K-theoretic crystals for set-valued tableaux of rectangular shapes

In earlier work with C.~Monical, we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials $L_{w\lambda}$ when $\lambda$ is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of Ross--Yong (2015) and Monical (2016) by constructing bijections with the respective combinatorial objects.


Introduction
In classical Schubert calculus, we can study the cohomology ring of the Grassmannian Gr(k, n), the parameter space for k-dimensional subspaces of C n , with respect to the basis given by the Poincaré duals of the Schubert varieties X λ that decompose Gr(k, n). In this context, the cohomology classes [X λ ] can be represented by Schur polynomials s λ , where the partition λ sits inside a k × (n − k) rectangle. A more modern approach is to study Gr(k, n) via connective K-theory, where the Schubert class [X λ ] is given as the push-forward of the class for any Bott-Samelson resolution of X λ . Here, polynomial representatives are given by symmetric (or stable) β-Grothendieck polynomials [FK94,Hud14].
The Schur polynomial s λ can be described combinatorially as a generating function for semistandard (Young) tableaux of shape λ (see, e.g., [Sta99,Ch. 7]). In addition, s λ has a representation-theoretic interpretation as the character of the highest weight representation V (λ) of the Lie algebra sl n of traceless n × n matrices (see, e.g., [Ful97,Ch. 8]). One way to compute s λ is by applying a product of Demazure operators π w0 corresponding to the reverse permutation w 0 to the monomial x λ := x λ1 1 · · · x λn n . Generalizing this formula refines Schur polynomials to the key polynomials κ wλ := π w x λ , which may be understood as characters of a Demazure modules V w (λ) [Dem74] (hence, κ wλ is also known as a Demazure character); geometrically, V m (λ) may be constructed as global sections of a line bundle on a flag variety [And85,LMS79].
One way to connect the combinatorial and representation-theoretic interpretations of key polynomials is through M. Kashiwara's theory of crystal bases for representations of quantum groups [Kas90,Kas91]. Indeed, Kashiwara showed that the Demazure module V w (λ) has a crystal basis and could be described as a subcrystal B w (λ), called a Demazure crystal, of the highest weight crystal B(λ) [Kas93,Lit95]. For U q (sl n ), the crystal B(λ) may be realized as the set of semistandard tableaux of shape λ, and the tableaux for the subcrystal B w (λ) are characterized by a combinatorial condition on their corresponding key tableaux [LS90].
In our previous paper with C. Monical [MPS21a], we initiated an analogous approach to Demazure crystals for Lascoux polynomials. We first gave a U q (sl n )crystal structure to the set of semistandard set-valued tableaux. Then we proposed an enriched crystal structure with the property that the Lascoux polynomials appear as the characters of our K-theoretic analogs of Demazure subcrystals. We coined this enriched structure a K-crystal. We established the existence of Kcrystals for single rows and columns, but we discovered that no such structure exists for general shapes. Nonetheless, we conjectured [MPS21a,Conj. 7.12] that K-crystals exist for all rectangular shapes. Our first main result is a proof of this conjecture. Our proof gives rise to a combinatorial formula for the class of Lascoux polynomials indexed by a weight in the Weyl group orbit of a multiple of a fundamental weight i.e., a rectangular shape partition). We then use this formula to establish the corresponding cases of the Ross-Yong-Kirillov and Monical conjectures. To our knowledge, these are the only proven combinatorial formulas for any class of Lascoux polynomials. 1 Let us remark on why our proposed K-crystal structure exists for a rectangular shape λ, but not for general shapes. In our work with C. Monical [MPS21a], we proposed a slightly weaker structure for general λ that depends on a choice of a reduced expression for w 0 . The key distinction appears to be that, in the rectangular case, the minimal-length coset representatives (such as the relevant parabolic w 0 ) that index Lascoux polynomials are all fully-commutative i.e., all reduced expressions differ only by commutations) [Ste96]. However, for more general shapes, such as λ = (2, 1) described in [MPS21a,Fig. 6,7], one needs to apply the braid relations s i s i+1 s i = s i+1 s i s i+1 to get all possible reduced expressions. Subsequently, we believe that, in general, K-crystal structures depend on choosing a commutation class of the reduced words for the appropriate parabolic w 0 (see 1 After this paper appeared as a preprint, subsequent work of V. Buciumas, the second author, and K. Weber [BSW20] gave another combinatorial formula for Lascoux polynomials. In particular, they proved the formula proposed here in our Conjecture 6.1. also [MPS21a,§7.3]). This fact seems related to an analogous dependence for Schubert classes in cohomology theories more general than connective K-theory (see, e.g., [BE90,GR13,LZ17]). Moreover, in the rectangular case, we have a flagging condition to characterize the tableaux in the K-Demazure crystal, and we expect an analogous key tableau condition to work for general shapes.
This paper is organized as follows. In Section 2, we recall the necessary background. In Section 3, we construct a K-crystal structure on set-valued tableaux of rectangular shapes. In Section 4 (resp. Section 5), we prove the conjectural combinatorial interpretation of Lascoux polynomials for rectangular shapes due to Ross-Yong-Kirillov (resp. Monical). In Section 6, we describe our conjecture for key tableaux of set-valued tableaux and their relationship with Lascoux polynomials.
Acknowledgements. OP is grateful for interesting conversations with Bob Proctor. TS would like to thank Takeshi Ikeda, Tomoo Matsumura, and Shogo Sugimoto for stimulating discussions. The authors thank Cara Monical for useful discussions. The authors thank the referees for their valuable comments and suggestions. In particular, we thank one of the referees for the bridging modification to the K-crystal operators. This work benefited from computations using Sage-Math [Sag19, SCc08].
2.1. Properties of symmetric groups. Let S n denote the symmetric group on {1, . . . , n} with simple transpositions {s i | 1 ≤ i < n}, where s i interchanges i and i + 1. Let w 0 ∈ S n be the reverse permutation n(n − 1) · · · 21. A reduced expression for a permutation w ∈ S n is an expression for w as a minimal-length product of simple transpositions. The length of a permutation is the length of any reduced expression for it; the element w 0 is the element with greatest length in S n . We recall that (strong) Bruhat order on S n is defined by v ≤ w if there exists a reduced expression for v that is a subword of a reduced expression for w.
Consider a partition λ (of length at most n) as a word of length n by appending 0's as necessary. Note that S n has a natural action on words of length n, which corresponds to the natural action on Z n of the Weyl group of sl n (which we can identify with the group of permutation matrices). Let Stab n (λ) = {w ∈ S n | wλ = λ} denote the stabilizer of λ. Recall that Stab n (λ) is a parabolic subgroup of S n and that every coset in the quotient of a Coxeter group by a parabolic subgroup has a unique minimal length representative. Thus, let S λ n denote the set of minimal length coset representatives of S n / Stab n (λ), and for any w ∈ S n , let ⌊w⌋ denote the corresponding minimal length coset representative of w in S n / Stab n (λ). For more on Coxeter groups, we refer the reader to, e.g., [BB05,Dav08,Hum90,Kan01].
2.2. Set-valued tableaux and their crystal structure. Let λ be a partition, which we often consider as a Young diagram. A (semistandard) set-valued tableau of shape λ is a filling T of the boxes of λ by finite nonempty sets of positive integers so that for every set A to the left of a set B in the same row, we have max A ≤ min B, and for C below A in the same column, we have max A < min C. (This is a setvalued generalization of the usual semistandard condition on tableaux.) For an integer a, we write a ∈ T if there exists a box of T containing a set A with a ∈ A. A semistandard set-valued tableau is a semistandard Young tableau if all sets have size 1. Let SV n (λ) denote the set of all set-valued tableaux of shape λ with entries at most n.
Next we recall the crystal structure on SV n (λ) from [MPS21a]. We first recall the crystal operators e i , f i : SV n (λ) → SV n (λ) ⊔ {0}, where i ∈ I := {1, . . . , n − 1}. We draw the crystals as a directed graph, where we have an i-colored edge T i − → U if and only if f i (T ) = U . For more details on crystals, we refer the reader to [BS17,Kas91].
The crystal operator f i acts on T ∈ SV n (λ) as follows: Write + above each column of T containing i but not i + 1, and write − above each column containing i + 1 but not i. Now cancel signs in ordered pairs −+. If every + thereby cancels, then f i T = 0. Otherwise let b correspond to the box of the rightmost uncanceled +. Then f i T is given by one of the following: • if there exists an adjacent box b → immediately to the right of b that contains an i, then remove the i from b → and add an i + 1 to b; • otherwise replace the i in b with an i + 1.
The action of e i is defined as follows: Construct the sequence + · · · +− · · · − as above. If there is not an uncanceled −, then e i T = 0. Otherwise let b correspond to the box of the leftmost uncanceled −. Then e i T is given by one of the following: • if there exists an adjacent box b ← immediately to the left of b that contains an i + 1, then remove the i + 1 from b ← and add an i to b; • otherwise replace the i + 1 in b with an i.
Identifying Z n with the multiplicative group generated by (x 1 , . . . , x n ), we define the weight function wt : SV n (λ) → Z n by wt(T ) = n i=1 x ci i , where c i is the number of A ∈ T such that i ∈ A. Define | wt(T )| = n i=1 c i . Let U q (sl n ) denote the Drinfel'd-Jimbo quantum group of the type A n−1 Lie algebra sl n , the Lie algebra of traceless n × n matrices over C. Let B(λ) be the highest weight U q (sl n )-crystal of all semistandard Young tableaux of shape λ [Kas90, Kas91, KN94].
Since ̟ w does not depend on the choice of reduced expression, we can define the Lascoux polynomials [Las01] as L a (x; β) := ̟ w x λ for any a ∈ Z n ≥0 , where λ is the sorting of a into a partition and w ∈ S λ n is the unique element such that wλ = a. The symmetric Grothendieck polynomial can be defined as the n variable truncation 2 of L w0λ (x; β) and is known [Buc02,Thm. 3.1] to be given combinatorially by The untruncated version is called a stable Grothendieck polynomial as it is the stable where A ranges over the entries of T . The statistic ex: SV n (λ) → Z is known as excess, and we call wt β (T ) the β-weight . There is currently no known geometric or representation-theoretic interpretation for general Lascoux polynomials. However, there are two conjectural combinatorial descriptions, which we now recall. The first conjectural combinatorial rule was introduced in [RY15]. To state it, we begin by recalling the notion of a K-Kohnert diagram to be a subset D of Z 2 >0 , which we realize as boxes, and a subset M ⊆ D of boxes that are marked. The conjectural rule to compute the Lascoux polynomial is as follows. Start with some a = (a 1 , . . . , a n ) ∈ Z n ≥0 and draw the initial K-Kohnert diagram as a skyline diagram by putting a box at each position {(i, y) | i ∈ [n], 1 ≤ y ≤ a i } (in Cartesian coordinates), marking no boxes. Then we successively apply any sequence of the following operations.  Example 2.3. Consider λ = 2 2 being a 2 × 2 rectangle and w = s 1 s 2 , so that a = (0, 2, 2). Then the set of K-Kohnert diagrams for a is Hence, Conjecture 2.2 (correctly) predicts that In the published version of [RY15], it is misstated that a Kohnert move could move the unmarked box through a marked box. See [Ros11,RY17].
The second conjectural combinatorial rule is from [Mon16]. We fill a skyline diagram with finite nonempty sets of positive integers that satisfy the following conditions. Call the largest entry in a box the anchor and the other entries free.
left column weakly taller right column strictly taller the anchors a, b, c of A, B, C, respectively, must satisfy either c < a or b < c. 4 (S.4) Every free entry is in the leftmost cell of its row such that the entry remains free and 2 is not violated. (S.5) Anchors in the bottom row equal their column index. We call such a tableau a (semistandard) set-valued skyline tableau. For a a weak composition (e.g., a finite string of nonnegative integers), let SLT a denote the set of set-valued skyline tableaux with shape a. We define the weight, excess, and βweight for a set-valued skyline tableau in the same way as for a set-valued tableau. Let Conjecture 2.4 ([Mon16, Conj. 5.2]). We have where the sum is taken over all permutations v less than or equal to w in Bruhat order. Thus, if Conjecture 2.4 holds, then we have another combinatorial interpretation of the Lascoux polynomial as By [MPS21b], this interpretation of Lascoux polynomials is equivalent to [Mon16, Conj. 5.3] .
2.4. K-crystals. We recall a proposed K-theory analog of crystals that was introduced in [MPS21a]. For a nilpotent operator ψ, we write that satisfy the following properties: (K.1) The set B is generated by a unique element u ∈ B that satisfies e i u = 0 and e K i u = 0 for all i ∈ I. That is to say, we can reach every element in B by applying a sequence of (K-)crystal operators from u. The element u is called the minimal highest weight element . (K.2) Let w = s i1 · · · s i ℓ ∈ S n be a reduced expression. The K-Demazure crystal 3) Let λ = wt(u) be the weight of the minimal highest weight element from 1. The is equal to the Lascoux polynomial L wλ (x; β).
A K-crystal is just a disjoint union of connected K-crystals. It is also strongly desirable for these operators to also satisfy e . Remark 2.5. This definition of a K-crystal is slightly more general than that given in [MPS21a], which was directly based on the combinatorics of set-valued tableaux. In particular, the extra condition that B w0 = B in 2 is implicit in the K-crystal definition given in [MPS21a] In [MPS21a], such K-crystals were constructed for the cases that λ is a singlerow [MPS21a, Thm. 7.5] or single-column [MPS21a, Thm. 7.9].
It was shown in [MPS21a] that there is no K-crystal (as defined here) solving Open Problem 2.6. Rather, in general, we believe that 2 should be relaxed, so that the K-crystal operators can depend on a choice of reduced word, giving a structure that we coined a weak K-crystal. Our main result is to prove [MPS21a, Conj. 7.12], thus giving an answer to Open Problem 2.6 for rectangular shapes, extending the single row and column results of [MPS21a]. In this rectangular case, the relevant Weyl group elements S λ n are all fully-commutative, so the expected dependence on reduced word does not appear.

K-crystals for rectangular shapes
In this section, we prove our main result: when λ is a rectangle, then SV n (λ) has a K-crystal structure. Thereby, we establish [MPS21a, Conj. 7.12], providing a solution to Open Problem 2.6 for rectangular shapes.
Our construction of the K-crystal operators is motivated by the heuristics given in [MPS21a], which come from the following K-theory analog of the decomposition of a crystal into i-strings i.e., restricting to the action of e i and f i for a fixed i ∈ I) based on the definition of the Demazure-Lusztig operators. Indeed, by considering only the action of a fixed i ∈ I, we predicted in [MPS21a] that the K-crystal should decompose into (maximal) subcrystals of the form where the solid (resp. dashed) arrow represents the f i (resp. f K i ) action and the top i-string has length one more than the bottom i-string. Such a subcrystal was coined an i-K-string in [MPS21a]. We say an i-K-string has length For the remainder of this section, we consider λ = s r to be an r × s rectangle.
Lemma 3.1. Let λ = s r be an r × s rectangle. For any w ∈ S λ n , there exists a reduced expression of w of the form for some −1 ≤ k < r and 1 ≤ i k < · · · < i 1 < i 0 ≤ n. (The case k = −1 means there are no terms in the product, so w = 1.) Example 3.2. To clarify our notation and symmetric group conventions, consider w = 2357146, so n = 7 and r = 4. Then the reduced expression for w in the form given by Lemma 3.1 is w = (s 1 )(s 2 )(s 4 s 3 )(s 6 s 5 s 4 ).
In particular, we have k = 3, i 0 = 6, i 1 = 4, i 2 = 2, and i 3 = i k = 1. For examples of these operators, see Figure 2; additional examples may be found in [MPS21a]. It is clear that if f K i T = 0 (resp. e K i T = 0), then f K i T ∈ SV n (λ) (resp. e K i T ∈ SV n (λ)). We give an example to illustrate the bridging condition.
Example 3.4. We apply the K-crystal operator e K 4 , which acts on the left box containing {4, 5}, which is unbridged, as the right box containing {4, 5} is bridged: 1 1 1 1 1 4 2 2 2 2 4,5 9 3 3 3 5 9 10 4 4 4,5 9 10 11 5 7 9 10 11 12 From our assumption, we have e i T ′ = 0 and for the rightmost unbridged box b with i, i + 1 ∈ b (note b exists by our assumption), there does not exist an i to the right of b that is either an uncanceled + or one paired with an i+1 to the left of b. Therefore, when we remove the i + 1 from b to obtain T , we create an uncanceled + for b. So e i T = 0 and f i T = 0. Furthermore, this added uncanceled + is the rightmost such uncanceled + and there are no other unbridged boxes to the right of b that contain i, i + 1 in T . Hence the action of f K i on T adds i + 1 to b, and thus From our assumption, we have e i T = 0 and there does not exist an unbridged box of T containing both i and i + 1 to the right of the rightmost box b corresponding to a uncanceled +. Since the + in b was uncanceled, removing it to form T ′ does not affect any other cancelations. Hence, it follows that e i T ′ = 0 as well since e i T = 0. The effect of adding an i + 1 to b cancels the + corresponding to b. Since b corresponded to the rightmost uncanceled +, there is no uncanceled + to the right of b in T . Moreover, there is not an i to the right of b that pairs with an i + 1 to the left of b as otherwise such an i + 1 would pair with the i in b. Hence e K i acts on T ′ by removing i + 1 from b, and thus e K i T ′ = T .
Lemma 3.6. Let λ = s r be an r × s rectangle. The restriction to any fixed i ∈ I decomposes SV n (λ) into i-K-strings.
Proof. Let T ∈ SV n (λ). Since f K i T = 0 and e K i T = 0 whenever e i T = 0, we cannot have the local situations around T be such an unbridged box, and since b contains the rightmost uncanceled +, there are no uncanceled + to the right of b ′ . Hence, we have e K i T = 0. Now we assume f K i T = 0. We have f ℓ i T = 0 if and only if f ℓ−1 i f K i T = 0 from the fact that to obtain f K i T , we removed the rightmost uncanceled + in T from b, which leaves the other uncanceled + and − unchanged. Consequently, we have Consider some w ∈ S λ n , and let i k < · · · < i 0 be from the reduced expression of w given by Lemma 3.1. For all k < j < r, define i j = r − j − 1. Define F (λ; w) to be the subset of SV n (λ) such that row r − j has all entries at most i j + 1. Equivalently, the entries in row j are at most w(j) for all 1 ≤ j ≤ r. We call such a set-valued tableau a flagged set-valued tableau. Diagrammatically, the flagging in each row is given by the labels on the right * * · · · * 1 . . . . . . . . . . . . . . . * * · · · * r − k − 1 * * · · · * i k + 1 . . . . . . . . . . . . . . . * * · · · * i 0 + 1 Example 3.7. Suppose w = s 2 and λ = 2×2. Then the flagged tableaux in F (λ; w) are those set-valued tableaux with shape 2 × 2 satisfying the flagging condition that entries of the first row are bounded by 1 and entries of the second row are bounded by 3. These bounds can be seen from the fact that The tableaux of F (λ; w) are precisely the shaded tableaux illustrated in Figure 2.
As the next lemma indicates, the flagging conditions characterize K-Demazure crystals. Recall that SV n w (λ) denotes the K-Demazure crystal of SV n (λ) corresponding to w ∈ S n .
Lemma 3.8. Let λ = s r be an r × s rectangle. For w ∈ S n , we have SV n w (λ) = SV n ⌊w⌋ (λ) = F (λ; ⌊w⌋). Proof. Let u be the minimal highest weight tableau of shape λ. That SV n w (λ) = SV n ⌊w⌋ (λ) is immediate from the fact that we have f i u = 0 for all i ∈ I \ {r}, i.e., whenever s i ∈ Stab n (λ). Hence, for the remainder of the proof we assume w = ⌊w⌋.
It remains to check that the tableaux U ∈ SV n w (λ) \ SV n w ′ (λ) differ from the elements of SV n w ′ (λ) exactly by the appropriate change in the flagging condition on row r −k. This check is identical to the proof of [MPS21a, Lemma 7.4] (which is the r = 1 case of the current lemma), except for being notationally more cumbersome.
It is a straightforward induction on the length i k − (r − k) of the leftmost factor of the reduced expression (3.1).
Theorem 3.9. Let λ = s r be an r × s rectangle. Then SV n (λ) is a K-crystal.
Proof. (K.1) is immediately from Lemma 3.8. (K.3) follows from Lemma 3.8 and that the properties of the Demazure-Lascoux operators imply ̟ w x λ = ̟ ⌊w⌋ x λ . To show (K.2), note that Lemma 3.8 implies SV n w (λ) only depends on the minimal length coset representative. By [Ste96,Thm. 6.1], every minimal length coset representative of S λ n is fully-commutative (see also [Ste96,Prop. 2.4]); in other words, they differ only by the commutation relations s i s j = s j s i for |i − j| > 1. It is clear that the (K-)crystal operators f i , f K i commute with f j , f K j for |i−j| > 1, and hence the K-Demazure crystal is independent of the choice of reduced expression.
We also have the following K-theoretic analog of [Kas93,Prop. 3

.3.4].
Corollary 3.10. Let λ = s r be an r × s rectangle. Consider an i-K-string S of SV n (λ), and let b be the highest weight element of S. Then, the set SV n w (λ) ∩ S is either empty, S, or {b}.
Proof. This follows immediately from Lemma 3.8, the semistandardness, and that the flagging on the rows is strictly increasing.
We also have the following interpretation of certain Lascoux polynomials as instances of (nonsymmetric) Grothendieck polynomials, indexed by some w ∈ S n . Recall from [Las90,LS82,LS83,FK94] that the (nonsymmetric) (β-)Grothendieck polynomial is defined by where s i1 · · · s i ℓ is a reduced expression.
Corollary 3.11. Let λ = s r be an r × s rectangle. Let w = (s k · · · s 2 s 1 )(s k+1 · · · s 3 s 2 ) · · · (s k+r−1 · · · s r+1 s r ) for some k ≥ 1, and let w = s m−1 (s m−2 s m−1 ) · · · (s r+1 · · · s m−1 )(s r · · · s k−1 ) · · · (s 1 · · · s k−1 ) ∈ S m where m = s + k + 1. Then, we have Proof. Theorem 3.9 shows that L wλ (x; β) is the character of the K-Demazure crystal and Lemma 3.8 shows that this character is a generating function for a class of flagged set-valued tableaux. It is clear that the permutations w 0 w −1 appearing in Corollary 3.11 are vexillary i.e. 2143-avoiding). Since the greatest term of L wλ (x; 0) in reverse lexicographic order is x wλ and the greatest term of G w0 w −1 (x; 0) in the same order is the Lehmer code of w 0 w −1 , we find that wλ is the Lehmer code of w 0 w −1 . Hence, the permutations w 0 w −1 are Grassmannian, and so the Grothendieck polynomials appearing in Corollary 3.11 are actually symmetric Grothendieck polynomials, but symmetric only in some initial segment of the variables x.
They are however the only Lascoux polynomials equal to a Grothendieck polynomial for which λ is a rectangle. T. Matsumura and S. Sugimoto have informed the authors that every flagged Grothendieck polynomial is a Lascoux polynomial by extending the proof of [Mat19, Thm. 3.3], which appears in their work [MS20]. (This is the K-theoretic analog of the fact that every flagged Schur function is a Demazure character [RS95].) Thus in particular, for w a vexillary permutation, the Grothendieck polynomial G w is known to be a flagged Grothendieck polynomial [KMY09, Thm. 5.8], so G w is also the Lascoux polynomial L a for a the Lehmer code of w. In the special case β = 0, A. Postnikov and R. Stanley showed that a Demazure character π w x λ is a flagged Schur function if and only if w ∈ S λ n is 312-avoiding [PS09]. These facts motivate the following conjecture. Consider an arbitrary flagging condition with Lemma 3.8. Then we can interpret these Lascoux polynomials as a Jacobi-Trudi-type determinant, where each part is the Segre class of a vector bundle [HIMN17,HM18]. (Although flagged set-valued tableaux also appear in [GK15], that use appears unrelated to ours, as the weights considered in the two contexts seem to be irreconcilable.)

Bijection with K-Kohnert diagrams
Recall that there is a natural bijection between the set of semistandard Young tableaux of shape 1 r with entries at most n and the collection of subsets of {1, . . . , n} of size r. For row i (starting from the bottom row and going up) of a K-Kohnert diagram D, consider the subset of {1, . . . , n} given by the horizontal coordinates of the unmarked boxes. Construct column i (from right to left) of a (a priori non-necessarily semistandard) tableau T by applying the natural bijection given above to this subset. Now, for every marked box in position (x, i) of D, there is a rightmost unmarked box (x ′ , i) to the left of (x, i). Insert x into the cell of column i containing x ′ . In other words, we insert x into the topmost (i.e. highest) possible cell of column i such that the resulting column is semistandard. (Note that a priori the rows may not be semistandard.) Write φ(D) for the resulting tableau T .
It is straightforward to see that the map φ is invertible and β-weight preserving. We will show below that φ(D) is in fact always a semistandard set-valued tableau. Indeed, our main effort in this section is to establish the following. Recall that D wλ is the set of K-Kohnert diagrams obtained from wλ.
Proposition 4.1. Let λ = s r be an r × s rectangle. For any w ∈ S λ n , φ restricts to a β-weight preserving bijection φ : D wλ → SV n w (λ). Example 4.2. Consider λ = 2 2 be a 2 × 2 square and w = s 2 . Under φ described above, we have where we have shaded in the selected boxes and put a • in the marked boxes. Observe that these tableaux are exactly SV n w (λ). Example 4.3. We continue Example 4.2 to w ′ = s 1 s 2 and obtain all of SV 3 w ′ (λ) = SV 3 (λ). We show the bijection φ on the remaining elements: In order to prove Proposition 4.1, we first construct the equivalent (K-)Kohnert moves on (semistandard) set-valued tableaux.
Definition 4.4 ((K-)Kohnert moves on set-valued tableaux). Let T ∈ SV n (λ). Consider an entry x ∈ Z such that x ∈ T . Let C be the leftmost column of T containing an x. Let b be the box in C containing x. Let x ′ be minimal such that x ′ + 1, x ′ + 2, . . . , x ∈ C, and let b ′ be the box in C containing x ′ + 1. If if any of x ′ + 1, x ′ + 2, . . . , x − 1 is not the only entry of its box in C), then we do not have a (K-)Kohnert move corresponding to x. Otherwise define the Kohnert move on T to (1) remove x from b; (2) if x ′ < x − 1, then moving all entries x ′ + 1, . . . , x − 1 in C down one row (which in particular moves x − 1 into b); and (3) inserting x ′ into b ′ . A K-Kohnert move is the same as a Kohnert move except we leave x ∈ b.
Lemma 4.5. Let T ∈ SV n (λ), and denote by T ′ the result of applying any (K-)Kohnert move to T . Then T ′ ∈ SV n (λ).
Proof. We consider only the Kohnert move as the K-Kohnert move is similar. We will use the notation from Definition 4.4.
The column increasingness condition for T ′ is satisfied since we only altered C and we took x ′ to be minimal i.e. any entries in or below b ′ must not contain an x ′ ). For the row increasingness condition of T ′ , we note that we are decreasing some entries in C, so it is sufficient to just consider the entries in the column C ← directly to the left of C. Let b ← be the box immediately to the left of b. By choice of C as leftmost, we must have x / ∈ C ← and hence by the row semistandardness of T , we must have max b ← < x. This implies that the box above b ← must have all entries strictly less than x − 1 by column semistandardness of T . By iterating this argument, the analogous statement holds for all boxes in C weakly between b and b ′ . Thus, T ′ ∈ SV n (λ). Now we prove Proposition 4.1 by using our flagging characterization of K-Demazure crystals from Lemma 3.8 and showing that φ intertwines the (K-)Kohnert moves on K-Kohnert diagrams with the (K-)Kohnert moves on set-valued tableaux.
Let D ∈ D wλ and T = φ(D). It is straightforward to see that a Kohnert move moving x in column y of T corresponds under φ to the Kohnert move on D that moves an unmarked box (x, y) to (x ′ , y). Note that taking the leftmost column in T is equivalent to taking a box in the top of the x-th column of D. We claim these are all possible Kohnert moves. Indeed, the condition that x ′ + 1, . . . , x − 1 are the only entry in their boxes corresponds to the fact that we did not cross a marked box in D. That x ′ ≥ 1 is equivalent to the moved box in D staying within Z 2 >0 after the move. That x = min b is equivalent to the box at (x, y) in D being unmarked. Hence, we obtain all possible Kohnert moves. The claim for K-Kohnert moves is similar.
For any T ∈ F (λ; w), any possible Kohnert or K-Kohnert move applied to T yields another element in F (λ; w) as entries in a particular row decrease. Thus we have φ(D wλ ) ⊆ F (λ; w) as the initial skyline diagram corresponds to the tableau T wλ with all entries in row r − k being {i k }.
To show φ(D wλ ) ⊇ F (λ; w), we will show that every set-valued tableau in F (λ; w) can be obtained from T wλ by applying (K-)Kohnert moves. For a given flagged setvalued tableau T ∈ F (λ; w), we start by applying (K-)Kohnert moves on the upper left box of T wλ until we get the entry in the upper left box of T . Recall that the corresponding (K-)Kohnert move acts on this entry/column since it acts on the leftmost applicable column. We repeat this process moving across the first i.e. topmost) row. As such, there will be no interactions between the different Kohnert moves. We then repeat this for the second row, then the third, and so on until all entries have been changed to T . All the tableaux produced along the way lie in SV n (λ) by Lemma 4.5.
We claim this procedure always generates a sequence of (K-)Kohnert moves in D wλ . Indeed, what we are doing is moving the first column from the skyline diagram to its appropriate spots for the final K-Kohnert diagram φ −1 (T ) starting from the top. It is easy to see that by doing this process, we are never moving a box across another box. 5 Hence, such (K-)Kohnert moves are always valid. Thus, φ(D wλ ) = F (λ; w).
Example 4.6. Let λ = 3 2 be a 2 × 3 rectangle and consider n = 4. We exhibit the sequence of (K-) Kohnert  Remark 4.7. We note that the proof of the intertwining of (K-)Kohnert moves did not require λ to be a rectangle. However, that φ is a bijection does require that λ is a rectangle as otherwise the image of the skyline diagram will not be a partition. For example, −→ 2 2 3 . Hence, the Ross-Yong-Kirillov Conjecture (Conjecture 2.2) holds for L a when a is any weak composition with a unique nonzero part size.
Proof. This follows from the definition of a K-crystal together with Lemma 3.8, Theorem 3.9, and Proposition 4.1.
To the best of our knowledge, no other cases of Conjecture 2.2 have been previously established.

Bijection with set-valued skyline tableaux
Consider a partition λ and permutation w. Define where the union is taken over all v strictly less than w in Bruhat order. We have Proof. Let i k < · · · < i 0 be from the reduced expression of w given by Lemma 3.1. By Lemma 3.8, we can characterize SV n w (λ) by T ∈ SV n w (λ) if and only if, for all 0 ≤ k < r, row k of T has at least one i k + 1 in it and no number strictly greater that i k + 1. By the fact that the rows of T are weakly increasing, an equivalent characterization is that the largest entry of the rightmost box in row k of T must be i k + 1.
Define a map ψ : SLT wλ → SV n w (λ) as follows. Consider some S ∈ SLT wλ and define T := ψ(S) by (1) sorting the anchor entries in each row in increasing order left to right and join the columns together; (2) placing each free entry f in the leftmost box of its row such that f is less than the anchor entry (i.e. so that the row is strictly increasing but the free entries remain free); (3) take the transpose of the result from the previous step; that is construct the i-th column of T from the (r + 1 − i)-th row as in Section 4.
To see that ψ is injective and that its image is contained in SV n (λ), we note that ψ is a restriction of the bijectionρ of [Mon16, Thm. 2.4] from set-valued skyline tableaux to reverse semistandard set-valued tableaux, except that we have reversed the rows and columns of the image tableaux. Indeed, reversing the rows and the columns for rectangular shapes is clearly a bijection from reverse semistandard set-valued tableaux to (ordinary) semistandard set-valued tableaux. To see that the image of ψ is SV n w (λ), we note that the anchors of the first row of S are already in increasing order. Thus, the anchor of the k-th column of S under ψ is the largest entry in the k-th row of ψ(S). By construction, the largest entry of the k-th column of S is i k + 1, and hence, ψ is surjective.
Finally, it is clear that ψ is β-weight preserving, so ψ is the desired bijection.
Example 5.2. Let λ = 2 2 be a 2 × 2 rectangle and n = 3. Then the set-valued skyline tableaux SLT s2λ and their corresponding element in SV To the best of our knowledge, the following theorem is the first to prove any case of Conjecture 2.4.
Theorem 5.4. For λ = s r an r × s rectangle, we have Hence, Monical's Skyline Conjecture (Conjecture 2.4) holds for L a when a is any weak composition with a unique nonzero part size.
Proof. This follows from the definition of a K-crystal together with Theorem 3.9, Equation (5.2), and Proposition 5.1.

K-key tableaux
A key tableau K is a semistandard tableau such that the entries in the j-th column of K are a subset of those in the (j − 1)-st column of K. One method to compute a Demazure character κ wλ is by summing over all semistandard tableaux of shape λ whose (right) key entries are less than corresponding entry in the unique key tableaux K wλ of weight wλ [LS90]. (This fact explains why Demazure characters are also known as key polynomials.) Furthermore, every semistandard tableau T has a unique (right) key tableau k(T ) associated with it (we refer the reader to [Wil13] for an algorithm), and a Demazure atom can be computed as a generating function for all semistandard tableaux T with k(T ) = K wλ [LS90]. (See [PW15] for much further discussion of these (and related) formulas.) Let ≺ denote the partial order on semistandard tableaux of shape λ such that T T ′ if and only if every entry of T is at most the corresponding entry in T ′ .
Based on the bijection from Proposition 5.1 and the (K-)Kohnert moves on setvalued tableaux (Definition 4.4), the following is a natural possible extension of key tableaux to the K-theory setting. For T ∈ SV n (λ), define K(T ) := k max(T ) , where max(T ) is semistandard tableau obtained by taking the greatest entry in each box of T . Thus Theorem 3.9 and Lemma 3.8 imply that for λ = r s L wλ (x; β) = T ∈SV n (λ) K(T ) K wλ wt β (T ), L wλ (x; β) = T ∈SV n (λ) K(T )=K wλ wt β (T ), (6.1) or equivalently summed over SV n w (λ) and SV n w (λ) respectively. However, these formulas do not work for general λ as, for example, should not be in the atom corresponding to w 0 .
Instead, we conjecture that Equation (6.1) should be modified by using the Lusztig involution to obtain a combinatorial interpretation of general Lascoux polynomials and atoms. Recall that the Lusztig involution on the highest weight crystal B(µ) is defined by sending the highest weight element U to the lowest weight element U * and extended to the remaining elements in B(µ) by Although at first glance Conjecture 6.1 looks rather different from our proved formulas in the rectangular cases, we now will show that Equation (6.1) establishes Conjecture 6.1 when λ is a rectangle. 6 To do so, we will construct a K-Lusztig involution ⋆ : SV n (λ) → SV n (λ) that also satisfies Equation (6.2), but is a twist of the Lusztig involution by an automorphism of the U q (sl n )-crystal SV n (λ) (i.e. it nontrivially permutes the irreducible components). Let λ = r s be a rectangle and T ∈ SV n (λ). Define T ⋆ to be the set-valued tableau obtained by rotating the tableau 180 • and then replacing each i → n + 1 − i. We note this is a well-known description of the Lusztig involution (also known as the Schützenberger involution or evacuation [Len07]) on semistandard tableaux of shape λ.
Proposition 6.2. Let λ be a rectangle. The K-Lusztig involution ⋆ satisfies Equation (6.2). Moreover, for T ∈ SV n (λ) as a tensor product of rows T = R 1 ⊗· · ·⊗R k , we have T ⋆ = R * k ⊗ · · · ⊗ R * 1 . Proof. The first claim follows from the definition of the crystal operators. We leave the details to the reader. For the second claim, we first note that T ⋆ = R ⋆ k ⊗ · · · ⊗ R ⋆ 1 , and so it is sufficient to show R ⋆ 1 = R * 1 . This follows from a straightforward induction on depth (i.e., the number of crystal operators applied from the highest weight element) and Equation (6.2).
Proposition 6.2 also suggests that Conjecture 6.1 holds for a definition of a (right) K-key tableau by K ′ (T ) := k(min(T † ) * ), where T † is constructed from T according to any automorphism of SV n (λ) such that wt(T † ) = w 0 wt(T ). However, given a (weak) K-crystal structure on SV n (λ), it would be preferable to have a T † construction that matches the labeling of tableaux T by K-keys K ′ (T ) with the decomposition of the K-crystal by K-Demazure subcrystals, as is the case with our K-Lusztig involution T ⋆ . Furthermore, it is likely that in general we want T ⋆ = R * k ⊗ · · · ⊗ R * 1 as in Proposition 6.2, but this would require an appropriate K-rectification or insertion scheme in order to obtain a result back in SV n (λ).
We also believe there exists an insertion scheme analogous to the one given by S. Mason in [Mas08, Sec. 3.3] to construct a bijection between SV n w (λ) and SLT wλ . This will possibly be a variant of the insertion given in [Buc02] similar to how Mason's map is a variant of the classical RSK algorithm.