-(4.2). For i ∈ [ℓ − 1], they have the following streamlined description. Let b ∈ B µ . Place a left parenthesis "(" below each letter i + 1 in b and a right parenthesis ")" below each letter i. Match parentheses in the usual way. The unmatched parentheses correspond to a subword consisting of i's followed by i + 1's. 6.3. RSK and crystals. We review the beautiful connection between U q (gl ℓ )-crystals and classical tableau combinatorics, which may be attributed to Kashiwara-Nakashima [38], and Lascoux-Schützenberger [57] who anticipated much of the combinatorics before the development of crystals. Other good references include [80] and [28,Chapter 7].
The crystals B µ are compatible with the following variant of the Robinson-Schensted-Knuth correspondence described in [ from left to right or by column inserting each letter from right to left. The recording tableau Q(b) of b is obtained by column inserting the bottom word of b from right to left and recording each newly added box with the corresponding top letter. More precisely, Q(b) is the tableau with the same shape as P (b) such that the skew shape shape(P (b i b i−1 · · · b 1 ))/shape(P (b i−1 · · · b 1 )) is filled with i's for all i.
Recall from §2.4 that SSYT ℓ (µ) denotes the subset of Tabloids ℓ (µ) consisting of tabloids with partition shape whose columns strictly increase from top to bottom. (This is the set of semistandard Young tableaux of content µ with at most ℓ rows, but with the fine print that we regard them as having ℓ rows some of which may be empty.) Theorem 6.4 (see [80,Theorem 3.6]). The decomposition of the U q (gl ℓ )-crystal B µ into highest weight U q (gl ℓ )-crystals is given by Here, B gl (ν) denotes the highest weight U q (gl ℓ )-crystal of highest weight ν.
6.4. The inv bijection and RSK. A biword can be thought of as a sequence of biletters ( v 1 w 1 )( v 2 w 2 ) · · · ( vm wm ) which is weakly decreasing for the order ( v w ) ≥ ( v ′ w ′) if and only if v > v ′ or (v = v ′ and w ≤ w ′ ). Then, for a biword b, define inv(b) to be the result of exchanging the top and bottom words of b and then sorting biletters to be weakly decreasing.
It is natural to regard inv as an involution on the set of biwords. However, as discussed in Remark 6.7 below, we prefer to think of inv as a bijection between biwords and tabloids, which we can do since biwords and tabloids may be naturally identified by equating blocks with rows (see the right side of (6.3)). Since the contents of the top and bottom words are exchanged by inv, it restricts to a bijection inv : B µ ∼ = ← → Tabloids ℓ (µ), which takes content to shape (we gave a direct description of the map B µ → Tabloids ℓ (µ) in §2.3). . The insertion (P ) and recording (Q) tableaux are exchanged by inv. In particular, for a biword b ∈ B µ , Q(b) = P (inv(b)) and for a tabloid T ∈ Tabloids ℓ (µ), P (T ) = Q(inv(T )). . Remark 6.7. Though it is possible to define a two-sided crystal structure on biwords in which crystal operators act on both a biword and its inverse, this is not the perspective we take here. Instead, we break the symmetry between the two sides by adopting the following conventions: crystal operators act only on the B µ side and not the Tabloids ℓ (µ) side; we are mainly interested in Q(b), not P (b), for b ∈ B µ , and P (T ), not Q(T ), for T ∈ Tabloids ℓ (µ) as these are the ones which identify inv of the highest weight element of a U q (gl ℓ )-component. Further, elements of B µ will be written as biwords and never tabloids; their inverses will be written as tabloids, though occasionally thought of as biwords for the purposes of computing inv.

Partial insertion andẽ max
i . In the remainder of Section 6, we match operations on the tabloids side with ones on the crystal side. The material in this subsection is similar to [81, §3.5], [54, §2] and perhaps can be considered folklore.
For an element b of a U q (gl ℓ )-crystal, definẽ i.e., the last element in the list b,ẽ i (b),ẽ 2 i (b), . . . which is not 0. For example, in the crystal B 432 ,ẽ max 1 (12 122 1222) = 12 112 1111. More generally, for w ∈ H ℓ , let w = s i 1 · · · s im be any expression for w as a product of s j 's; defineẽ max w =ẽ max i 1 · · ·ẽ max im ; by Proposition 6.11 (ii) below, this is independent of the chosen expression for w.
Recall that T i denotes the i-th row of a tabloid T .
Definition 6.8 (Partial insertion). Given a tabloid T , P i (T ) is the tabloid obtained from T by replacing rows i and i + 1 of T by the tableau P (T i+1 T i ) (if P (T i+1 T i ) has only one row, then the i + 1-st row of P i (T ) is empty). More generally, for w = s i 1 · · · s im ∈ H ℓ , define P w = P i 1 · · · P im ; by Proposition 6.11 (iii) below, this is independent of the chosen expression for w. For how this is related to Definition 2.13, see Remark 6.16.
For example, P 2 .
The following commutative diagrams give a summary of §6.4-6.5 (the left holds by Proposition 6.9 and the right by Propositions 6.5, 6.9, and 6.11 (iv)). Proof. Set T = inv(b). Recall from §6.2 thatẽ max i (b) is obtained by viewing i + 1's and i's as left and right parentheses and then changing all unmatched i + 1's to i's. We claim that inv(P i (inv(b))), computed using the row bumping algorithm, is obtained by the same rule except with the following greedy parentheses matching in place of the ordinary one: read ")"s from right to left and match each with the rightmost unmatched "(". To see this, first note that the letters in top(b) above the i's (resp. i + 1's) in bottom(b) are the values of T i (resp. T i+1 ). The row bumping algorithm computes P i (T ) by processing the letters of T i from left to right; each letter x of T i bumps the smallest entry of T i+1 greater than x not already bumped (if it exists). Each bump corresponds to a greedy-matched pair in b and the unmatched i + 1's of b correspond to the entries of T i+1 not bumped, which are exactly the ones that move from T i+1 to (P i (T )) i in computing P i (T ).
It remains to show that, given a string w 1 · · · w m in the letters "(" and ")", the ordinary and greedy matching rules produce the same unmatched "("s. We proceed by induction on m. Consider the subword w i · · · w m where w i is the rightmost matched "("; it must look like ()) · · · )(· · · (. Let (w i , w j ) (resp. (w i , w i+1 )) be the greedy (resp. ordinary) matched pair in this subword. Though these pairs typically differ, deleting the greedy-matched pair yields the same string as deleting the ordinary matched pair. Since the position of the "(" in both pairs is the same, the result follows by the inductive hypothesis. Proposition 6.10. Let b ∈ B µ and set T = inv(b) ∈ Tabloids ℓ (µ). Then b is a U q (gl ℓ )highest weight element if and only if any of the following equivalent conditions holds: Proof. Condition (a) is the definition of b being a U q (gl ℓ )-highest weight element. The equivalence (a) ⇐⇒ (b) is by Proposition 6.9, and (b) ⇐⇒ (c) is clear from computing P (T i+1 T i ) by column insertion.
The streamlined version of katabolism from Definition 2.15 agrees with this one in the setting of Theorem 2.18, as we now verify. Proof. We first verify the following claim: for any tabloid T such that its subtabloid T [i,ℓ−1] is a tableau, P i · · · P ℓ−1 (T ) can be obtained by column inserting T ℓ into T [i,ℓ−1] , i.e., P i · · · P ℓ−1 (T ) = P i,ℓ (T ) in the notation of Definition 2.13. To ease notation, assume i = 1, as this easily implies the general case. We have P 1,ℓ (T ) = P (T ), the unique tableau with reading word Knuth equivalent to that of T . Then by Proposition 6.11 (iv), P 1,ℓ (T ) = P (T ) = P w 0 (T ) = P 1 · · · P ℓ−1 P w [1,ℓ−1) (T ) = P 1 · · · P ℓ−1 (T ), where the last equality uses that T [ℓ−1] is a tableau.
Proof. We must show that for any T ∈ Tabloids ℓ (µ), T is w-katabolizable if and only if inv(T ) ∈ B µ;w . We prove this by induction on p + i length(w i ). The base case p = 1, w 1 = id is clear. Now suppose w 1 = id. We can write B µ;w = F w 1 (B µ;(id,w 2 ,...,wp) ) and then inv(T ) ∈ B µ;w ⇐⇒ẽ max where the second equivalence uses Proposition 6.9 and the inductive hypothesis.

Schur and key positivity
We connect charge to sl ℓ -weights and then combine the results of Section 6 with Corollary 2.7 and Theorem 2.11 to give several character formulas for DARK and AGD crystals; this yields our katabolism formula Theorem 2.18 upon combining with the rotation theorem. Stronger key positivity results are then obtained via the restriction Theorem 4.1.  Thus, just as we discussed for F w in §4.7, D w makes sense for any w ∈ H ℓ and D w D w ′ = D ww ′ for all w, w ′ ∈ H ℓ . The character of a subset S of a U q ( sl ℓ )-crystal is char(S) := b∈S e wt(b) ∈ Z[P ]. Kashiwara [35] gave a Demazure operator formula for the character of any Demazure crystal, and Naoi extended this to encompass the action of Σ, as follows: ℓ ] by permuting the variables) and has kernel (x 1 · · · x ℓ − 1). It is an extension of the map ζ from (2.7) to a larger domain.
We wish to recover an element of A[x ±1 1 , . . . , x ±1 ℓ ] given its image under ζ, and this is possible if we know it to be homogenous of a given degree. Accordingly, let Let µ be a partition and set m = |µ|. Suppose G is a U q ( sl ℓ )-crystal such that e −µ 1 Λ 0 char(G) ∈ ζ(X m ) (by the proof of Theorem 7.5 below, this holds for G = AGD(µ; w), our main case of interest). We define the x-character of G by In other words, if we find f ∈ X m such that ζ(f ) = char(G)e −µ 1 Λ 0 , then f = char x;µ (G). We will need two facts which relate π i and D i , and Φ and τ via ζ. First, it is straightforward to show from the S ℓ -equivariance of ζ that Second, we claim that for any f ∈ X m , ζ(Φ(f )) = e −mδ/ℓ τ (ζ(f )). (7.4) Since ζ, τ, and Φ are ring homomorphisms, it is enough to prove ζ(Φ(x i )) = e −δ/ℓ τ (ζ(x i )). This is readily verified from the computation where m i := d, Λ i and the last equality is by (4.4).

7.2.
Charge and sl ℓ -weights. The pairing d, · on sl ℓ -weights gives a statistic on U q ( sl ℓ )crystal elements, which is not available for U ′ q ( sl ℓ )-seminormal crystals. Naoi [70] showed that the strict embedding Θ µ matches this statistic to energy, thereby effectively allowing the full information of sl ℓ -weights to be seen on the DARK side. Since energy on B µ matches charge on Tabloids ℓ (µ) = inv(B µ ) [69], the charge and d, · statistics agree. Remark 7.2. It is actually more natural to connect charge and d, · directly as they both essentially measure the number off 0 -edges required to construct the crystal element, whereas energy is a more complicated statistic. In the interest of space, we just give the idea: for b =f a 1 i 1 · · ·f a k i k u Λ ∈ F i 1 · · · F i k {u Λ } with i j ∈ I, −d, wt(b) − wt(u Λ ) is the number off 0 's appearing inf a 1 i 1 · · ·f a k i k . A similar statement can be made for AGD crystals. Charge also has a similar flavor sincef 0 -edges are related to property (C3) below by the inv map-see [79, §4.2].
Charge is a statistic on words of partition content which is commonly defined by a circular-reading procedure (see, e.g., [81, §3.6]). We prefer to take the following theorem of Lascoux and Schützenberger as its definition. (C4) Charge is constant on Knuth equivalence classes.
We will view charge as a statistic on tabloids by setting charge(T ) = charge(T ℓ · · · T 2 T 1 ) for any T ∈ Tabloids ℓ , where the concatenation T ℓ · · · T 2 T 1 is the row reading word of T .
We now prove (7.5). Set m i = d, Λ i for i ∈ I. Since Θ µ commutes with the P cl -valued weight functions, cl(wt Θ µ (b ⊗ u µ 1 Λ 0 )) = cl(µ 1 Λ 0 ) + wt(b). Thus (7.5) is equivalent to and C is a constant that depends on µ and ℓ but not b. Further, D(b) = charge(inv(b)) by [69] (see also [80,Proposition 4.25]). Hence, to pin down the constant, we need only verify (7.7) for a single b ∈ B µ . We choose b hw : . The third equality is by (4.4) and a direct computation of the charge of inv(b hw ), the tabloid of content µ with all letters i in row mod ℓ 1 (i).

7.3.
A Schur positive formula for Catalan functions: proof of Theorem 2.18.
Corollary 7.6. In the case w 1 = w 0 (the longest element in H ℓ ), the characters in Theorem 7.5 have the following Schur positive expansion: Proof. Combine Theorems 6.21 and 7.5, noting that each component C U of the U q (gl ℓ )crystal B µ;w contributes q charge(U ) times b∈C U x content(b) = b∈B gl (shape(U )) x wt(b) = s shape(U ) (x) to the left side of (7.9); this last (well-known) equality follows from Proposition 4.8.  [81] in the parabolic case: Proposition 7.7. When Ψ is the parabolic root ideal ∆(η) for some composition η of ℓ (see (2.12)), a tableau T of partition content µ is n(Ψ)-katabolizable if and only if it is R(η, µ)-katabolizable in the sense of [81, §3.7].

Combining
Proof. Checking whether T is n(Ψ)-katabolizable begins with the computation U = P 1,ℓ • kat · · · P η 1 −1,ℓ • kat •P η 1 ,ℓ • kat(T ). The key observation is that each row T 1 , T 2 , . . . , T η 1 of T is never touched by the column insertions until it is rotated to become the new ℓ-th row. Hence the computation of U amounts to the following: check whether T 1 contains µ 1 1's, remove these 1's, then column insert the result into T [η 1 +1,ℓ] to obtain a new tableau V , then check whether T 2 contains µ 2 2's, remove these 2's, column insert the result into V , and so on. These checks are equivalent to checking whether T contains the superstandard tableau Z of shape (µ 1 , . . . , µ η 1 ). Thus, T is not rejected in this computation if and only if T contains Z, and if so, U is obtained by column inserting T [η 1 ] \ Z into T [η 1 +1,ℓ] one row at a time, which is the same as the rectification of the skew tableau formed by placing T [η 1 ] \ Z and T [η 1 +1,ℓ] catty-corner. This is exactly the first step in the katabolism algorithm of [81]. Continuing in this way with η 2 , η 3 , . . . gives the result. 7.4. Key positivity. We generalize the results above to key positive formulas for characters of AGD and DARK crystals and tame nonsymmetric Catalan functions. To do this, we address the algorithmic problem of obtaining explicit key expansions for characters of subsets which we know to be disjoint unions of U q (gl ℓ )-Demazure crystals; some of this material, in particular Proposition 7.9, is similar in spirit to [5, §4].
Let B be a U q (gl ℓ )-crystal. The weight function takes values in Z ℓ and we write wt(b) = (wt 1 (b), . . . , wt ℓ (b)) for the entries of wt(b). The crystal reflection operators S i : B → B, i ∈ [ℓ − 1], are given by Note that s i (wt(b)) = wt(S i (b)). The operators S i were first studied by Lascoux and Schützenberger [57], and later generalized by Kashiwara [36]. They satisfy the braid relations and therefore generate an action of S ℓ on B. For 1 ≤ i < j ≤ ℓ, let s ij = s i s i+1 · · · s j−2 s j−1 s j−2 · · · s i ∈ S ℓ denote the transposition swapping i and j, and S ij = S i S i+1 · · · S j−2 S j−1 S j−2 · · · S i the corresponding reflection operator.
Proposition 7.8. The relation β > α is a covering relation in Bruhat order on Z ℓ if and only if there exist 1 ≤ i < k ≤ ℓ such that α = s i k β with α i > α k , and α j / Proof. For permutations α and β, this is a well-known combinatorial description of the Bruhat order covering relations of S ℓ (see, e.g., [7,Lemma 2.1.4]). The general case can be deduced from this one by a standardization argument and the fact that any covering relation in the Bruhat order poset restricted to minimal coset representatives is actually a covering relation in the full Bruhat poset (by, e.g., [7,Theorem 2.5.5]).
For example, β = 32812852 > 52812832 = α is a covering relation and α = s 1 7 β. The next proposition is motivated by the following algorithmic problem: suppose we have access to the elements of a U q (gl ℓ )-Demazure crystal G and want to determine the γ ∈ Z ℓ for which G = BD(γ) (see §4.8 for the definition of BD(γ)).
Proposition 7.9. Let G be a U q (gl ℓ )-Demazure crystal. There is a unique element u lw ∈ G such that, setting γ = wt(u lw ), (1) γ + is the highest weight of G, and (2) S ij (u lw ) / ∈ G for all covering relations γ < s ij γ in Bruhat order on Z ℓ . Moreover, G = BD(γ).
For a tabloid T ∈ Tabloids ℓ (µ) and i ∈ [ℓ − 1], define S ′ i := inv •S i • inv(T ) and S ′ ij = inv •S ij • inv(T ). In fact, we only need this action on the set of row frank tabloids: RowFrank ℓ (µ) := {T ∈ Tabloids ℓ (µ) | shape(T ) is a rearrangement of shape(P (T ))}, which is also the set of inverses of the extremal weight elements of the crystal B µ . Since shape(S ′ i (T )) = s i (shape(T )) for any T ∈ RowFrank ℓ (µ), the S ′ i preserve the set RowFrank ℓ (µ). This also gives a simple description of S ′ ij (T ) for T ∈ RowFrank ℓ (µ): S ′ ij (T ) is the unique row frank tabloid Knuth equivalent to T with shape obtained from shape(T ) by exchanging the i-th and j-th parts.
is not w-katabolizable for all i < j such that shape(T ) < s ij (shape(T )) is a covering relation in Bruhat order on Z ℓ .
Theorem 7.11. The DARK crystal B µ;w is isomorphic to a disjoint union of U q (gl ℓ )-Demazure crystals, with decomposition given by Proof. By Corollary 2.7 and Theorem 4.1, the U q (sl ℓ )-restriction of AGD(µ; w) is isomorphic to a disjoint union of U q (sl ℓ )-Demazure crystals. So the same is true of B µ;w ⊗ u µ 1 Λ 0 (by Theorem 2.11) and therefore B µ;w as well. Hence by Remark 4.2, B µ;w is isomorphic to a disjoint union of U q (gl ℓ )-Demazure crystals; this decomposition can be written as B µ;w = C U ∩ B µ;w , where C U ranges over the U q (gl ℓ )-components of F w 0 B µ;w (see Theorem 6.21). Then by Proposition 7.9 and Theorem 6.20, each set inv(C U ∩ B µ;w ) contains a unique T ∈ RowFrank ℓ (µ) which is extreme w-katabolizable, and C U ∩ B µ;w = {b ∈ B µ;w | Q(b) = U} =C T ∼ = BD(shape(T )).
On the left of Figure 4 are the inverses of the U q (gl ℓ )-highest weight elements of B µ;w obtained by computing P (T ) of the tabloids on the right. This is also the set of U ∈ SSYT ℓ (µ) which are (id, s(Ψ))-katabolizable (= n(Ψ)-katabolizable), providing an example of Theorem 2.18 and Corollary 7.6 as well: reading off their shapes and charges yields the following Schur positive expression for H(Ψ; µ; w 0 ) = b∈B µ;(w 0 ,s(Ψ)) q charge(inv(b)) x content(b) .
Let us check that the tabloid T =  We must also show that S ′ ij (T ) is not w-katabolizable for all covering relations shape(T ) < s ij (shape(T )). We have shape(T ) = 42011, and there are three covering relations corresponding to (i, j) = (1, 2), (2,3), and (2, 4). We show that the t = 0 specialized nonsymmetric Macdonald polynomials are characters of AGD crystals and equal to certain nonsymmetric Catalan functions. We thus obtain a key positive formula for these polynomials as a special case of Corollary 7.12.
(ii) (τ s(d)) d is a reduced expression for y ̟ d in S ℓ , and thus y ̟ d = (τ s(d)) d in H ℓ .
We will need the observation that affine generalized Demazure crystals AGD(µ; w) for constant µ are just affine Demazure crystals.
Proof. As µ i = 0 for i = m, the first statement is immediate from the definition of AGD(µ; w) in (2.6). The second follows from the fact thatf i u aΛ 0 = 0 for i ∈ [ℓ − 1].
Recall from (7.2) the definition of the x-character of a crystal. The next result is partially a restatement of Sanderson's theorem [76] (specifically,Ẽ α = q p(p−ℓ) 2ℓ char x;µ (B v (Λ 0 ))). However, we now have the advantage of seeing it as part of the more general Theorem 7.5 and can make it combinatorially explicit in a way which encompasses earlier work of Lascoux [53] and Shimozono-Weyman [81] on cocharge Kostka-Foulkes polynomials.
The proof is given in §8.3, along with a similar result for m > ℓ.
The companion result to Theorem 8.8 for m > ℓ is more technical and requires a crystal version of setting x ℓ+1 = · · · = x m = 0, which we now describe.
Let B be a U q (gl m )-crystal which is a isomorphic to a disjoint union of highest weight crystals B gl (ν) for ν = (ν 1 ≥ · · · ≥ ν m ≥ 0). The weight function takes values in Z m ≥0 and we write wt(b) = (wt 1 (b), . . . , wt m (b)) for the entries of wt(b). Let S ⊂ B be isomorphic to a disjoint union of U q (gl m )-Demazure crystals. Let Res J B denote the U q (gl ℓ )-restriction of B corresponding to Dynkin node subset J = [ℓ − 1] ⊂ [m − 1] (see §4.2). By Theorem 4.1 Res J S is isomorphic to a disjoint union of U q (gl ℓ )-Demazure crystals. Define Sincef j , j ∈ [ℓ − 1], fixes wt i for i > ℓ, R m ℓ S is also a disjoint union of U q (gl ℓ )-Demazure crystals and its character is obtained from that of S by setting x ℓ+1 = · · · = x m = 0.
Below we work with H m and its submonoid H ℓ generated by s 1 , . . . , s ℓ−1 (ℓ ≤ m); denote by ι : H ℓ ֒→ H m the inclusion and w [1,m) and w [1,ℓ) their longest elements.  Proof. Let s i 1 · · · s i k be a reduced word for w [1,m) . For any b ∈ B, m i=ℓ+1 wt i (f ℓ (b)) > m i=ℓ+1 wt i (b) and for j = ℓ, m i=ℓ+1 wt i (f j (b)) = m i=ℓ+1 wt i (b); also, m i=ℓ+1 wt i (b) = 0 impliesf j (b) = 0 for j > ℓ. It follows that an arbitrary elementf where v is the product of the s i j with i j < ℓ. Since s i 1 · · · s i k contains a reduced word for w [1,ℓ) , it follows from Remark 4.4 that F v R m ℓ S = F w [1,ℓ) R m ℓ S. Lemma 8.14. Given n = (n 1 , . . . , n p−1 ) ∈ [ℓ] p−1 and z ∈ H ℓ , define the tuples w = (z, s ℓ−1 · · · s n 1 , . . . , s ℓ−1 · · · s n p−1 ) ∈ (H ℓ ) p and w = (ι(z), s m−1 · · · s n 1 , . . . , s m−1 · · · s n p−1 ) ∈ (H m ) p . Then for any partition µ = (µ 1 , . . . , µ p ), R m ℓ B µ; w = B µ;w . Proof. By Theorem 6.20, this is equivalent to showing that T ∈ Tabloids ℓ is w-katabolizable if and only if T is w-katabolizable, where T is the same as T but regarded as an element of Tabloids m . One checks easily by induction that these two katabolism computations are essentially identical, the only difference being that whenever kat is applied in the w-katabolism algorithm, it matches the application of P s ℓ ···s m−1 • kat in the w-katabolism algorithm; this holds because at every step (in either algorithm) just before kat is applied, the input tabloid is empty in rows ℓ + 1, . . . , m.