Unique rectification in $d$-complete posets: towards the $K$-theory of Kac-Moody flag varieties

The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of"unique rectification targets"in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's"$d$-complete posets"to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting.


Introduction
The 1970s saw a major advance in the combinatorial approach to enumerative geometry when M.-P. Schützenberger proved the Littlewood-Richardson rule for describing the cohomology rings of Grassmannians. Since then, the modern Schubert calculus has turned to extending this understanding in two different directions: on the one hand to replace the Grassmannian with a more complicated homogeneous space, and on the other hand to replace ordinary cohomology with a richer generalized cohomology theory. Along these lines, the goal of this paper is to begin unraveling the K-theoretic Schubert calculus of Kac-Moody homogeneous spaces. Our results are purely combinatorial in nature, but allow us to conjecture explicit Littlewood-Richardson-style rules in this geometric context.
Let G be a complex Kac-Moody group with Borel and opposite Borel subgroups B + and B − , respectively. Let B + ⊆ P ⊂ G be a parabolic subgroup. The homogeneous space X = G/P is a Kac-Moody flag variety. The Zariski closures of the B − -orbits are the Schubert varieties {X w } w∈W P and give a cell decomposition of X; here, W P denotes the set of minimal-length representatives of the quotient W/W P , where W is the Weyl group of G and W P is the parabolic Weyl group for P . The cohomology ring H ⋆ (G/P ) thereby has a distinguished Schubert basis {σ w } w∈W P , where σ w is Poincaré dual to X w . Thus, to determine multiplication in H ⋆ (X), it suffices to determine the Schubert structure constants c w u,v defined by In the case that X = Gr k (C n ) is a Grassmannian, the parameter space of kdimensional linear subspaces of C n , this problem is solved in a positive combinatorial manner by any of the various Littlewood-Richardson rules (e.g., [LR34,Sch77,Vak06]). For a general Kac-Moody flag variety X, these c w u,v are also non-negative integers, but it is generally a major open problem to give an analogous Littlewood-Richardson-style rule to determine them.
For X = Gr k (C n ), M.-P. Schützenberger's Littlewood-Richardson rule is stated in terms of the jeu de taquin for standard Young tableaux [Sch77] fitting inside a k × (n − k) rectangle. One may realize this rectangle as a subposet of positive roots for GL n (C) in such a way that the inversion set of w ∈ W P is an order ideal in this subposet. One may further realize standard Young tableaux as linear extensions of intervals in this poset. Using this perspective, H. Thomas and A. Yong [TY09a] gave a uniform extension of Schützenberger's rule to compute all cohomological Schubert structure constants for the larger family of minuscule varieties. This was further extended by P.-E. Chaput and N. Perrin [CP12] to a positive combinatorial formula for computing certain Λ-minuscule Schubert structure constants for general Kac-Moody X. In the Chaput-Perrin rule, the role of the k × (n − k) rectangle is played by the d-complete posets introduced by R. Proctor [Pro99a,Pro99b]; d-complete posets are exactly those posets encoding the containment relations among Λ-minuscule Schubert varieties.
Much work in the modern Schubert calculus has been devoted to studying homogeneous spaces through richer cohomology theories. In these theories, there are Schubert bases analogous to the cohomological σ w and the structure constants defined analogously to Equation (1.1) enjoy various positivity properties. Hence, it makes sense to attempt to develop positive combinatorial formulas for these structure constants in the style of the classical Littlewood-Richardson rules. In the Grassmannian case, one has, for example: the equivariant cohomology rule of A. Knutson and T. Tao [KT03], the K-theory rule of A. Buch [Buc02]; the equivariant K-theory rule of O. Pechenik and A. Yong [PY17a]; the quantum cohomology rule of A. Buch, A. Kresch, K. Purbhoo, and H. Tamvakis [BKPT16]; and the equivariant quantum cohomology rule of A. Buch [Buc15]. Our interest is in the ordinary K-theory ring K(X) of the Kac-Moody flag variety X, where the K-theoretic Schubert classes {[O Xw ]} w∈W P are represented by the structure sheaves of the Schubert varieties. Specifically, we are interested in the structure constants K w u,v of K(X) defined by For Grassmannians, various alternatives to Buch's original rule [Buc02] for K w u,v are now known [Vak06,TY09b,PY17b]. However, only the rule of H. Thomas and A. Yong [TY09b] is currently known to extend to all of the minuscule varieties [BR12,CTY14,BS16]. This Thomas-Yong rule is based on a jeu de taquin theory for increasing tableaux. This combinatorial theory displays a number of additional subtleties when compared to Schützenberger's jeu de taquin for standard tableaux.
In particular, a key ingredient is the need to identify increasing tableaux with the unique rectification target property. (These combinatorial notions are reviewed in Section 2.) In [TY09b,Problem 9.1] and [BS16,Remark 3.24], the authors ask to what extent their combinatorial theory extends to the case of d-complete posets. The missing ingredient is that it is not currently known whether general d-complete posets have "enough" unique rectification targets. We conjecture, however, that they do.
Conjecture 1.1. Let P be a d-complete poset and let λ ⊆ P be an order ideal. Then there is an (explicitly-defined) unique rectification target supported on λ.
We initiate a study of the existence and structure of unique rectification targets in the d-complete posets. As shown by R. Proctor [Pro99a], every d-complete poset can be constructed by gluing together (in prescribed ways) certain irreducible d-complete posets. These irreducible pieces are classified in [Pro99a] and include all of the minuscule posets (i.e., the posets describing the Schubert stratification of minuscule varieties). The informal version of our main result is the following special case of Conjecture 1.1.
Theorem 1.2. Conjecture 1.1 holds in the case that P is built from minuscule posets.
For an example of such a poset covered by Theorem 1.2, see Figure 1. We also demonstrate the extent to which Conjecture 1.1 is sensitive to the poset P being d-complete. We establish general results on the failure of Conjecture 1.1 to extend to posets that are slight deformations of d-complete posets.
For any P satisfying Conjecture 1.1, one obtains (as in [BS16, §3.5]) a corresponding combinatorially-defined associative commutative unital algebra K(P) with a basis {λ} indexed by order ideals of P. The structure constants t ν λ,µ of K(P) are defined in such a way as to transparently alternate in degree. (This construction is discussed in Section 6.) For w ∈ W P Λ-minuscule, the interval [id, w] in Bruhat order is isomorphic to the poset of order ideals of a certain d-complete poset P w constructed from w. Building on Conjecture 1.1, we propose the following. Conjecture 1.3. Let X = G/P be a Kac-Moody flag variety and let m ∈ W P be Λ-minuscule. Then, for u, v, w ≤ m in Bruhat order, we have the equality of structure constants K w u,v = t ν λ,µ , between the rings K(X) and K(P m ), where the order ideals λ, µ, ν ⊆ P m correspond to the the Weyl group elements u, v, w ∈ W P , respectively.
In Section 6, we give the precise versions of Conjecture 1.1 and Theorem 1.2, as well as the details necessary for a precise understanding of Conjecture 1.3. In light of Conjecture 1.3, Theorem 1.2 should be understood as giving a conjectural positive combinatorial rule for certain K-theoretic Schubert structure constants of Figure 1. The Hasse diagram of a representative d-complete poset P that is "built from minuscule posets" in the sense of Theorem 1.2. In P, every order ideal λ ⊆ P has a unique rectification target, provided by Theorem 1.2.
Kac-Moody flag varieties. Several cases of Conjecture 1.3 are known to be true or have been previously conjectured. If the flag variety X is minuscule, then Conjecture 1.3 reduces to the main theorem of [BS16]. If, on the other hand, X is general but |ν| − |λ| − |µ| = 0, then Conjecture 1.3 reduces to [CP12, Conjecture 1.1], many cases of which are proved in [CP12, Theorem 1.3]. Assuming one followed the general structure utilized by [CP12,BS16], the main ingredients one would need in a proof of Conjecture 1.3 are (1.) a proof of the remaining cases of Conjecture 1.1 and (2.) ad hoc geometric verifications of Conjecture 1.3 for special u lying in a generating set of classes. For a large class of such Schubert problems, Theorem 1.2 provides the necessary first ingredient, so it only remains to establish the second in those cases.
Another potential application of Theorem 1.2 (or more generally Conjecture 1.1) is to establishing plane partition identities. In [HPPW16], the authors use the existence of unique rectification targets in minuscule posets to give bijective proofs of the equinumerosity of various classes of plane partitions, in particular resolving a 1983 question of R. Proctor [Pro83]. The main technology of [HPPW16] applies equally to any d-complete poset satisfying Conjecture 1.1; hence, we expect Theorem 1.2 to yield analogous identities. Further discussion may appear elsewhere. This paper is organized as follows. In Section 2, we fix notation for posets and describe the Thomas-Yong theory of jeu de taquin for increasing tableaux. We then recall the definition of unique rectification targets (URTs). Section 3 studies the behavior of URTs when two posets are combined via Proctor's slant sum operation. Section 4 builds on Section 3 by introducing the notion of a p-chain URT, a stronger version of a URT that we will later need. Section 5 establishes the necessary technical fact that all increasing tableaux of straight shape in a double-tailed diamond poset are p-chain URTs. In Section 6, we first recall background on d-complete posets. We also recall needed notions to make Conjectures 1.1 and 1.3 precise. We then apply the results from Section 4 and Section 5 to the study of d-complete posets and prove Theorem 1.2, our main result.

posets, skew shapes, and rectifications
All posets in this paper will be finite, nonempty, and connected. These assumptions are made for convenience and clarity only; many of our results do not fundamentally rely on these properties, although the statements and proofs become messier without them. Moreover, the original definition of d-complete posets (which we follow in this paper) requires finiteness. Although there is now a more general notion of infinite dcomplete posets (see the "Added Notes" at the very end of [PS17] for discussion), we will not explicitly consider such objects. In this section, P will denote an otherwise arbitrary poset.
We begin by fixing necessary terminology regarding posets. For x, y ∈ P, we say that z covers x (written x ⋖ z), if x < z and there does not exist a y ∈ P with x < y < z. Let x, y ∈ P. If x < y, we say that x is an ancestor of y and that y is a descendant of x. If x ⋖ y, we say that x is a parent of y and that y is a child of x. Adding "weak" to any of these terms also allows for equality, e.g. x is a weak descendant of y if x ≥ y. We denote the minimum element (if it exists) of the poset P by0 P . We say a poset P has a0 P to mean that it has a minimum, which is0 P .
We often visualize posets using Hasse diagrams, where each element is represented by a circle, and a ⋖ b if there is a line that goes up from a to b.
Example 2.1. Let Q be the poset on the elements {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)} of Z 2 under the natural order (a, b) ≤ (c, d) if both a ≤ c and b ≤ d. As a Hasse diagram, we have Q = We will use this Q as a running example throughout this section. ♦ A shape ν of P is any subset of P. The shape ν has a natural poset structure given by restricting that of P. A shape ν of P is called an order ideal of P if it is closed downwards, i.e. if y ∈ ν and x < y together imply x ∈ ν. Similarly, an order filter of P is a subset that is closed upwards. For historical reasons, we will also refer to the order ideals of P as straight shapes.
Example 2.2. The following are all the straight shapes of the poset Q from Example 2.1. For greater visual context, we represent elements not in the straight shape with solid black circles.

♦
If λ ⊆ ν are straight shapes of P, then the shape ν \ λ is called a skew shape of P and is denoted ν/λ. Note that every straight shape λ can also be realized as the skew shape λ/∅.
An element x ∈ λ is called an inner corner of the skew shape ν/λ if x is maximal in λ. We write IC(ν/λ) to denote the set of inner corners of ν/λ.
Clearly, we have the following.
For a skew shape ν/λ of P, a function T : ν/λ → Z >0 is called a skew increasing P-tableau of shape ν/λ if T is a strictly order preserving map, i.e. if x < y implies T (x) < T (y). If, in addition, T is a bijection onto an initial segment of Z >0 , we say T is a skew standard P-tableau. In both cases, if ν/λ is a straight shape, we drop the word "skew." We depict a skew increasing P-tableau T using Hasse diagrams with labels. For an element x ∈ P, we put the value of T (x) in the circle of the Hasse diagram corresponding to x. Also, to make clear what the ambient poset is we represent skewed out elements (the elements in λ) with unlabeled hollow circles.
Example 2.4. If P is the numbers 1, 2, 3, 4 with the usual order and ν/λ = P and T (x) = x + 5, then T can be visualized as . ♦ For a skew shape ν/λ of P, we say a function T : ν/λ → Z >0 ∪ {•} is a skew dotted increasing P-tableau of shape ν/λ if there is a rational number q such that T becomes a strictly order preserving map (ν/λ → Q) when we replace each • with that fixed q.
Example 2.6. The following is a skew dotted increasing Q-tableau ♦ Definition 2.7. Let T be a skew increasing P-tableau of shape ν/λ. If γ is a nonempty set of inner (or outer) corners of ν/λ, then AddDots γ (T ) is the skew increasing P-tableau S of shape ν/λ ∪ γ defined by Example 2.8. For ♦ Definition 2.9. Let T be a skew dotted increasing P-tableau. For n ∈ Z >0 , Swap •,n (T ) is the skew dotted increasing P-tableau S defined by if T (x) = n and T (y) = • for some y ⋗ x; T (x), otherwise.
Example 2.10. We have ♦ Definition 2.11. Let T be a skew dotted increasing P-tableau. Let Q be the subset of P which T maps to an integer, i.e.
Example 2.14. As an example, let T be an increasing P tableau For a tableau T of shape ν/λ, we use the notation IC(T ) to mean IC(ν/λ).
The rectifications of T are the elements of the rectification set rects(T ) = {U : U ∈ S n for some n ∈ Z ≥0 and U is of straight shape}.
To denote that U is a rectification of T given by sliding the sequence of sets of inner corners (γ 1 , . . . , γ n ), we write T Another rectification follows this path . One can check that these two are the only We say a skew increasing P-tableau rectifies uniquely if it has exactly one rectification. We say an increasing P-tableau T of straight shape is a unique rectification target (URT) if every skew increasing P-tableau which rectifies to T rectifies uniquely.

Unique rectification in slant sums
In this section, we explore the structure of URTs in posets that are built out of smaller posets by a slant sum operation.
Definition 3.1. [Pro99a] Let P, Q be disjoint posets. Assume Q has a minimum element0 Q . Let p ∈ P. The slant sum of Q to P at p, denoted P p /0 Q Q is the poset on P ⊔ Q induced by imposing0 Q ⋗ p together with the orders on P and Q. Because a poset's minimum is unique, we will usually drop the0 Q and denote the slant sum as P p / Q. If Q 1 , . . . , Q n are pairwise disjoint posets which each have minima, then P p / (Q 1 , . . . , Q n ) denotes the iterated slant sum of posets at p. (Clearly, the order of the Q i s does not matter.) Finally, given p 1 , . . . , p m ∈ P and pairwise disjoint posets Q j i with minima, we write . . , Q m rm ) to denote the result of slant summing each Q j i onto p j (in any order). We say that a poset P is a chain if all pairs of elements of P are comparable, that is, if P is a total order. The size of a chain P is the number of its elements.
Proposition 3.2. Let P be a poset with0 P and let C = {c} be a chain of size 1.
If T is a a skew increasing P-tableau that rectifies uniquely (in P), then the skew increasing R-tableau T R of shape ν ⊔c/λ⊔c defined by T R (x) = T (x) rectifies uniquely (in R).
Proof. Suppose U is the unique rectification of T . Let p be the minimum of P.
When we rectify T R in R, since c is covered only by p, the last step of rectification is to slide c. Just before this final step, one has a tableau S whose shape is an order ideal in P ⊂ R. Clearly, a process of slides producing S from T R (in R) corresponds to a sequence of slides rectifying T (in P). Thus, since T rectifies uniquely to U, we have S(x) = U(x) for all x ∈ P.
The final slide is necessarily Slide {c} , since c is the only inner corner of S. Hence, there are no further choices to make and thus T R rectifies uniquely. is not a unique rectification target in the poset .
We now introduce some useful notation and terminology that we will need. Suppose Q is a shape of P, and T is a skew increasing P-tableau. Then, the restriction of T to Q, denoted T | Q is the increasing Q-tableau given by restricting the domain of T to those elements that are in Q.
Definition 3.5. Suppose F is an order filter of a poset P with a0 F . We say F is a funnel if, for all p ∈ P \ F with p < f for some f ∈ F , we have p <0 F . Note that, in particular, the embedded copy of Q in any slant sum P p /0 Q Q forms a funnel.
Definition 3.6. Let F be a funnel of a poset P. Suppose T is a skew increasing P-tableau of shape ν/λ with a rectification U. Then, the corresponding skew increasing F -tableau of T for U, denoted as ( Proposition 3.7. Let F be a funnel of a poset P and let T be a skew increasing P-tableau. If U is a rectification of T , then U| F is a rectification of (T → U)| F .
Proof. For concision, write F for (T → U)| F . The proposition is trivial if F is the empty tableau, so we assume F is supported on at least one element of F . Let T γ 1 ,...,γn − −−− → U. Using the sequence of sets of inner corners γ 1 , . . . , γ n , we will recursively construct sets of inner corners θ 1 , . . . , θ n that rectify F to S| F . We write T j i to represent T after i slides and just before the jth swap; that is, In particular, T 0 0 = T and T 1 0 = AddDots γ 1 (T ). Let F 0 := F . For 1 ≤ i ≤ n, recursively define θ i and F i as follows Finally, we define certain subsets of θ i which will be useful in our analysis. Let θ 0 i = ∅ and, for k > 0, let Note that θ 1 i = θ i . Set m := U(0 F ) = min Range(F ). For the remainder of this proof, we say F j i and T j i N-agree if F j i and T j i agree on all numeric labels within F greater than or equal to m; that is, for Additionally, for the remainder of this proof, we say that F j i and T j i agree if they satisfy all the conditions: . We now establish inductively that F j i and T j i agree for all i and j. First, note that, since F 0   (Case 2: j = 0): In this case, . Proof of (A.0): Since F 0 i and T 0 i N-agree by assumption and AddDots θ i+1 does not affect numerical labels, it is clear that where the first and last equalities are by adding •s not affecting numeric labels and the middle equality is by ( . First, we prove some helpful claims.
There are two cases to consider: either j ⋆ = j or j ⋆ > j.
We will obtain a contradiction by deriving that p ∈ IC(F i ).
Thus, for any p ′ ⋗ p with p ′ / ∈ Dom(F i ), we then have, for all p ′′ > p ′ , that p ′′ / ∈ Dom(F i ). Thus, p ∈ IC(F i ), which is our desired contradiction.
Proof of (A.0): By the inductive (A.0), F j i and T j i agree on numeric labels greater than or equal to m. Since F j+1 i and T j+1 i are just F j i and T j i after applying Swap •,j , it suffices to consider the movement of the j labels. If j < m, we are done. Hence, assume j ≥ m. We know by (  (A.0). As this last set is the definition of θ j+1 i , this completes the proof of (A.2) and hence the induction.
As a consequence of our induction, we have F 0 n = T 0 n | F = U| F . Hence, We now derive a few straightforward corollaries of Proposition 3.7. We will not use these corollaries in the sequel; however, they seem interesting to us for elucidating some of the structure of URTs among collections of related posets.
Corollary 3.8. Let T be a skew increasing P-tableau and let F be a funnel of P. We construct the shapes C i recursively. Let C 0 := min(P) and let C i+1 := C i ∪ min(P \ C i ). Finally, the bottom chain of P is Proof. In any rectification W of T , the labels of W | C must be just the smallest numbers in the range of T in increasing order.
For U a straight-shaped increasing R-tableau, if U| P k is an URT in P k for each 1 ≤ k ≤ n, then U is a URT in R.
Proof. Suppose U, V are rectifications of some skew increasing R-tableau T . Since c is in the bottom chain of R, U| C = V | C by Lemma 3.10. Let m := U(c) = V (c). It remains to show that U| P k = V | P k for all k. Fix some P k . By the increasingness of U and V , we have U(0 P k ) > m and V (0 P k ) > m, but both U(0 P k ) and V (0 P k ) must also be less than all the other labels in their respective tableaux for elements in P k that are greater than0 P k . Thus, since0 P k only covers c and all the P k are disjoint funnels, it is easy to see that Finally, since U(0 P k ) = V (0 P k ), we have U| P k = V | P k by applying Corollary 3.8.
We will say a poset T is a tree if T has a0 T and if each other x ∈ T has exactly one parent.
Corollary 3.12. Let T be a tree. Let t 1 , . . . , t n be (not necessarily distinct) elements of T . Let P 1 , . . . , P n be pairwise disjoint posets with minimum elements. Define Proof. By repeated application of Corollary 3.11.

Restricted rectifications and p-chain URTs
Most of the results in this section are fairly technical lemmas that we will need later. (1) Fix p ∈ P and let k := |{x ∈ P : x ≤ p}| be the cardinality of the principal order ideal generated by p. Let C be a chain poset of size k. If for pairwise disjoint posets Q 1 , . . . , Q n with minimum elements and T is a skew increasing R-tableau, then there is a skew increasing (P p / C)-tableau C such that rects| P (T ) = rects| P (C).
Proof. We first prove (1). Since there are k weak ancestors of p, there are at most k distinct labels from h Q h that can swap into P during rectification of the skew increasing R-tableau T . Let q 1 < · · · < q m ∈ Z >0 be the m smallest labels of elements in T | h Q h , where m is the lesser of k and the number of distinct labels in T | h Q h . Fix a chain C = {c 1 < · · · < c k } of size k. Define the skew increasing P p / C-tableau C as follows: T (x), x ∈ P ∩ Dom(T ); q r , x = c r and 1 ≤ i ≤ m.
Write T j i to represent T after i slides and just before the jth swap; that is, Similarly, let C j i be For the rest of this proof, we say T j i and C j i are similar if they satisfy both of the following conditions: We show by induction that T j i and C j i are similar for all i and j. In particular, this will yield T j i | P = C j i | P , proving that U| P = T 0 n | P = C 0 n | P ∈ rects| P (C), as desired. By construction, we have that T 0 0 | P = T | P = C| P = C 0 0 | P so T 0 0 and C 0 0 satisfy (S.0). Condition (S.1) for T 0 0 and C 0 0 is also by construction. Now, inductively assume T j i and C j i are similar. For concision, write S j i for the set (Case 1: j = M + 1): We have that Before turning to the next case, note that if none of the weak ancestors of p are skewed out of T 0 i | P = C 0 i | P , then (S.0) and (S.1) continue holding in perpetuity since no elements of h Q h or C will be involved in any swaps. Thus, for the remaining cases, assume at least one weak ancestor of p is skewed out.
Further, note that if S j i = ∅, then since at least one weak ancestor of p is skewed out, we have {T j i (q) ∈ Z : q ∈ h Q h } = {C j i (c) ∈ Z : c ∈ C} = ∅ by the definition of q m . Hence if S j i = ∅, then (S.0) and (S.1) continue to hold in perpetuity, since neither tableau has labels outside of P. Thus, for the remaining cases, we may further assume S j i = ∅. (Case 2: j = 0): First, we verify that θ i+1 ⊆ IC(C 0 i ). Let t ∈ θ i+1 . Since t ∈ θ i+1 , we have that t ∈ γ i+1 ∩ P, so t is an inner corner in T 0 i | P = C 0 i | P . (This equality is by the inductive (S.0).) Hence, since C has skewed out nodes only in P, it follows that t ∈ IC(C 0 i ). Thus, θ i+1 ⊆ IC(C 0 i ). Since θ i+1 = γ i+1 ∩ P, we have Similarly to the previous case, since adding •s does not affect the numerical labels in h Q h or C, (S.1) for T 1 i and C 1 i is immediate from (S.1) for T 0 i and C 0 i . (Case 3: 0 < j < M + 1): We show (S.0) first. Observe that the structure of R, P p / C, and P easily ensures that if q ∈ P, then (F.1) for allq ⋖ q (in either R or P p / C),q ∈ P, and (F.2) if q = p, then for all q ′ ⋗ q (in either R or P p / C) we have that q ′ ∈ P. Here, the first and third equalities are by the definition of the swapping process and the increasingness of the tableaux. The second equality is because min {T j i (p ′′ ) ∈ Z : p ′′ ∈ R and p ′′ > p} = min {C j i (p ′′ ) ∈ Z : p ′′ ∈ P p / C and p ′′ > p}, as follows from the inductive (S.0), (S.1), and the assumption that S j i = ∅. This proves (S.0).
It remains to show (S.1). The swapping process only affects labels with value j. We already have so it just remains to show that Otherwise, T j+1 i (p) = j. Then, nothing could have been swapped into p at this stage. Thus, since p is the only element connecting P to Q h or C, in this situation (S.1) for T j+1 i and C j+1 i is immediate from (S.1) for T j i and C j i . This completes the induction and shows that rects| P (T ) ⊆ rects| P (C).
To show the reverse containment rects| P (C) ⊆ rects| P (T ), we follow the same strategy, except that we first remove any skewed out nodes in h Q h . Suppose C θ 1 ,...,θn − −−− → V . We must find a sequence of inner corners that yields a rectification U of T such that U| P = V | P . First, we remove any skewed nodes in h Q h , as follows. Let T 0 = T . Recursively define and Let k be least such that α k = ∅. Then T k has no skewed out nodes in h Q h . Finally, let γ i := θ i ∩ P. Then Slide γ 1 ,...,γn (T k ) = U| P . The proof is exactly the same as before except C, V , T k , θ i , and γ i play the respective roles of T , U, C, γ i , and θ i . This completes the proof of (1). The proof for (2) is by induction on n. The base case n = 1 is the previously proven statement (1). For n > 1, let P ′ = P p 1 / (Q 1 1 , . . . , Q 1 m 1 ) and let S = P ′ p 2 / C 2 p 3 / . . . pn / C n . By the inductive hypothesis, there is a skew increasing S-tableau T S such that rects| P ′ (T S ) = rects| P ′ (T ), so furthermore by restriction we have rects| P (T S ) = rects| P (T ).
Definition 4.3. Let P be a poset and fix p ∈ P. Let U be a URT in P. Then U is a p-chain unique rectification target in P if U is a URT in P p / C for every chain poset C. More generally, U is a {p 1 , . . . , p n }-chain URT in P if U is a URT in P p 1 / C 1 p 2 / · · · pn / C n for all pairwise disjoint chains C 1 , . . . , C n .
Being a p-chain URT is a strictly stronger notion than being a URT. For an example of a URT that is not a p-chain URT, see Remark 3.4.
Proposition 4.4. Let R be the slant sum R := P p 1 / (Q 1 1 , . . . , Q 1 m 1 ) p 2 / · · · pn / (Q n 1 , . . . , Q n mn ), for p i distinct and Q j i all pairwise disjoint with minimum elements. Let T be a skew increasing R-tableau with rectifications U and V . If U| P is a {p 1 , .., p n }-chain URT in P, then U| P = V | P .
(1) Let R be the slant sum P p / Q and let U be an increasing R-tableau of straight shape. Suppose A ⊆ P and B ⊆ Q. If p ∈ A and U| P is an A-chain URT in P and U| Q is a B-chain URT in Q, then U is an A ∪ B-chain URT in R.
Let C be a poset formed by taking R and slant summing chains on top of elements in A ∪ B. We must show that U is a URT in C. Hence, suppose some skew increasing C-tableau C rectifies to U and V . Then, we must show U = V . Since U is an A-chain URT in P, by Proposition 4.4, we have that U| P = V | P .
It is easy to see that since U and V agree on P, they must also agree on any chain C a slant summed onto an element a ∈ A ⊆ P. This is since, in any such chain, the labels of C a in U and V must be exactly those labels of C a in T that have values greater than U(a) = V (a). By increasingness of U and V , these are necessarily written in increasing order along C a in both tableaux. Thus, U| Ca = V | Ca .
Let Q C be the principal order filter of C generated by0 Q . Since Q C is a funnel in C, we may consider the tableaux (T → U)| Q C and (T → V )| Q C . By Definition 3.6, we have (T → U)| Q C = (T → V )| Q C , since both tableaux defined to the the restriction of T to the set where the second equality is by recalling U(p) = V (p) (since p ∈ P) and noting that p is the only element of C covered by0 Q . By Proposition 3.7, Thus, we have shown that U(c) = V (c) for all c ∈ C, so U = V , as desired.
The following result follows inductively from Corollary 3.11; we, however, prove it here as a useful demonstration of working with p-chain URTs in preparation for more sophisticated uses later.
Corollary 4.6. Let T be a tree. Let U be any increasing T -tableau of straight shape. Then U is a URT in T .
Proof. Let n = |T |. Define T 1 ⊆ · · · ⊆ T n such that for all i, T i is an order ideal of T and |T i | = i. In particular, we have T 1 = {0 T } and T n = T . We claim that for all i, U| T i is a T i -chain URT in T i . We work by induction on i. First, we note that in any singleton poset P, every increasing P-tableau of straight shape is a P-chain URT by Lemma 3.10. Thus, since |T 1 | = 1, U| T 1 is a T 1 -chain URT in T 1 . Now suppose i > 1. Let t be the unique element in T i \ T i−1 and let p be the unique parent of t in T . Then T i = T i−1 p / {t}. By the inductive hypothesis, U| T i−1 is a T i−1 -chain URT in T i−1 . Because |{t}| = 1, U| {p} is a {t}-chain URT in {t}. Thus by Proposition 4.5, U| T i is a T i -chain URT in T i . This completes the induction. Hence U| Tn = T is a URT in T n = T .
Trees are a particularly simple subfamily of the d-complete posets studied in this paper. Corollary 4.6 should be understood as a particularly strong version of Theorem 1.2 for this special subfamily.

Double-tailed diamonds
In this section, we investigate the p-chain unique rectification targets of certain posets, called double-tailed diamonds. This special family of d-complete posets plays a central role in the study of general d-complete posets. We will apply the results developed here to the general case in Section 6.
For k ≥ 3, a double-tailed diamond D(k) has 2k − 2 elements, two of which are incomparable elements in the middle with chains of size k − 2 above and below them. Figure 2 illustrates the Hasse diagrams of some of these posets. It is easy to work out that any increasing tableau on any order ideal of a double-tailed diamond is a URT. (This is even explicitly observed in [BS16, Proof of Theorem 3.12].) For application in Section 6, we need to strengthen this observation to the setting of p-chain URTs. To study the p-chain URTs of double-tailed diamonds, we introduce a chained double-tailed diamond. A chained double-tailed diamond is formed by slant summing a chain onto each of the two middle elements of the double-tailed diamond. We index the elements of a chained double-tailed diamond as shown in Figure 3. We refer to the set of elements indexed as ℓ k as the left chain of the poset, and those indexed as r k as the right chain. In this notation, a chained double-tailed diamond corresponds to a triple of positive integers m, n, p ≥ 1. We denote the chained double-tailed diamond for (m, n, p) by D(m, n, p). In particular, D(k) = D(1, k, 1).
Proposition 5.1. Let T be a skew increasing D(m, n, p)-tableau. Then, T rectifies uniquely.
Proof. Suppose T has shape ν/λ. If {ℓ 1 , r 1 } ⊆ λ, then there are no choices to be made during rectification and hence T rectifies uniquely. Thus, assume {ℓ 1 , r 1 } ⊆ λ. We can repeatedly perform slides on inner corners not equal to ℓ 1 or r 1 because all the descendants of ℓ 1 and r 1 are in disjoint chains and hence these slides clearly commute. Thus, we may assume T has exactly two inner corners ℓ 1 and r 1 . Write I := {ℓ 1 , r 1 }. Let s ∈ Z be largest such that t s ∈ ν (set s = 0 if t 1 / ∈ ν). We induct on s. If s = 0, then ν is a tree, and so T rectifies uniquely by Corollary 4.6. Assume s ≥ 1 and that the proposition holds for smaller s. (Case 1: T (t 1 ) = min Range(T )): Without loss of generality, we may assume T (ℓ 2 ) < T (t 1 ). There are three possibilities for γ ⊆ I: either γ = {ℓ 1 }, γ = {r 1 }, or γ = {ℓ 1 , r 1 }. If we choose γ = {ℓ 1 }, then after Slide {ℓ 1 } is completed, r 1 will be the unique inner corner; thus, the following slide is necessarily at r 1 . Similarly, if we choose γ = {r 1 }, the next slide is necessarily at ℓ 1 . By routine case analysis, one checks that in any of the various cases: T (t 1 ) < T (r 2 ), T (t 1 ) = T (r 2 ), or T (t 1 ) > T (r 2 ). Thus, any choice made at this first step of rectifying T yields the same tableau after one or two slides. That latter tableau has a unique rectification, as there are no further choices to be made. Hence, T rectifies uniquely.
Consider any rectifications U and V of T . By Lemma 3.10, for all k we have U(b k ) = V (b k ) is the (n − k + 1)st smallest element of Range(T ). Thus, since T (t 1 ) = min Range(T ) by assumption, we have U(b n ) = T (t 1 ). Hence, t 1 / ∈ Dom(R) and so U looks like: . . .

R(r p )
Consulting these pictures, one observes that and similarly for V . Since U(b 1 ) = V (b 1 ) by Lemma 3.10, this means U(ℓ 1 ) = V (ℓ 1 ). Clearly, the labels of U in the left chain of D(m, n, p) are exactly the labels on the left chain of D(m, n, p) in T that are at least U(ℓ 1 ) written in increasing order. Since the same is true for V , we have U(ℓ q ) = V (ℓ q ) for all q. The same argument shows U(r q ) = V (r q ). Thus, U = V and T rectifies uniquely.
(Case 2.2: s ≥ 2): Let U, V be rectifications of T . As in Case 2.1, we have U(b n ) = V (b n ) = T (t 1 ). Let Q := D(m, n, p) \ {b n }. We must show U| Q = V | Q . Since n ≥ s ≥ 2, Q has a minimum and is a funnel of D(m, n, p). Then by Proposition 3.7, U| Q is a rectification of S U := (T → U)| Q and V | Q is a rectification of S V := (T → V )| Q . Since U(b n ) = V (b n ) = T (t 1 ), it follows from the definition of corresponding tableaux that S U = S V . In fact, S U and S V are merely T restricted by deleting all labels of value T (t 1 ). Hence, write S := S U = S V . It remains to show that S rectifies uniquely in Q.
Since S is T restricted by deleting all labels of value T (t 1 ) = min Range(T ) and the inner corners of T are exactly {ℓ 1 , r 1 }, it follows that the inner corners of S are exactly those elements q ∈ Q with T (q) = T (t 1 ). The structure of Q ensures that ℓ 2 and r 2 are the only two nodes q besides t 1 that could possibly have the label T (t 1 ). Let J ⊆ {t 1 , ℓ 2 , r r } be the set of inner corners of S. Clearly, since the various slides only affect disjoint chains, for any set partitions (γ 1 , . . . , γ h ) and (δ 1 , . . . , δ k ) of J, we have Slide γ h • · · · • Slide γ 1 (T ) = Slide δ k • · · · • Slide δ 1 (T ).
Hence, without loss of generality, we may assume that we perform Slide {t 1 } first. That is, set S ′ := Slide {t 1 } (S) and observe that Rects(S ′ ) = Rects(S).
Finally, we must show that S ′ rectifies uniquely. Let the shape of S ′ be η/θ. Recall, s is defined to be the largest integer with t s ∈ Dom(T ), so by the construction of S, we also have that s is the largest integer with t s ∈ Dom(S). Hence, since S ′ := Slide {t 1 } (S), we have that t s / ∈ Dom(S ′ ). Since s ≤ n, this ensures that η is an order ideal of where the first equality is by the definition of Q and the second equality follows from n ≥ s ≥ 2. Hence, S ′ is a skew increasing D(m, n − 1, p)-tableau. Moreover, the largest i such that S ′ (t i ) is defined is s − 1, so by the inductive hypothesis, S ′ rectifies uniquely in D(m, n − 1, p). Thus, S rectifies uniquely in Q, and so W | Q is the same for all rectifications W of T , so T rectifies uniquely.
Corollary 5.2. Every increasing D(m, n, p)-tableau of straight shape is a URT.
Proof. Immediate from Proposition 5.1. In this section, we recall the definition of d-complete posets following [Pro99a], and prove our main result Theorem 1.2 regarding slant sum trees of minuscule posets. (We use, however, the convention of [Pro99b] regarding the orientation of our posets; the paper [Pro99a] uses the opposite convention, so the posets in [Pro99a] are the duals of ours.) The proofs presented in this section are all straightforward, relying on the technical results of the previous sections. We also develop appropriate terminology here to give precise interpretations of Conjectures 1.1 and 1.3. If x, y ∈ P, the interval [x, y] is the set {z ∈ P : x ≤ z ≤ y}. We call an interval [x, y] in P a Q-interval if it is isomorphic to the poset Q. We will be especially interested in D(k)-intervals (k ≥ 3). Let D 0 (k) := D(k) \ {t}, where t is the minimal element of D(k). We will also be interested in D 0 (k)-intervals (k ≥ 4). Examples of D 0 (k)-intervals are shown in Figure 4; the corresponding posets D(k) are shown in Figure 2.  Definition 6.1. A poset P is D(3)-complete if it satisfies the following three conditions: (1) anytime an element z covers two distinct elements x and y, there exists a fourth element w that x and y both cover; (2) if [w, z] is a D(3)-interval in P with elements {w, x, y, z}, then w is only covered by x and y in P; and (3) in such a D(3)-interval, there is no w ′ = w that both x and y cover.
is also a D 0 (k)-interval, then we say that [x, z] and [x ′ , z] overlap. Definition 6.2. For any k ≥ 4, a poset P is D(k)-complete if it satisfies the following three conditions: (1) there are no incomplete D 0 (k)-intervals; (2) if [w, z] is a D(k)-interval, then w is covered by only one element in P; and (3) there are no overlapping D 0 (k)-intervals. A poset P is d-complete if it is D(k)-complete for every k ≥ 3.
Briefly, the algebraic context of d-complete posets is as follows. (For further details, see [Ste96,Pro99b,CP12].) Let Λ be a dominant integral weight of a Kac-Moody Lie algebra g with (generally infinite) Weyl group W . The Weyl group element w ∈ W is called Λ-minuscule if it can be written as a reduced word in the simple reflections as w = s i 1 s i 2 · · · s i ℓ , so that for all j (s i j+1 · · · s i ℓ − s i j · · · s i ℓ )Λ = α i j , where α i j is the simple root for s i j . (In fact, this property is independent of the choice of reduced word [Ste01, Proposition 2.1].) Now, if w is Λ-minuscule, then the interval [id, w] in Bruhat order is a distributive lattice. A poset P is d-complete if and only if it is isomorphic to the poset of join irreducibles of such a 'Λ-minuscule distributive lattice'; equivalently, a poset Q is isomorphic to a Bruhat interval [id, w] for some Λ-minuscule w if and only if Q is isomorphic to the poset of order ideals of a d-complete poset. Since Bruhat order on W also describes containment of Schubert varieties in the Kac-Moody homogeneous space X = G/B, we have for u, v ≤ w all Λ-minuscule that the inclusion of Schubert varieties X u ⊆ X v is equivalent to the reverse inclusion λ v ⊆ λ u of the corresponding order ideals in the d-complete poset for w. In addition to their algebraic relations, d-complete posets enjoy a number of beautiful combinatorial properties, including an analogue of the classical hook-length formula (for a full proof of this fact, see [KY17]). Figure 1 shows an example of a reasonably large d-complete poset.
Say a d-complete poset is irreducible if it is not the slant sum of two d-complete posets. R. Proctor [Pro99a] showed that all d-complete posets can be uniquely decomposed as a slant sum of irreducible d-complete components. In this decomposition, irreducible components are only slant summed onto special nodes of other irreducible components, called acyclic nodes [Pro99a]; that is, if P = Q q / R is d-complete and R is irreducible, then q is an acyclic node of its irreducible component. (We avoid giving the somewhat technical definition of acyclic nodes, as it is sufficient for our purposes to use Proctor's explicit identification [Pro99a] of all acyclic nodes of all irreducible d-complete posets.) The irreducible d-complete posets are classified into 15 (mostly infinite) families; we follow Proctor's numbering and naming conventions for these families from [Pro99a]. Of these 15 families, only the components from families 1-9 and 11 have any acyclic nodes.
For a poset P, we say an increasing P-tableau T of straight shape λ ⊆ P is minimally-labeled if it is minimal among all increasing P-tableaux of shape λ under nodewise comparison of labels; that is, if U is another increasing tableau of shape λ, then U(x) ≥ T (x) for all x ∈ λ. It is easy to see that there exists a unique minimally-labeled P-tableau of each straight shape λ. We write M λ for this unique tableau. The precise version of Conjecture 1.1 is the following. Conjecture 6.3. Let P be d-complete and let λ ⊆ P be an order ideal. Then, the minimally-labeled increasing P-tableau M λ of shape λ is a unique rectification target.
In light of the slant sum structure of d-complete posets, Conjecture 6.3 would follow from Proposition 4.5 together with information about (p-chain) URTs in the 15 families of irreducible d-complete posets. Specifically, it remains to show that • for each irreducible d-complete poset Q with acyclic nodes, that M λ is a pchain URT for each order ideal λ ⊆ Q and each acyclic node p ∈ Q, and that • for each irreducible d-complete poset Q without acyclic nodes, that M λ is a URT for each order ideal λ ⊆ Q. Unfortunately, we are unable to establish the necessary results for some of these families; hence, we can only leverage Proposition 4.5 to prove a weaker version of Conjecture 6.3, namely Theorem 1.2. First, we recall the minuscule posets, a special subset of d-complete posets. Except for some trivial instances, all minuscule posets are irreducible.
Algebraically, one obtains the minuscule posets as follows. Suppose the Kac-Moody group G is in fact complex reductive. Put a partial order on the positive roots Φ + of G by taking the transitive closure of the covering relation α⋖β if and only if β −α is a simple root. The simple root δ is a minuscule root if for every positive root α ∈ Φ + , the multiplicity of δ ∨ in the simple coroot expansion of α ∨ is at most 1. For each minuscule root, one obtains a corresponding minuscule poset P δ by restricting the partial order on Φ + to those positive roots that use δ in their simple root expansion. There is also a corresponding minuscule variety obtained as the quotient G/P δ , where P δ is the maximal parabolic subgroup associated to the minuscule root δ. The minuscule poset P δ encodes the Schubert stratification of G/P δ ; specifically, the Schubert varieties are naturally indexed by the order ideals of P δ , and inclusions of order ideals correspond to reverse inclusions of Schubert varieties.  Table 1, with examples shown in Figure 5. One infinite family of minuscule posets is the rectangles; combinatorially, these are the products C i × C j of two chain posets. Another infinite family is the double-tailed diamonds studied in Section 5. The final infinite family is the shifted staircases; identifying the chain C i with the natural order on {1, . . . , i}, shifted staircases are of the form with the order structure restricted from C i × C i . For convenience, we will assume that shifted staircases have at least 10 nodes, as the smaller shifted staircases coincide with small rectangles/double-tailed diamonds. Lastly, for the definitions of the exceptional Cayley-Moufang swivel and bat, see their Hasse diagrams depicted in the second row of Figure 5. The acyclic nodes of the minuscule posets are also shown in Figure 5; we will use the indexing of these nodes as L and R, as in that figure.
We will only use the following proposition in the case k = 1 of rectangles; however, for possible future use, we note that it is equally true for four of Proctor's other Lemma 6.4. Let k ∈ {1, 3, 5, 6, 7}. Let P be an irreducible d-complete poset from family k and let A ⊆ P be the set of all acyclic nodes in P. If a straight-shaped increasing P-tableau U is a URT for all posets in family k, then U is a A-chain URT in all such posets.
Proof. Suppose A = {a 1 , ..., a k }. Let i 1 , ..., i k be arbitrary positive integers. Let R be the iterated slant sum P a 1 / C i 1 a 2 / . . . a k / C i k of P with a collection of chains. Observe that R is an order ideal of a larger poset in the same irreducible family. Thus, U is a URT in R, as desired.
Theorem 6.5 ( [BS16]). Let P be a minuscule poset. Then, for every order ideal λ ⊆ P, the minimally-labeled increasing P-tableau M λ of shape λ is a URT in P.
Corollary 6.6. Let P be a rectangle. Let M λ be an minimally-labeled P-tableau of straight shape. Then, M λ is an {L, R}-chain URT in P.
Proof. This follows from Lemma 6.4 and Theorem 6.5.
Corollary 6.7. Let P be a shifted staircase with at least 10 nodes. If M λ is a minimally-labeled P-tableau of straight shape, then M λ is an {R}-chain URT in P.
Proof. If S is the slant sum P R / C j of P with a chain, then S is an order ideal of a larger shifted staircase, in which minimally-labeled tableaux are URTs by Theorem 6.5.
In order to state the following, we adopt the convention that an ∅-chain URT in P is just a URT in P.
Proposition 6.8. Let P be a minuscule poset. Let A be the set of acyclic nodes in P. Let M λ be a minimally-labeled increasing P-tableau of straight shape. Then, M λ is an A-chain URT in P.
Proof. If P is the Cayley-Moufang swivel or the bat, then it has no acyclic nodes, so A = ∅. Hence, in these cases, it suffices to verify that M λ is a URT in P. This fact is a special case of Theorem 6.5.
If P is a rectangle, then A = {L, R}, and M λ is a {L, R}-chain URT in P by Corollary 6.6. If P is a double-tailed diamond, then A = {L, R}, and M λ is a {L, R}chain URT in P by Corollary 5.3. Finally, if P is a shifted staircase with at least 10 nodes, then M λ is an A-chain URT in P by Corollary 6.7.
Proposition 4.5 allows us to extend Proposition 6.8 to show that minimally-labeled tableaux are unique rectification targets in iterated slant sums of minuscule posets. Theorem 6.9. Let P be a d-complete poset. If P is an iterated slant sum of minuscule posets, then all minimally-labeled increasing P-tableaux of straight shape are unique rectification targets.
Proof. We prove the stronger statement that all minimally-labeled increasing Ptableaux of straight shape are A-chain URTs in P, where A denotes the set of acyclic nodes in P. We induct on the number n of irreducible components in the slant sum decomposition of P. For Q an irreducible component of P, write A Q for the set of acyclic nodes of Q.
The base case, n = 1 is provided by Proposition 6.8.
Otherwise, P is the slant sum of irreducible components. One of these components contains the minimum0 P ; call this component M. By R. Proctor's classification of acyclic nodes [Pro99a], M has at most two acyclic nodes. Then, where L and R are the acyclic nodes of M (if L or R is not an acyclic node, then we have ℓ = 0 or r = 0 respectively), and L 1 , . . . , L ℓ and R 1 , . . . , R r are disjoint d-complete posets that are slant sum trees of minuscule components. Note that each R i and L j is a slant sum of strictly fewer than n irreducible components.
Suppose T is a minimally-labeled increasing P-tableau of straight shape. Then, T | R i is a minimally-labeled R i -tableau of straight shape (modulo shifting the alphabet), so by the inductive hypothesis, T | R i is an A R i -chain URT in R i for all i. Similarly, T | L i is an A L i -chain URT in L i for all i. Finally, T | M is a minimally-labeled M-tableau of straight shape, so by the inductive hypothesis it is A M -chain URT in M. Thus, by Proposition 4.5, we have that T is an A-chain URT in P, where A is the set of acyclic nodes in P.
The following is the precise version of Theorem 1.2.
Corollary 6.10. Let P be a d-complete poset. If P is an iterated slant sum of minuscule posets and Q ⊆ P is an order ideal, then all minimally-labeled increasing Q-tableaux of straight shape are unique rectification targets.
Proof. Let M λ be a minimally-labeled increasing Q-tableau of straight shape. Since Q is an order ideal of P, M λ is also a minimally labeled increasing P-tableau of straight shape. Hence by Theorem 6.9, M λ is a unique rectification target in P, so it is a unique rectification target in Q.
Finally, we recall the construction necessary to make precise sense of Conjecture 1.3. Let P be any poset satisfying the conclusion of Conjecture 6.3. Then, as in [BS16, §3.5], we construct a combinatorial K-theory ring associated to P. Let K(P) be the free abelian group on the set of order ideals of P. Define a product structure on K(P) by setting where the Greek letters denote order ideals of P and t ν λ,µ is defined to be (−1) |ν|−|λ|−|µ| times the number of skew increasing P-tableaux of shape ν/λ that rectify to the minimally-labeled tableau M µ . (Since M µ is by hypothesis a URT in P, this number is well-defined.) By [BS16, Proposition 3.17], this product structure makes K(P) into a commutative associative algebra with the empty order ideal as multiplicative identity. Conjecture 1.3 claims then that, when P is d-complete, the structure constants of the algebra K(P) coincide with corresponding Λ-minuscule Schubert structure constants of the K-theory ring K(X), where X = G/P is a Kac-Moody homogeneous space, w ∈ W P is a Λ-minuscule Weyl group element for P , and P is the poset of join irreducibles of the distributive lattice [id, w].