Saturation for Flagged Skew Littlewood-Richardson Coefficients

We define and study a generalization of the Littlewood-Richardson (LR) coefficients, which we call the flagged skew LR coefficients. These subsume several previously studied extensions of the LR coefficients. We establish the saturation property for these coefficients, generalizing work of Knutson-Tao and Kushwaha-Raghavan-Viswanath.


INTRODUCTION
The Littlewood-Richardson (LR) coefficients are among the most celebrated numbers in algebraic combinatorics.They are the multiplicities in the decomposition into irreducibles of tensor products of irreducible polynomial representations of general linear groups.As such, they are the structure constants for the multiplication of Schur polynomials (which form a basis for the ring of symmetric polynomials).These coefficients also determine the branching of irreducible representations of symmetric groups on restriction to Young subgroups.They also come up in several other places.For example, they occur as structure constants for the multiplication of Schubert cohomology classes of Grassmanians.
Several generalizations of these coefficients can be found in the literature: e.g., [Zel81], [RS95a], [KRV21].Our broader goal is to investigate, for some of these generalizations, the analogue of the saturation theorem of Knutson-Tao for the LR coefficients.In this paper, we consider a simultaneous generalization of Zelevinsky's skew LR coefficients [Zel81] and the flagged LR coefficients of [KRV21].We prove that these flagged skew LR coefficients exhibit the saturation property.
In order to do this, we need to lift the main key-positivity result of [RS95a] to crystals; we do this using a recent result of Assaf [ADG22].We also give a hive-like model for the flagged skew LR coefficients.Although a skew Gelfand-Tsetlin (GT) polytope need not be isomorphic to a GT polytope, it turns out that every flagged skew hive polytope is isomorphic to some flagged hive polytope.4 1 3 2 3 1 2 FIGURE 1.A skew tableau of shape (4, 3, 2, 1)/(2, 1), weight (2, 2, 2, 1) and reverse-row reading word 2132314.

PRELIMINARIES AND STATEMENTS OF MAIN THEOREMS
A partition λ = (λ 1 , λ 2 , • • • ) is a weakly decreasing sequence of non-negative integers with finitely many non-zero terms (or parts).The length of the partition λ is defined to be the largest integer i such that λ i is non-zero and we denote it by l(λ).The weight of λ is the sum of its parts and we denote it by |λ|.We denote by P[n] the set of all partitions of length at most n.The Young diagram of the partition λ is the left and top justified collection of boxes such that row i contains λ i boxes.We denote it again by λ.For Young diagrams λ and µ such that λ ⊃ µ (i.e., λ i ≥ µ i ∀i), the skew diagram λ/µ is obtained by removing the boxes of µ from those of λ.
Given partitions λ ⊃ µ, a semi-standard skew tableau of shape λ/µ is a filling of the skew diagram λ/µ that is weakly increasing along the rows (from left to right) and strictly increasing along the columns (from top to bottom).A semi-standard tableau of shape λ is just a semi-standard skew tableau of shape λ/empty .We denote by Tab(λ/µ) the set of all semi-standard skew tableaux of shape λ/µ and Tab(λ/µ, n) the subset of Tab(λ/µ) where the fillings are all ≤ n.It will be convenient to let Tab(λ/µ) be the empty set if λ, µ are partitions with λ ⊃ µ.The weight of a tableau T ∈ Tab(λ/µ, n) is defined as wt(T ) = (t 1 , t 2 , • • • , t n ), where t i is the number of times i occurs in T .A standard (skew) tableau T is a semi-standard (skew) tableau of the same shape in which 1, 2, • • • , k appears exactly once, where k is the number of boxes in T .For T ∈ Tab(λ/µ), we write b T to denote the reverse-row reading word of T which is the word obtained by reading T right to left and from top to bottom.
Fix n a positive integer.A flag Φ = (Φ 1 , Φ 2 , • • • , Φ n ) is a weakly increasing sequence of positive integers such that 1 Φ n = n.For λ, µ ∈ P[n] and flag Φ = (Φ 1 , Φ 2 , • • • , Φ n ), Tab(λ/µ, Φ) is the set of all elements T in Tab(λ/µ) such that the entries in row i of T are at most Φ i for 1 ≤ i ≤ n.Following Reiner-Shimozono [RS95a], we define the flagged skew Schur polynomial where T varies over Tab(λ/µ, Φ) and for t ), these reduce to the skew Schur polynomials 1 In the literature, a flag need not have Φn = n, but for our purposes it is sufficient to consider only such flags.
When µ is the empty partition, they become the flagged Schur polynomials s λ (X Φ ), which coincide with key polynomials corresponding to 312-avoiding permutations [PS09, theorem 14.1].The flagged skew Schur polynomials s λ/µ (X Φ ) also have a representation theoretic interpretation as characters of certain Borel modules called flagged Schur modules [RS98].
For 1 ≤ i ≤ n − 1, define the Demazure operator T i on the ring of polynomials in the variables x 1 , x 2 • • • , x n as follows: For w ∈ S n (the symmetric group), we define a reduced expression for w.This is well-defined because the T i 's satisfy the braid relations.For α ∈ Z n + , let α † be the partition obtained by sorting the parts of α in descending order and let ω ∈ S n be any permutation such that ωα † = α (here, the action of ω is the usual left action of S n on n-tuples).We recall that the key polynomial κ α is defined to be κ α = T ω (x α † ), and that this is independent of the choice of ω.A polynomial f is said to be key-positive if it is a sum of key polynomials.If f is key-positive, then x λ f and T w (f ) are also key-positive, for all λ ∈ P[n] and all w ∈ S n .The former follows from a theorem of Joseph [Jos03] and the latter from the fact that a composition of Demazure operators is itself a Demazure operator.Theorem 20] showed that the flagged skew Schur polynomial s λ/µ (X Φ ) is key-positive.Now, if w 0 denotes the longest element of S n , we have the Schur polynomial indexed by α † .More generally, (since the key polynomials form a Z-basis of the polynomial ring in n variables [RS95a, corollary 7]) given any polynomial we have that T w 0 (f ) is a symmetric polynomial, which can therefore be expanded in the basis of Schur polynomials.If further f is key-positive, then equation (1) shows that T w 0 (f ) is Schur-positive, i.e., a sum of Schur polynomials.This leads us to the main objects of our study.
By the remarks preceding equation (1), it follows that the LHS of (2) is Schur positive, and thus the flagged skew Littlewood-Richardson coefficients are non-negative integers.It is clear by definition that these coefficients are zero if µ ⊃ γ.It will follow from Theorem 1 below that they are also zero if ν ⊃ λ.
If we set λ = (0, . . ., 0), we have where the sum runs over all compositions α that are obtained by permuting the parts of ν.The coefficients on the right are the ones which appear in the flagged Littlewood-Richardson expansion of [RS95a, section 7].
Our first result provides two combinatorial models for flagged skew LR coefficients (see §3, 4 for undefined terms) that generalize those for LR coefficients: is the number of the integral points of the flagged skew hive polytope SHive(λ, µ, γ, ν, Φ).
Our proof of Theorem 1 hinges on understanding the crystal structure on the set of flagged skew tableaux Tab(µ/γ, Φ).We show in particular that this set is a disjoint union of Demazure crystals; this lifts the key-positivity result of Reiner-Shimozono [RS95a, Theorem 20] from the level of characters to that of crystals.

THE CRYSTAL Tab(µ/γ, Φ)
The purpose of this section is to prove that the subset Tab(µ/γ, Φ) of the type A n−1 crystal Tab(µ/γ, n) is a disjoint union of Demazure crystals.This is the key step in proving the first part of theorem 1 which we will see at the end of this section.By a crystal of type A n−1 (see section 2.2 of [BS17]), we mean a finite and non-empty set B together with maps e i , f i : B → B ⊔ {0} wt : B → Z n where i ∈ {1, 2, • • • , n − 1} and 0 ∈ B is an auxiliary element, satisfying the following conditions: (1) If x, y ∈ B then e i (x) = y if and only if f i (y) = x.In this case it is assumed that where ǫ 1 , ǫ 2 , • • • , ǫ n are the standard orthonormal vectors in R n .
(2) For all x ∈ B and i ∈ {1, 2, • • • , n − 1} we require that where ε i (x) (resp.φ i (y)) is the maximum number of times e i (resp.f i ) can be applied to x (resp.y) without making it 0.
The maps e i and f i are called the raising and lowering operators respectively.

Example: The
Also, wt(i) = ǫ i ∈ Z n .This crystal can be depicted by the following "crystal graph": The crystal graph associated to a type A n−1 crystal C is an edge-coloured (the colours being 1, 2, • • • n − 1) directed graph whose vertex set is the underlying set of the crystal.An edge with colour k originates from x ∈ C and terminates at y ∈ C if and only if f k (x) = y.We say a crystal is connected if its crystal graph is connected (viewed as an undirected graph).
A subset C ′ of a crystal C which is a union of connected components of C inherits a crystal structure from that of C. In this case, we call C ′ a full-subcrystal of C.

Tensor Product of Crystals.
If B and C are type A n−1 crystals, there is a natural notion of the tensor product crystal B ⊗ C. As a set it is {x ⊗ y : x ∈ B, y ∈ C} (where x ⊗ y is just a symbol).We define wt(x ⊗ y) to be wt(x) + wt(y).The raising and lowering operators are defined as follows: and We define the character of a crystal C to be ch(C) := u∈C x wt (u) .From the definition of the tensor product crystal, it is elementary to observe that the character of the tensor product is equal to the product of the characters.The tensor product is associative2 -see [BS17, §2.3 and remark 1.1].
The set B ⊗k is usually called the crystal of words (of length k).An element ζ of B ⊗k is said to be a dominant word if e i ζ = 0 for all i.
by the following embedding into B ⊗k (where b T is the reverse-row reading word of T , defined in §2): The image under this embedding is a full-subcrystal of B ⊗k (see [BS17, Section 3.1]).

Demazure Crystals. Let λ ∈ P[n]
and ω a permutation in the symmetric group S n .For any reduced expression where T 0 λ is the unique dominant tableau of shape λ (i.e., with shape and weight both equal to λ).
Remark 2. For any ω ∈ S n and λ ∈ P[n], T 0 λ is the unique element in B ω (λ) such that e i (T 0 λ ) = 0 for all i.In (5) above, we could replace T 0 λ with any other dominant word b λ ∈ B ⊗|λ| of weight λ.We thereby obtain a subset of B ⊗|λ| : which is isomorphic to B ω (λ) as crystals, i.e., there is a weight-preserving bijection between these sets which intertwines the crystal raising and lowering operators (where defined).We also refer to B ω (b λ ) as a Demazure crystal in what follows, and write (by abuse of notation) The following proposition is the refined Demazure character formula in [Kas93]. Examples: We will denote this Demazure crystal by B k .
A subset S of a crystal C is said to have the string property if ∀i ∈ {1, 2, • • • , n − 1} and ∀x ∈ S such that e i (x) = 0, we have Remark 3. The converse of proposition 2 is not true in general (see [ADG22]).
A characterization of when a tensor product of Demazure crystals decomposes into Demazure crystals was given in [Kou20].Following [Kou20], a different characterization was obtained in [ADG22] as follows: Subsets of crystals that exhibit the string property are referred to as extremal in [ADG22].The following proposition is a strengthening of [ADG22, Proposition 8.1].
, for each i, X 2 contains a head of some i-string).If X 1 ⊗ X 2 has the string property, then X 1 has the string property.
Therefore by the string property of Proof.We prove by induction on k.The case k = 1 is straight forward.Suppose k > 1.Then it follows from proposition 3 that Let x⊗y ∈ D⊗D k such that e i (x⊗y) = 0 and f i (x⊗y) = 0.It follows that f i (x⊗y) ∈ D⊗D k because e i (f i (x ⊗ y)) = x ⊗ y and the fact that the decomposition Let Φ be a flag and ρ ∈ Z n + be a composition of k ∈ Z + (i.e., Suppose that e i (u) = 0. Then e i (u) ∈ B ρ Φ , because of proposition 2 and the fact that B Φ i 's are Demazure crystals.Now suppose furthermore that f i (u) = 0. Let t be the index where e i acts in u. i.e., Theorem 4. Let Φ be a flag and µ, γ ∈ P[n] such that µ ⊂ γ.Then Tab(µ/γ, Φ) is a disjoint union of Demazure crystals.
Proof.Define the composition ρ by Tab(µ/γ, n) is a full subcrystal of B ⊗k , the theorem follows from lemma 1 because [RS95a] proves the above theorem at the character level (i.e., the key positivity of s µ/γ (Φ)).Theorem 4 is however implicit in [RS95a] and allows us to compute the explicit decomposition of Tab(λ/µ, Φ) into Demazure crystals.A short sketch is deferred to the appendix.
Remark 5.It is important that we assume Φ to be weakly increasing.For example, if µ = (3, 2), γ = (1, 0) and Φ = (3, 2), then Tab(µ/γ, Φ) does not even have the string property.This fact also follows from theorem 3 because Tab(λ, Φ 0 ) ⊗ D has the string property as we show below: ).This implies that ε i (v) > φ i (u).By the tensor product rule we therefore have We now prove the first part of theorem 1.
Proof.By Proposition 4 (and Remark 2), for all ν ∈ P[n] there exists a multi-subset Taking characters, we obtain  6), each B w (ν) has a unique element ξ such that e i ξ = 0 for all i; this element has weight ν.Thus, the number of elements ζ as above is also equal to |W(ν)|.Putting all these together establishes Theorem 5.

A HIVE MODEL FOR FLAGGED SKEW LITTLEWOOD-RICHARDSON COEFFICIENTS
In this section we define the skew hive polytope and its faces corresponding to flags Φ.We then prove the second part of theorem 1. 4.1.Skew GT patterns.Given m, n ≥ 1, a skew Gelfand-Tsetlin pattern is an array of real numbers {x ij : 0 ≤ i ≤ m, 1 ≤ j ≤ n} satisfying the following inequalities: The above inequalities simply mean that the consecutive rows interlace.Hence we arrange the rows in the shape of a parallelogram as follows (shown for m = n = 4): x 01 x 02 x 03 x 04 x 11 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 34 x 41 x 42 x 43 x 44 For µ, γ ∈ P[n], such that γ ⊂ µ, the skew Gelfand-Tsetlin polytope GT(µ/γ, m) is the set of all skew Gelfand-Tsetlin patterns (x ij ) with 0 ≤ i ≤ m, 1 ≤ j ≤ n satisfying x 0j = γ j , x mj = µ j for all j.Define, In the sequel, we will only have occasion to consider the case when m = n.Consider the map The skew GT pattern on the left maps to the skew tableau on the right under the map Υ.
well-known -see for instance [Lou,§3] (whose pattern drawing convention differs from ours by a vertical flip).
Lemma 2. The map Υ is a bijection.

4.2.
Flagged skew GT patterns.We keep the notation of the previous subsection, but in addition assume that we are given a flag Φ = (Φ 1 , • • • , Φ n ).Define the set of flagged skew GT patterns: and let GT Z (µ/γ, Φ) denote the set of integer points in this polytope.We have: Lemma 3. The map Υ restricts to a bijection between GT Z (µ/γ, Φ) and Tab(µ/γ, Φ).
Otherwise let k be the maximum such that Φ k = n.Then the number of i (Φ j < i ≤ n) that appear in the j th row (1 ≤ j ≤ k) of Υ(X) is x ij − x (i−1)j = 0. Since ∀j > k, Φ j = n so for those j in j th row of Υ(X) all entries are ≤ Φ j (= n).Thus Υ(X) ∈ Tab(µ/γ, Φ).Also, if T ∈ Tab(µ/γ, Φ) then (i, j) th entry of Υ −1 (T ) is the number of entries ≤ i that appear in the j th row of T .So Υ −1 (T ) ∈ GT Z (µ/γ, Φ).
In the next two subsections, we give a hive model for the flagged skew Littlewood-Richardson coefficients.
4.3.Skew hives.The (n + 1) × (n + 1) array of nodes in figure 3 is called the n-hive parallelogram.Observe that the small rhombi3 in the n-hive parallelogram are oriented in the following three different ways:

Northeast (NE):
, Southeast (SE): and Vertical: .(2) The contents of all the small rhombi are non-negative.We recall that the content of a small rhombus is the sum of the labels on its obtuse-angled nodes minus the sum of the labels on its acute-angled nodes.
We denote the set of integer points in SHive(λ, µ, γ, ν) by SHive Z (λ, µ, γ, ν).An element of SHive(λ, µ, γ, ν) is called a skew hive with boundary (λ, µ, γ, ν).The rows of a skew hive h ∈ SHive(λ, µ, γ, ν) (from top to bottom) give a sequence of vectors h Consider the parallelogram array ∂h with (n + 1) rows (and n nodes in each row) whose Proof.We follow closely the proof of Proposition 4 in [KRV22].Let h ∈ SHive Z (λ, µ, γ, ν).By the discussion in the previous para and lemma 2, it is clear that T = Υ(∂h) ∈ Tab(µ/γ, n).Now we will show that T ∈ T ab ν λ (µ/γ, n).Let ∂h = (x ij ).Then x ij = h ij − h i(j−1) .So the number of times i appears in row j of T = x ij − x (i−1)j .Now we have to prove that b where b T k is the reverse reading word of the k-th row of T .Also, let N ik be the number of times i appears in the word b . Labelling of NE oriented rhombi in 4-hive parallegram and the shaded region is a typical configuration of R(Φ) (for Φ = (1, 2, 2, 4)) We will show that h ik = h ′ ik for all i by induction on k.Clearly, h 00 = h ′ 00 = 0 and 4.4.Flagged skew hives.Given a flag Φ, consider the face of the skew hive polytope defined by SHive(λ, µ, γ, ν, Φ) := ∂ −1 (GT(µ/γ, Φ)).We call this the flagged skew hive polytope.We let SHive Z (λ, µ, γ, ν, Φ) denote the set of integer points in this polytope.It is elementary to observe that SHive Z (λ, µ, γ, ν, Φ) = ∂ −1 (GT Z (µ/γ, Φ)).Lemma 3 and Theorem 6 together give us the second part of theorem 1: The NE rhombi of the n-hive parallelogram are labelled R ij with 1 ≤ i, j ≤ n as shown in the example in Figure 5.To the flag Φ, we associate the set forms a bottom-left-justified region of NE rhombi (shown in purple in Figure 5).It is easy to see from equation (7) that SHive Z (λ, µ, γ, ν, Φ) is obtained from SHive Z (λ, µ, γ, ν) by imposing the condition that all rhombi in R(Φ) are flat.For example, the hive in figure 4(b) is such that every NE oriented rhombus in R(Φ) is flat where Φ = (2,2,3,4).

FLAGGED SKEW LR COEFFICIENTS ARE w-REFINED LR COEFFICIENTS
In this section we prove that any skew hive polytope is affinely isomorphic to some hive polytope (albeit in twice as many ambient dimensions).Moreover, this isomorphism maps the flagged skew hive polytope to a hive Kogan face corresponding to some 312-avoiding permutation [KRV21, §2.4] and preserves the integral points.
The horizontal strings of nodes ("rows") of the triangular array are termed the zeroth row, first row, second row, etc starting from the top.Consider the labelling of the NE oriented small rhombi as shown in the example in figure 6.Given a flag Φ, consider the set of NE rhombi R(Φ) = {R ij | n > i ≥ Φ j }.We define the face Hive(α, β, γ, Φ) of the polytope Hive(α, β, γ) as the collection of those hives in which all the rhombi in R(Φ) are flat.As we vary Φ, these run over the hive Kogan faces corresponding to 312-avoiding permutations, in the terminology of [KRV21, §2.4].Refer [KRV21] for more details.Lemma 4. Let α, β, γ ∈ R n + be weakly decreasing sequences such that either α or β is a constant sequence.Then Hive(α, β, γ) is either empty or a singleton set.The latter is true if and only if  From the two rhombus inequalities in this picture, we conclude that if a − b = b − c, then each is also equal to e − d.Now β i = b for all i implies that the successive differences of labels on the bottom (i.e., n th ) row of h are all equal to b.The observation above means that the successive differences of labels on the (n − 1) th row are also all b.We proceed by induction, moving up the hive triangle, to conclude that the successive differences of labels along every row of h is equal to b.In particular, all labels of h are uniquely determined from those on left boundary alone.An additional compatibility condition arises from summing the differences in each row -this gives k i=1 κ i = kb for each k, or equivalently that κ = β as claimed.The proof for the case that α is constant is similar.One considers instead the trapezia of the form: a Then there exists an affine linear isomorphism between SHive(λ, µ, γ, ν) and Hive( λ, μ, ν).Moreover, the isomorphism preserves integral points and maps SHive(λ, µ, γ, ν, Φ) onto Hive( λ, μ, ν, Φ), where Proof.For h ∈ SHive(λ, µ, γ, ν) we describe ψ(h) as follows (see figure 7 for a representative example): (1) ψ(h) is a labelling of the (2n + 1)triangular array, with boundary labels coinciding with those of hives in Hive( λ, μ, ν).
(2) The bottom-left-justified (n + 1) × (n + 1) parallelogram in ψ(h) (white, with a red border in figure 7) is labelled by h+n•ν 1 , i.e., the labels of the nodes of h are translated by the constant n • ν 1 .(3) The labels of the top n rows of ψ(h) (highlighted in yellow in figure 7) are determined by the boundary conditions of Hive( λ, μ, ν) and the choice of parallelogram labels in (2) above.This follows from lemma 4, using the fact that the first n components of λ are equal (4) Again, using (2) and the fact that the last n components of μ are equal, lemma 4 implies that the labels of the bottom-right-justified (n + 1)-triangular subarray in ψ(h) (highlighted in blue in figure 7) are determined by the boundary conditions of Hive( λ, μ, ν).
Here, q = 2ν 1 + γ 1 and for x a label in h we write x ′ to denote x + 3ν 1 .
By the above description, to verify the well definedness of the map ψ, it suffices to check the rhombus inequalities in ψ(h) for the following 2n rhombi: • The n SE rhombi each of which straddles the regions described in (2) and (4).
• The n vertical rhombi each of which straddles the regions described in (2) and (3).
The first n rhombi inequalities hold because of the fact that the entries of h increase along the rows (this follows form the Southeast rhombi inequalities and the fact that µ t ≥ 0 ∀t).The second n rhombi inequalities hold because the edge labels 4 of the Northeast edges (green edges in figure 7) originating from the (n + 2) th row is bounded above by ν 1 , which is equal the edge label of the Northeast edges originating from the (n + 1) th row (blue in the figure).This follows from the Northeast rhombi inequalities and the boundary condition on h.For example, in figure 7 we have e ′ − a ′ ≤ f ′ − b ′ ≤ g ′ − c ′ = ν 1 .Therefore the map ψ is well defined.
It is clear from the definition of the map that it is injective, affine linear and sends integral skew hives to integral hives.We now establish surjectivity.Given a hive in Hive( λ, μ, ν), consider its triangular subarrays marked in yellow and blue in Figure 7.These are themselves hives, and lemma 4 implies that these hives are uniquely determined (since the corresponding hive polytopes are non-empty).In particular, the labels on the bottom row of the yellow triangle are γ + n • ν 1 where γ denotes the vector of partial sums of γ as defined in §4.3.Likewise, the labels on the left edge of the blue triangle must be |γ| + ν + n • ν 1 .These are also edges of the the white parallelogram.The othe two edges of the parallelogram have edge labels λ + n • ν 1 and |λ| + μ + n • ν 1 .This proves surjectivity of ψ. y is defined to be x − y.
Finally, since the map ψ does not change rhombus contents, it does not alter any flatness conditions within the white parallelogram.Thus if the left and bottom justified region R(Φ) is flat in h, then it remains flat in ψ(h); however since ψ(h) is a triangular hive in twice as many ambient dimensions, this region would now correspond to R( Φ) in ψ(h), where We remark that Hive( λ, μ, ν, Φ) coincides with the hive Kogan face K Hive ( λ, μ, ν, w( Φ)) of [KRV21], where w( Φ) ∈ S 2n is the unique Theorem 9 can also be proved by working directly with the skew hive polytope, rather than with its isomorphic hive polytope.This involves mimicking all the arguments of [Buc00; KRV21] for skew hives.While we have chosen a shorter approach in this paper, this alternate approach naturally suggests numerous other refinements of the LR coefficients with the saturation property.These will be considered in a future publication.

APPENDIX: DECOMPOSITION OF Tab(λ/µ, Φ) INTO DEMAZURE CRYSTALS
In this section, we make theorem 4 effective, describing algorithmically the Demazure crystals which occur in the decomposition.The arguments below are implicit in the character level proof of [RS95a], and so we content ourselves with sketching their broad contours.
We start with a brief discussion of the Burge correspondence [Ful97].We use the standard notation [n] = {1, 2, • • • , n}.Consider a matrix M = (m ij ) of size r × n with non-negative integer entries.We associate a biword to M as follows such that for any pair (i, j) that indexes an entry m ij of M , there are m ij columns equal to i j in w M , and the columns of w M are ordered as follows: (1) (2) i k+1 > i k whenever j k+1 > j k .
In other words, form the biword w M by reading the entries m ij of M from left to right within each row starting with the bottom row and proceeding upwards, recording each i j with multiplicity m ij .We will often denote the row and column indices of the biword by i = i 1 i 2 • • • i t and j = j 1 j 2 • • • j t .Additionally, given a flag Φ, if i k ≤ Φ j k for all k (in particular, the matrix M is block upper-triangular) then i is said to be (j, Φ)-compatible (see [RS95a]).
Theorem 10. [Ful97, Appendix A, Proposition 2] The Burge correspondence gives a bijection between the set of all r × n matrices with non-negative integer entries M at r×n (Z + ) and the set of pairs (P, Q) of semistandard tableaux with the same shape where entries of P are in [n] and entries of Q are in [r].We use the notation The reverse filling of the skew shape λ/µ, denoted RF (λ/µ), is defined to be the filling of the boxes of the shape λ/µ by 1, 2, 3, • • • , |λ/µ|, sequentially from right to left within each row, starting with the top row and proceeding downwards.
We say a standard (skew) tableau Q with |shape(Q)| = |λ/µ| is λ/µ-compatible if it satisfies the following: (1) If i+ 1, i are adjacent in a row of RF (λ/µ) then i+ 1 appears weakly north and strictly east of i in Q.
(2) If i occurs directly above j in a column of RF (λ/µ) then j appears weakly west and strictly south of i in Q. A(Q, λ/µ, Φ) where the union is over all λ/µ-compatible standard tableaux Q [RS95a].We will show that A(Q, λ/µ, Φ) is isomorphic to some Demazure crystal as crystals, i.e., there is a weight-preserving bijection between these sets which intertwines the crystal raising and lowering operators (where defined).
For a composition α, key(α) is the semi-standard tableau of shape α † whose first α k columns contain the letter k for all k.One can see that key(α) is the unique tableau of shape α † and weight α.We define W(α, Φ) as the set of all words u = • • • u 2 •u 1 , where each u i is a maximal row word of length α i together with the properties that each letter in u i can be atmost Φ Let a = a 1 a 2 • • • a t be a word.Then by an ascent of the word a we mean a positive integer 1 ≤ k ≤ t − 1 such that a k < a k+1 .We recursively define the essential subword ess L (a) of a with respect to a positive integer L or ∞ to be the following indexed subword of a: (1) ess L (a) is the empty word if t = 0.
Define the essential subword of a as ess(a) = ess ∞ (a).Proof.The proof is similar to that of Lemma 8 of [RS95a].
For a semi-standard tableau T , let T | <L denote the subtableau of T consisting of the entries of T which are less than L, and let K (T ) denote the left key tableau of T .For more details on computing left and right keys, see [RS95a], [Wil13] and [RS95b].Lemma 6.Let L ≥ 1. Suppose that (1) P, P ′ and Q are semi-standard tableaux of the same shape such that wt(Q) = (1, 1, • • • , 1) and K (P )| <L = K (P ′ )| <L .
( Then ess L (a) = ess L (a ′ ) and a, a ′ have the same ascents.
Proof.The proof is similar to that of Lemma 9 of [RS95a].Now we have the following proposition: 2.[Kas93, Proposition 3.3.5]For any λ ∈ P[n] and ω ∈ S n , the Demazure subcrystal B ω (λ) of Tab(λ, n) has the string property.
e., β i = b (say) for all i.If Hive(α, β, γ) is non-empty, let h be an element.Define κ = γ − α ∈ R n .The labels along the left and right edges of h are k i=1 α i and k i=1 γ i for 0 ≤ k ≤ n.Now consider the following types of trapezia formed by two overlapping unit rhombi, one NE and the other SE.

Lemma 5 .
Let Φ be a flag and i= i 1 i 2 • • • i t be a word.If a = a 1 a 2 • • • a t , b = b 1 b 2 • • • b t are words in [n]having the same essential subword and ascents, then i is (a, Φ)-compatible if and only if i is (b, Φ)-compatible.