A Demazure Character Formula for the Product Monomial Crystal

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type $A$, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on $\mathbf{R}$.

family of finite dimensional representations of G which appear in the study of slices to Schubert varieties in the affine Grassmannian. Our main theorem provides a Demazure-type character formula for these representations. In type A we show that these representations are related to generalised Schur modules, and give an explicit realisation for the crystal of a generalised Schur module. We now discuss our motivations and results in more detail.
Our first motivation comes from the representation theory of algebras which quantise slices to Schubert varieties in the affine Grassmannian. Let G be a reductive algebraic group with Langlands dual G ∨ . The affine Grassmannian Gr of G ∨ is a Poisson ind-variety which plays an important role in geometric representation theory (e.g. [MV07]). Gr is stratified by spherical orbits Gr = λ∈P + Gr λ , where λ ranges over the dominant weights P + of G. The closure relation on these strata is given by the positive root ordering on P + : Gr λ = µ≤λ Gr µ . Fixing a pair µ ≤ λ in X + , consider the transversal slice W λ µ to Gr µ in Gr λ . These slices inherit a Poisson structure, and under the geometric Satake correspondence they (non-canonically) geometrise weight spaces of the irreducible representation of g [BF10].
In [Kam+14] the authors initiate a program to construct quantisations of W λ µ , and study the representation theory of the resulting algebras. These algebras are called truncated shifted Yangians and denoted Y λ µ (R). Here R ∈ Π i∈I C λi /S λi is a deformation parameter, where λ i = λ, α ∨ i . Modules over truncated shifted Yangians naturally afford a highest weight theory, leading to a category O(Y λ µ (R)), which is the "algebraic category O" in the sense of [Bra+14]. This category plays an important role in the "symplectic duality" program of Braden-Licata-Proudfoot-Webster. In [Kam+19b] O(Y λ µ (R)) was recently used to prove the categorical symplectic duality between Nakajima quiver varieties and the slices in the affine Grassmannian.
The representation theory of Y λ µ (R) easily reduces to the case where R consists of integers satisfying certain parity conditions (see Section 3 below). It is conjectured in [Kam+19a], and proven in [Kam+19b], that the sum V(R) = µ O(Y λ µ (R)) carries a categorical g-action in the sense of Khovanov-Lauda and Rouquier [KL09;Rou08;KL11]. Therefore the (complexified) Grothendieck group of this category is a representation of g. This representation is our main object of study: V (R) = K C (V(R)).
While it is known that for generic (respectively singular) parameters V (R) is isomorphic to a tensor product of fundamental representations (respectively a single irreducible representation), the representation V (R) in general is quite mysterious. Let B(R) be the crystal of V (R), which is called the product monomial crystal due to its realisation as a subcrystal of Nakajimas's monomial crytal [Kam+19a]. Our first main result provides an explicit character formula for the crystal B(R), in terms of multiplications by dominant weights and application of isobaric Demazure operators (Theorem 5.9).
Our proof relies on defining truncations of the crystal B(R), which are certain subsets of B(R) (Section 5.1). Each of these truncations is described globally, however we show that "nearby" truncations are related via a crystal-instrisic operation, the extension of strings. Determining a path of nearby truncations from the smallest truncation to the largest gives a Demazure character formula (Section 5.2). We also show that each truncation is a disjoint sum of Demazure crystals (Section 5.3), which is not obvious from the global description of the truncation.
As a consequence of the categorification described above, the elements of weight µ in B(R) are in bijection with the simple highest weight modules of Y λ µ (R). In fact the crystal structure provides even more refined data, for example the highest-weight elements of weight µ in the crystal correspond precisely to the finite dimensional simple Y λ µ (R)-modules [Kam+19a, Proposition 3.17]. Thus, our character formula (Theorem 5.9) provides combinatorial information about the representation theory of Y λ µ (R). Our second motivation comes from the study of generalised Schur modules. Recall that for each partition λ, there is an endofunctor on vector spaces called the Schur functor S λ , and when V is the basic representation of GL(V ) over C, the representation S λ (V ) is irreducible, of highest weight λ. The modules S λ (V ) are well-studied and are called Schur modules.
The partition λ can be thought of as a Young diagram, a configuration of boxes in the plane, or a finite subset of N × N of a special form. In fact, to any finite subset D ⊆ N × N, called a diagram, we can associate a generalised Schur functor S D , and hence a generalised Schur module S D (V ). Considerably less is known about these functors in complete generality, though there have been some combinatorics developed for the class of "percentage-avoiding" diagrams [Mag98b;RS98].
Our second main result describes the crystal of S D in the case where D is column-convex (see Remark 6.16). More precisely, we associate a parameter R to D and prove that B(R) is the crystal of S D (Theorem 6.23). This provides a model for the crystal of S D in terms of Nakajima monomials (the other known model for this crystal is due to Laksmibai, Littelmann, and Magyar [LLM02] using Demazure operators).
To prove Theorem 6.23, we compare our character formula (Theorem 5.9) for B(R) in type A n to a character formula for S D (C n+1 ) given by Reiner and Shimozono [RS98, Theorem 23], which shows that they have the same character when n is taken large enough compared to D. We then derive some stability results for B(R) (Section 6.1), which imply that B(R) is the character of S D (C n+1 ) for any n.
Theorem 6.23 implies that in type A, the category V(R) defined by the truncated shifted Yangians are categorifications of generalised Schur modules associated to column-convex diagrams.
We note that our result shows that any skew Schur module is categorified by V(R), for some R.
1.1. Acknowledgements. I would like to thank my advisor Oded Yacobi for suggesting this project, and I am grateful for his support, guidance and patience throughout. I'd like to thank Peter Tingley and Alex Weekes for many helpful conversations while they were visiting Sydney. I'd also like to thank Nicolle Gonźalez and Travis Scrimshaw for many illuminating conversations concerning Demazure crystals and Victor Reiner for answering all of my questions about generalised Schur modules. This research was supported by an Australian Postgraduate Award; the results of this paper will appear in my PhD thesis.
2. Background 2.1. Notation. Let G be a reductive algebraic group over C, equipped with a pinning T ⊆ B ⊆ G where B is a Borel subgroup and T is a maximal split torus. Let I be a set such that |I| is the semisimple rank of G. This determines the following combinatorial data: (1) The weight lattice P = Hom(T, G m ).
(2) The simple roots α i ∈ P for all i ∈ I.
Because of the proof of Theorem 3.2 relies on the theory of Nakajima quiver varieties, we assume throughout that G is simply-laced , meaning that the Cartan matrix α ∨ i , α j i,j is symmetric with all off-diagonal entries either 0 or −1. Define the Dynkin diagram of G as the simple graph with vertex set I, where i ∼ j if and only if α ∨ i , α j = −1. We may then fix a two-colouring I = I 0 ⊔I 1 of the Dynkin diagram, and we say that the parity of a vertex i ∈ I is even if i ∈ I 0 and odd if i ∈ I 1 .
A finite-dimensional module V over G decomposes into weight spaces denote the group algebra of P written multiplicatively, so that e λ e µ = e λ+µ . Then the formal character of the module V is the sum For each dominant weight λ ∈ P + , let V (λ) denote the irreducible module of highest-weight λ. The category of finite-dimensional G-modules is semisimple, with the modules {V (λ) | λ ∈ P } forming a complete irredudant list of simple objects. The characters of the V (λ) for λ ∈ P + form a basis for Z[P ] W , and hence the isomorphism class of any finite-dimensional G-module is determined entirely by its character.
Remark 2.1. We have restricted to the case of reductive G in order to simplify the exposition, however our character formula Theorem 5.9 holds in the more general Kac-Moody setting provided that the Dynkin diagram is still simply-laced and bipartite. In order to state the analagous results in this more general setting, one would replace the category of finite-dimensional Gmodules by the category O int for the corresponding quantum group. In particular, our results apply to finite types A, D, E and their untwisted affinisations, excluding A (1) 1 which is not simplylaced, and A (1) n for n odd, which is not bipartite. 2.2. Crystals. Crystals were unearthed by Kashiwara [Kas90;Kas91;Kas94]. There is a rather general notion of a G-crystal, however we will only require the notion of an upper-seminormal crystal, for which we can give some simplified axioms and definitions. We follow the exposition of [Jos03, Section 2] for this section.
An upper-seminormal abstract G-crystal is a set B, together with a weight function wt : B → P , and for each i ∈ I, crystal operators e i , f i : B → B ⊔ {0} and maps ε i , ϕ i : B → Z satisfying the following axioms: ( By the above axioms, the data of an upper-seminormal abstract crystal is entirely determined by (B, wt, (e i ) i∈I ). The upper-seminormal abstract crystal B is called a seminormal abstract crystal if it additionally satisfies ϕ i (b) = max{k ≥ 0 | f k i b ∈ B} for all i ∈ I, b ∈ B. Each crystal defines a crystal graph, a directed graph on the vertex set B, with an i-labelled edge from b to b ′ whenever f i (b) = b ′ . The edge-labelled graph is equivalent to the data of the e i or f i , and hence an upper-seminormal abstract crystal is determined entirely by its weight function and graph. We say that B is connected if the underlying undirected graph of its crystal graph is connected. An element b ∈ B is called primitive if e i (b) = 0 for all i ∈ I, or equivalently if it has no incoming edges in the crystal graph. An element b ∈ B is called highest-weight if it is both primitive, and there is a directed path in the crystal graph from b to every element of B.
There are two different rules for forming the tensor product of two abstract crystals, we use the convention from [Kas94]. If B 1 , B 2 are abstract G-crystals, then their tensor product B 1 ⊗B 2 has underlying set the Cartesian product B 1 × B 2 , with pairs of elements written b 1 ⊗ b 2 , the convention that 0 ⊗ b 2 = b 1 ⊗ 0 = 0, and maps given by This gives the category of abstract crystals the structure of a monoidal category (the tensor product of two upper-seminormal crystals is again upper-seminormal).
It has been shown [Kas91] that for each λ ∈ P + , the highest-weight module V (λ) admits a seminormal crystal base B(λ). We say that an abstract G-crystal is simply a G-crystal if it is a disjoint union of various B(λ). Each B(λ) is connected as a graph, with a unique vertex b λ ∈ B(λ) satisfying e i (b λ ) = 0 for all i ∈ I. Such a vertex b λ is called highest weight , since it is both primitive (meaning it is killed by all the e i ) and also generates the whole of B(λ) under the f i operators.
The subcategory of g-crystals is closed under tensor product, and the decomposition multiplicities of the B(λ) agree with those of the V (λ): For each λ ∈ P we denote by B λ = {b ∈ B | wt b = λ} the λ-weight elements of B. The formal character of an abstract G-crystal is the sum ch B = λ∈P |B λ | e λ ∈ Z[P ]. If the abstract G-crystal is seminormal, then ch B ∈ Z[P ] W . For each λ ∈ P + , we have ch V (λ) = ch B(λ).
2.3. Demazure modules and Demazure crystals. Fix a λ ∈ P + . The elements of the Weyl group orbit W · λ are called the extremal weights of V (λ), and their corresponding weight spaces V (λ) wλ are all one-dimensional. The B-submodule generated by V (λ) wλ is called the Demazure module V w (λ). As vector spaces we have V e (λ) = V (λ) λ , the one-dimensional highest-weight space, and V w• (λ) = V (λ) the whole module. We say that V w (λ) is the Demazure module of Demazure lowest weight wλ.
For each µ ∈ P , let D(µ) denote the Demazure module of Demazure lowest weight µ. The collection {D(µ) | µ ∈ P } is a complete irredundant list of Demazure modules, and furthermore the Demazure characters {ch D(µ) | µ ∈ P } form a basis for Z[P ]. (The triangularity property ch D(µ) = e µ + ν>µ c ν e ν shows that the Demazure characters are linearly independent, while the fact that they span follows from the fact that Z[P ] is a limit of finite-dimensional subspaces of the form Span Z X for X ⊆ P Weyl-invariant).
A character formula for the Demazure modules was first given in [Dem74], with a more recent proof in the arbitrary symmetrisable Kac-Moody setting appearing in [Kas93]. For each i ∈ I, define the Z-linear Demazure operator π i : Z[P ] → Z[P ] by The Demazure operators define a 0-Hecke action on Z[P ], that is to say each is idempotent (π 2 i = π i for all i ∈ I) and they satisfy the braid relations (if i ∼ j then π i π j π i = π j π i π j , and if i ∼ j then π i and π j commute). The Demazure character formula states that when (i 1 , . . . , i r ) is a reduced decomposition for w, then It was shown [Kas93] that each Demazure module V w (λ) for λ ∈ P + admits a crystal base B w (λ), called a Demazure crystal (although B w (λ) is in general an abstract G-crystal rather than a crystal). Moreover, the crystal B w (λ) can be obtained from B(λ) in the following way. Define the extension of i-strings operator D i , which acts on a subset X ⊆ B of an abstract G-crystal: The operators D i satisfy X ⊆ D i X ⊆ B, and D i (D i X) = D i X. Then let (i 1 , . . . , i r ) be a reduced decomposition for w ∈ W , and we have where b λ ∈ B(λ) is the highest-weight element. Note that Eq. (10) only defines a subset of B(λ): we equip this subset with its canonical abstract upper-seminormal crystal structure coming from the restrictions of wt and e i for i ∈ I.
2.4. Multisets. We will use multisets throughout the paper. Multisets will always be denoted using boldface type, such as R, S, T, or Q. Given a set X, a multiset based in X is a function R : X → N, where we write R[x] for the value of R at x ∈ X, henceforth called the multiplicity of x in R. The support of R is the subset Supp(R) = {x ∈ X | R[x] > 0} ⊆ X, and a multiset is finite if its support is finite. Any summations or products over R are taken with multiplicity, so for example if f : X → G is a function into an abelian group (written multiplicatively) and R is a finite multiset based in X, . If R and Q are multisets based in X, their multiset union is the function for all x ∈ X, and in this case, their multiset difference is the function R − Q.
The notation we use for multisets is similar to set notation, with exponents denoting multiplicity. For example, if X = {x, y, z} is a base set, then R = {x 2 , y} denotes a multiset R based in X where x appears with multiplicity 2, and y appears with multiplicity 1 (or treating R as a function R :

Definition of the product monomial crystal
The product monomial crystal is defined as a certain subcrystal of the Nakajima monomial crystal. The Nakajima monomial crystal is not only a crystal, but also has an abelian group operation given by multiplication of monomials. The product monomial crystal will be the monomial-wise product of certain subcrystals of the Nakajima monomial crystal.
3.1. The Nakajima monomial crystal. Let G be a pinned reductive group as in Section 2.1, we now define the Nakajima monomial crystal M(G) as in [HN06, Section 2]. Let Z{I×Z} denote the free abelian group of monomials in the variables {y i,c | i ∈ I, c ∈ Z}. Let A(G) = P ×Z{I×Z} be the product of abelian groups, written such that a typical element p ∈ A is of the form for some element wt(p) ∈ P and coefficients p[i, c] ∈ Z, finitely many of which are nonzero. For each i ∈ I and c ∈ Z, define the auxiliary monomial Definition 3.1. The Nakajima monomial crystal M(G) is defined to be the submodule of A(G) satisfying the two conditions . The crystal structure on M(G) is then defined by wt, ε i , and ϕ i as above, and Because of the parity condition (2) appearing in Definition 3.1, it is convenient to introduce the following notation. Let I×Z ⊆ I × Z denote the subset of parity-respecting pairs The monomial crystal without condition (2) coincides with the crystal defined in [Kas02, Section 3]. Restricting to monomials y i,c for (i, c) ∈ I×Z forms a "good" subset in the sense of [Kas02, Proposition 3.1], hence Theorem 3.2. We often picture elements of M(G) in the following way. Place the Dynkin diagram I in the plane, then place the grid I×Z above the Dynkin diagram as an infinite strip of points. A monomial p ∈ M(G) is a finitely supported assignment (i, c) → p[i, c] of points to integers, together with a weight wt p ∈ P . The statistics ϕ k i (p) and ε k i (p) can then be pictured as a sum over points in a half-infinite column i. An example for G = SL 6 is shown in Fig. 1. The group G = SL 6 has Dynkin diagram A 5 , a path on 5 vertices. Pictured above is the monomial p = e 2̟2+5̟3+2̟4 · y −2 2,22 · y 2 2,24 · y 5 3,23 · y 6 4,20 · y −4 4,24 . The two shaded regions are showing computations of ϕ 20 2 (p) and ϕ 22 4 (p) respectively.
3.2. The product monomial crystal. From this point onwards, we fix a system {̟ i } i∈I of fundamental weights, meaning any collection of elements satisfying α i , ̟ j = δ ij . Note that in general the ̟ i are members of P ⊗ Z Q rather than the weight lattice P , and for general G such a system is not unique, a notable exception being when G is semisimple.
Definition 3.3. A fundamental subcrystal of M is a subcrystal generated by an element of the form e n̟i y n i,c for some n ≥ 0 such that n̟ i ∈ P + . Denote this subcrystal by M(i, c) n . It is straightforward to check that such a monomial p = e n̟i y n i,c is highest-weight, and therefore generates a subcrystal of M isomorphic to B(n̟ i ) by Theorem 3.2.
As the monomial crystal M is a subgroup of A, it inherits the group operation given by multiplication of monomials. Explicity, for p, q ∈ M we have wt(p · q) = wt(p) + wt(q), and For subsets X, Y ⊆ M, we define their product X · Y = {x · y | x ∈ X, y ∈ Y } as usual, and call this the monomial-wise product of subsets.
Definition 3.4. For any finite multiset R based in I×Z, define the product monomial crystal as the monomial-wise product of subsets of the fundamental subcrystals Remark 3.5. After fixing the system of fundamental weights, the weight of a monomial p ∈ M(R) is given by wt p = i,c p[i, c]̟ i . So we can safely omit the e λ term from monomials from now on, instead relying on our fixed system of fundamental weights to reconstruct λ from the monomial p.
This subset is closed under the crystal operators, and its only highest-weight element is y 2 1,1 , showing that M(R) ∼ = B(wt y 2 1,1 ) = B(2̟ 1 ) as SL 3 -crystals. We may denote the monomials above pictorially, as in Fig. 1: We then obtain the following pictures for the three connected crystals M(∅), M(1, 1) and M(1, 1) 2 : It is perhaps surprising that the monomial-wise product M(1, 1)·M(1, 1) turns out to be again a G-subcrystal of M(G), since the monomial-wise product is not obviously related to the crystal operators. In fact, the subset M(R) is always a subcrystal of M, justifying the name product monomial crystal. The proof of Theorem 3.7 uses an explicit isomorphism between M(R) and the crystal defined by a graded Nakajima quiver variety depending on R. The author does not know of a purely combinatorial proof, and the use of quiver varieties is the reason for the simply-laced restriction on G. Remark 3.8. Our notation differs from [Kam+19a] in three ways. Firstly, they use the symbol B(R) rather than M(R) to denote the product monomial crystal. Secondly, they define the monomial crystal only for semisimple Lie algebras, and hence the e λ weight term is missing as explained in Remark 3.5. Thirdly, they use a collection (R i ) i∈I of multisets where R i is a multiset based in 2Z + parity(i); to go between these notations, set and that furthermore, by varying R while keeping wt R fixed, both extremes M(R) ∼ = cB(wt(R)) and M(R) ∼ = (i,c)∈R B(̟ i ) can be achieved.
Example 3.9. Let G = SL 4 , and fix wt(R) = 2̟ 2 . Then depending on R, there are three possibilities for the isomorphism class of M(R):

Analysis of the product monomial crystal
In this section we give a high-level analysis of the product monomial crystal, which will lay the foundation for the more precise analysis in Section 5 leading to the character formula. The results of this section appear in [Kam+19a], however we go into more detail here. 4.1. Labelling elements of the product monomial crystal. Let R and S be finite multisets based in I×Z, and define the auxiliary monomials By the definition of the monomial crystal M(G), each element p ∈ M(i, c) n is of the form p = y n i,c z −1 S for some finite multiset S based in I×Z. Hence each element p of the product monomial crystal M(R) is of the form y R z −1 S for some finite multiset S. In fact, by the linear independence of the z i,k in the abelian group M(G), the multiset S is uniquely determined by p. Using this labelling scheme, when Remark 4.1. In type A, the S multisets arising in M(R) may be interpreted in terms of partitions "hung from pegs", with each peg corresponding to an element of R. For more details (and a picture of this), see [WWY17, Section 2.5.3]. While we do not apply this interpretation explicitly in this paper, the author found it invaluable to make the connection between M(R) and the generalised Schur modules.

4.2.
A partial order, upward-closed and downward-closed sets. We assume from now on that the Dynkin diagram I is connected. Define a partial order ≤ on the set I×Z as the transitive closure of (i, c) ≤ (i, c + 2) and (i, c) ≤ (j, c + 1) for all j ∼ i. A subset J ⊆ I×Z is called upward-closed if whenever x ∈ J and y ∈ L satisfy x ≤ y, then y ∈ J. If J is upward-closed, a minimal element x ∈ J is one such that for all y ∈ L, either x ≤ y or x and y are incomparable. Every upward-closed set is a union of the upward-closed sets generated by its minimal elements. For any subset X ⊆ I×Z, define up(X) = {y ∈ I×Z | x ≤ y} to be the upward-closed set generated by X, and define down(X) similarly. For each i ∈ I, define the i-boundary of an upward-closed set J to be A minimal point of J is a boundary point, but boundary points are not necessarily minimal.
A good reason to introduce this order is that the fundamental subcrystals M(i, c) n always "grow downwards" from the point (i, c) with respect to this partial order, as made precise in the following lemma.
The claim is vacuous for the highest-weight element y n i,c since its associated S-multiset is empty. As M(i, c) n is connected, it is enough to show that the f i operators preserve the above property.
Suppose that p = y n i,c z −1 S ∈ M(i, c) satisfies Supp S ≤ (i, c − 2), and f j (p) = 0, so f j (p) = pz −1 j,k−2 (i.e. f j (p) adds the point (j, k − 2) to S). In particular, the definition of ϕ j gives that p[j, k] > 0; this inequality together with Eq. (18) then implies , and in this case (j, k − 2) = (i, c − 2) and so the property holds. Otherwise, R[j, k] = 0 and since l∼j S[l, k − 1] is strictly positive, at least one upward neighbour of (j, k − 2) is already included in Supp S, and the claim follows by the transitivity of ≤.

4.3.
Supports of monomials, highest-weight monomials. Given a monomial p = y R z −1 S ∈ M(R), its R-support is defined to be the set Supp R (p) = Supp R ∪ Supp S. Note that the R-support is defined in terms of the S-labelling, and can be large compared to the support of the original monomial in terms of the y i,c , as the next example shows.

Truncations and a character formula
Recall from Section 2.3 the definition of the Demazure operator π i and the extension-of-strings operator D i . The following theorem of Kashiwara establishes a commutation property of the character function with π i and D i .
Theorem 5.7. [Kas93;BS17]. If X is a subset of an abstract g-crystal B, and X satisfies the string property, then ch(D i X) = π i (ch X) for all i ∈ I.
It is not true that if some subset X has the string property, then D i (X) has the string property -for a counterexample, see [BS17,Chapter 13]. However, we can verify directly that all of the truncations we have been considering have the string property.
By definition of ϕ i , k is largest such that ϕ k i (p) = ϕ i (p), and hence ϕ l+2 i (p) < ϕ k i (p) for all l ≥ k. But since p[i, r] = 0 for all r < k, we have that ε l i (p) = ϕ k i (p) − ϕ l+2 i (p) > 0 for all l ≥ k, and hence ε i (p) = 0 and e i (p) = 0.
We then arrive at an inductive character formula for any truncation M(R, J) by interpreting both Lemma 5.2 and Lemma 5.3 in terms of characters, using Theorem 5.7.
Theorem 5.9. The following rules give an inductive character formula for any truncation M(R, J): (1) ch M(∅, J) = 1 for any J.
(2) Suppose J is an upward-closed set containing R, and Q lies along the boundary ∂J.

5.3.
Truncations are Demazure crystals. The procedure given in Theorem 5.9 looks quite similar to the construction of a Demazure crystal (Section 2.3), and we will show that each truncation M(R, J) is in fact a Demazure crystal. We rely on the main result of [Jos03], which states that if X is a Demazure crystal and b is a highest-weight element of some crystal, then {b} ⊗ X is a Demazure crystal. We use this result by formulating Lemma 5.3 in purely crystaltheoretic terms.
Lemma 5.11. Let J be an upward-closed set containing R, and let Q be a multiset supported along the boundary of J, so Supp Q ⊆ ∂J. Set µ = wt(Q), and write b µ ∈ B(µ) for the highest-weight element. There is a bijective, weight-preserving map Proof. The map is defined as a consequence of Lemma 5.3, and is bijective and weight-preserving, so all that remains to be seen is the e i -equivariance. Let us recall the rule (Eq. (5)) for applying e i to a tensor product of two crystal elements: Fix an i ∈ I, and let (i, k) ∈ ∂J be that unique point on the boundary of J lying in column i. Let p ∈ M(R + Q, J) be arbitrary. Since the support of Q lies in ∂J, the only element of Q in column i which could have nonzero multiplicity is (i, k), where it has multiplicity Q[i, k] = α ∨ i , µ . Since p[i, l] = 0 for all l < k, we then have If e i (p) = 0, then ε l i (p) ≥ 0 for all l, and hence α ∨ i , µ ≥ −ε l i (p/y Q ) for all l ≥ k, and hence α ∨ i , µ ≥ max l −ε l i (p/y Q ) = ε i (p/y Q ). Then we are in the first case of Eq. (32), and e i (Φ(p)) = 0.
On the other hand, if e i (p) = z i,r p for some r ≥ k, then 0 < −ε r i (p) = ε i (p), and applying Eq. (33) gives that α ∨ i , µ < −ε r i (p/y Q ) ≤ ε i (p/y Q ). So we are in the second case of Eq. (32), and all that remains to check is that e i (p/y Q ) = z i,r p/y Q . However, this is clear from Eq. (33) since adding a constant α ∨ i , µ to the values of the ε · i will not change the value r, as r is defined as the least l such that ε l i (p) is minimised. for any upward-closed set J containing R.
Proof. By the Demazure character formula, a Demazure character is of the form π x (e λ ) for some x ∈ W and λ ∈ P + . As the π i operators braid and are idempotent, they define a 0-Hecke action on Z[P ], in other words a representation of the Hecke algebra associated to W with defining relation E 2 i = E i for all i ∈ I. The standard basis element E w• of this Hecke algebra satisfies E i E w• = E w• = E w• E i for all i ∈ I, and hence so do the Demazure operators: π w• π x = π w• . We then have that π w• π x (e λ ) = π w• e λ = ch V (λ) by the Demazure character formula. Now, M(R, J) is a disjoint sum λ,w c λ,w B w (λ) of Demazure crystals. Since every highestweight element of M(R) appears in the truncation M(R, J) we have that M(R) ∼ = λ,w c λ,w B(λ) as G-crystals. On the level of characters, this becomes

The product monomial crystal in type A
We show that in type A, the product monomial crystal M(R) is isomorphic to the crystal of a generalised Schur module S D , a GL n -module defined by a diagram D depending on R. The generalised Schur modules have a stable decomposition when D is fixed and n increases, and the character of each is given by a Demazure-type formula [RS95;RS98]. We first show that the product monomial crystal M(R) has a stable decomposition when R is fixed and n increases, and then compare characters to show that the stable decompositions of S D and M(R) agree.
Remark 6.2. For the root datum GL n , the associated Kac-Moody algebra is gl n , and the Weyl group is isomorphic to the symmetric group on n letters, with s i : P n → P n swapping ǫ i and ǫ i+1 and leaving the other ǫ j fixed. The weight λ = n i=1 λ i ǫ i is dominant if and only if λ 1 ≥ · · · ≥ λ n , and furthermore we call λ a polynomial weight if it is dominant and λ i ≥ 0 for all 1 ≤ i ≤ n.
Let I n = {1, . . . , n − 1} and I ∞ = {1, 2, 3, . . .}. For a finite multiset R based in I ∞× Z, we will say that R lives over I n if Supp R ⊆ I n× Z. If R lives over both I n and I m , it defines a GL n crystal M(GL n , R) and a GL m crystal M(GL m , R). Given two finite multisets R and S living over I n , we define the Nakajima monomial where we have included the full formula to make it clear that the z −1 S term depends on n. In general we have v(GL n , R, S) = v(GL m , R, S) because of this dependence on n. However, for n ≤ m there is a map of sets from the smaller crystal to the larger one, defined by the S-parametrisation. Lemma 6.3. Suppose that n ≤ m and let R be a finite multiset living over I n . Then there is an injective map of sets Furthermore, the image of this map is precisely Proof. The claim is trivial for n = m, so assume that n < m. The map is a priori an injective map into the monomial crystal M(GL m ), so we need to show that the image is contained in M(GL m , R), and the description of the image is correct. From the definition of the monomial crystal B(GL n ), we have for p = v(GL n , R, S) that Only the last summand depends on n, and we see that for q = v(GL m , R, S) that Since S lives over I n we must have S[n, −] = 0 and hence ϕ k i (p) = ϕ k i (q) for all i ∈ I n . This implies that if f i v(GL n , R, S) = v(GL n , R, S ′ ) then f i v(GL m , R, S) = v(GL m , R, S ′ ) for all i ∈ I n . Now consider the case of a fundamental subcrystal, where R is a single element R = {(i, c)}. Clearly Ψ n,m sends the highest-weight element y R ∈ M(GL n , R) to y R ∈ M(GL m , R), and since M(GL n , R) is generated by the f i for i ∈ I n , the previous paragraph shows that Φ n,m (M(GL n , R)) ⊆ M(GL m , R). Furthermore, since the crystal is connected it is clear that the subset of M(GL m , R) of monomials living over I n is precisely those generated under only f i for i ∈ I n , i.e. those monomials q ∈ M(GL m , R) such that Supp R (q) lives over I n .
The claim follows for general R by factorisation into a product of monomials coming from various fundamental subcrystals.
The image of the inclusion map Ψ n,m can be described purely in terms of weights, rather than monomials.
Lemma 6.4. Let n ≤ m and R be a finite multiset living over I n . Then Proof. Let p = v(GL n , R, S). Then since wt(p) is a linear combination of ǫ 1 , . . . , ǫ n it is certainly true that ǫ ∨ i , wt Ψ n,m (p) = 0 for all i > n. Conversely, suppose that q = v(GL m , R, S) satisfies ǫ ∨ i , wt q = 0 for all i > n. Writing wt q = wt R − i∈Im k i α i for some integers k i > 0, we have (42) showing that k n = 0 and hence S[n, −] = 0. The same trick can be applied to show that k i = 0 for all i ≥ n, showing that Supp R q lives over I n . Hence the claim follows by the second statement of Lemma 6.3.
If p is highest-weight then so is Ψ n,m (p), and therefore the map Ψ n,m (p) restricts to an injection on the highest-weight elements of each crystal: where we use the notation B h.w. for the highest-weight elements of the crystal B.
Lemma 6.5. Fix a finite multiset R based in I ∞× Z, let X be the intersection of up(R) and down({(i, c − 2 | (i, c) ∈ R)}), and let n ≥ 1 be smallest such that X lives over I n . Then: (1) For all m ≥ n, the inclusion ψ n,m is bijective and weight-equivariant, under the inclusion P n ֒→ P m taking ǫ i to ǫ i . (2) For k ≤ n, the image of the inclusion ψ k,n is described purely in terms of weights, by Proof.
(1) If q ∈ M(GL m , R) h.w. , then by both Lemma 4.2 and Corollary 4.5 we have Supp R (q) ⊆ X, and hence the ψ n,m is surjective by the description of Im Ψ n,m in Lemma 6.3. The weight-equivariance follows from the definition of Ψ n,m . (2) Follows from Lemma 6.4.
The previous lemma shows that the decomposition of M(GL n , R) stabilise when R is held fixed and n is allowed to grow, with the smallest n guaranteed to agree with the stable decomposition given in the statement of the lemma.
Definition 6.6. Let R be a finite multiset living over I ∞ , and let n be such that ψ n,m is bijective for all m ≥ n. Define the decomposition multiplicities c λ R ∈ N by the equation where the sum is over all weights λ ∈ P + n .
The fact that the constants c λ R are well-defined is a consequence of the fist part of Lemma 6.5. The second part of Lemma 6.5 gives us the following rule for using the coefficients c λ R to decompose M(GL n , R) when n is not stable for R.
Corollary 6.7. The decomposition of M(GL n , R) for any n ≥ 1 is given in terms of the c λ R as where only partitions with length at most n appear in the sum.
Recall that a weight λ = λ 1 ǫ 1 + · · · + λ n ǫ n of GL n is called polynomial if λ i ≥ 0 for all 1 ≤ i ≤ n, and is both dominant and polynomial if and only if λ 1 ≥ · · · ≥ λ n ≥ 0, or in other words if (λ 1 , . . . , λ n ) is a partition with at most n parts. Let Part n denote the set of partitions with at most n parts, Part = n≥0 Part n denote the set of all partitions. We briefly remind the reader how to go between partitions and weights of GL n .
A partition λ = (λ 1 ≥ · · · ≥ λ k > 0) is a weakly decreasing list of positive integers, and the length of the partition λ is ℓ(λ) = k, the length of the list. We draw partitions as Young diagrams using English notation, so that for example λ = (4, 3, 1, 1) is represented as the diagram A partition λ of length at most n may be interpreted as a dominant weight of GL n , by taking λ, α ∨ i to be the number of columns of length i for each 1 ≤ i ≤ n − 1, and taking the number of columns of length n to be the multiplicity of the determinant. For example, the same partition λ = (4, 3, 1, 1) would represent the weight ̟ 1 + 2̟ 2 + det of GL 4 .
Example 6.8. We give a worked example of starting from a finite multiset R, determining the stable coefficients c λ R , and specialising those stable coefficients to any GL n . Let R = {(1, 5), (3, 1), (4, 6)}. The set up(R) ∩ down({(i, c − 2) | (i, c) ∈ R}) are depicted as the green shaded regions in Fig. 7, showing the smallest stable n for this particular multiset is n = 6.
Using a computer, we determine the decomposition of M(GL 6 , R) to be In terms of partitions, the λ for which c λ R = 1 are (49) , and c λ R = 0 for all other partitions λ. Since R lives over I 5 it makes sense to specialise to M(GL 5 , R), whose decomposition we can compute from the stable coefficients c λ R by Corollary 6.7: we simply need to throw away the two partitions whose length is more than 5. In terms of fundamental weights, we obtain the decomposition  Fig. 8. The row-stabilising subgroup R T is generated by the transposition (24), while the column-stabilising subgroup C T is generated by (23) and (45). The diagram D may be made into a skew diagram D ′ by applying the permutation (234) to the rows, followed by (132) to the columns. We then use the theory of skew Schur functions to compute that Σ D ∼ = Σ D ′ ∼ = Σ (2,1,1,1) ⊕ Σ (3,1,1) ⊕ Σ ⊕2 (2,2,1) ⊕ Σ (3,2) . For example, the generalised Littlewood-Richardson coefficient c  {(1, 1), (2, 2), (3, 2), (2, 3), (4, 3)}, pictured as a collection of squares in the plane. We index positions in the plane like matrices, so that the first coordinate goes down the page, and the second coordinate goes right along the page. In the middle is a tableau T : D → {1, . . . , 5}, and on the right is a rearrangement of the rows and columns of D to a second diagram D ′ which is skew, of shape (3, 2, 2, 1)/(2, 1).
6.3. Schur modules associated to arbitrary diagrams. Let V be a finite-dimensional Cvector space, and D be a diagram with d boxes. The tensor power V ⊗d is naturally a (GL(V ), S d ) bimodule, and we define the generalised Schur module to be (up to isomorphism) the left GL(V )- As a consequence of Schur-Weyl duality, the generalised Schur module decomposes as S D (V ) ∼ = ℓ(λ)≤dim V c λ D S λ (V ). The restriction on partitions having length at most dim V is not strictly necessary, since in this case we would have S λ (V ) = 0.
The main theorem to be shown is that in type A, the product monomial crystal is always a crystal of a generalised Schur module.
6.4. Schur and Flagged Schur modules. In order to study the characters of the Schur module S D (V ), it is convenient to introduce a more concrete definition of the Schur module (note that our previous definition was only up to isomorphism), as well as a quotient of the Schur module, called the flagged Schur module. While the Schur module is a module for the whole of GL(V ), the flagged Schur module will be a module for a Borel subgroup of GL(V ), and will only be defined when dim V is large enough compared to D. In the following discussion, we adopt the definitions from [RS99].
For a diagram D, let col j (D) ⊆ D denote the subset of boxes in column j, and row i (D) ⊆ D denote the subset of boxes in row i. Let k (V ), T k (V ), and S k (V ) denote the exterior, tensor, and symmetric algebras of degree k of V . We define the map ψ D as the composition where the first map is comultiplication in each exterior algebra, the second map is the natural rearrangement, and the third map is multiplication in the symmetric algebra. The Schur module S D (V ) is defined as the GL(V )-submodule Im ψ D . Note that rearranging the columns of D leaves S D (V ) invariant, while rearranging the rows of D yields a different (but isomorphic) Schur module. Now, suppose that the diagram D satisfies D ⊆ {1, . . . , r} × N (the diagram fits within rows 1 through r), and n = dim V ≥ r. Fix a full flag of quotient spaces V • = (V n → V n−1 → · · · → V 1 ), by which we mean that dim V i = i and each map V i → V i−1 is surjective. We may postcompose ψ D with the projection and define the flagged Schur module S flag is the subgroup fixing the flag of quotients, then S flag D (V • ) will be a B(V • ) module, but rarely a GL(V ) module. Note that again, the module S flag D (V • ) is unchanged under column permutations of D, but is no longer invariant under row permutations. This construction (and its dual, the Weyl and flagged Weyl modules) are given in full detail in [RS99], sections 2 and 5.
Example 6.12. Let λ be a partition with at most r rows, and D be its Young diagram, placed so that the longest row of λ is in the first row of D. Then S D (C r ) ∼ = V (λ), the irreducible GL rmodule of highest weight λ, and S flag D (C r → C r−1 → · · · → C 1 ) ∼ = V (λ) λ , the highest-weight space. If the diagram is placed upside-down, so that the longest row of λ is in row r, then both the Schur and flagged Schur modules are isomorphic to V (λ).
Many of the results known about the generalised Schur modules S D (V ) are due to geometric constructions of this module as sections of a line bundle over a (generally singular) variety, in [Mag98b;Mag98a]. In this setting, the flagged Schur module (or its dual, the flagged Weyl module) naturally arise, and in [RS98; RS95], a Demazure-type character formula is given for the characters of the flagged Schur modules of percentage-avoiding diagrams D. Fortunately, the diagrams we will encounter are northwest, which are automatically percentage-avoiding, and so these results apply. (A diagram D is northwest if whenever (j, k), (i, l) ∈ D with i < j and k < l, then (i, k) ∈ D). 6.5. Diagrams and multisets defined by partition sequences. It is quite awkward to directly state the map from a multiset R to a corresponding diagram D. Instead, we will define each of R and D from a common partition sequence.
(2) For i > 0, D(λ i ) is obtained from D(λ i−1 ) by shifting the contents of D(λ i−1 ) down one row, and placing the Young diagram of λ (i) to the right of the previous diagram, with the longest row of λ (i) in row 1. Remark 6.16. All of the results of [RS98] apply to the class of column-convex diagrams, which are diagrams where the columns have no gaps: if (i 1 , j) ∈ D and (i 2 , j) ∈ D for i 1 < i 2 , then all of (i 1 , j), (i 1 + 1, j), . . . , (i 2 , j) ∈ D. We note that diagrams of the form D(λ) are always column-convex, and conversely that every column-convex diagram D is of the form D(λ) after applying a column permutation.
Lemma 6.17. Let λ be a partition sequence of length r, and let V be a vector space of dimension n ≥ r, with a fixed full quotient flag V • . Then the characters of the flagged Schur modules S flag D(λ i ) (V • ) satisfy the following recurrence: (1) For i = 0, ch S flag D(λ 0 ) (V • ) = 1.
Remark 6.21. If λ is a partition sequence of length r, then R(λ) is stable for GL r and hence M(GL r , λ) may be used to compute the stable coefficients c λ R of Definition 6.6. Lemma 6.22. Let λ be a partition sequence of length r. Then the characters of the truncated product monomial crystals M(GL r , R(λ i ), J(λ i )) satisfy the following recurrence: (1) For i = 0, ch M(R(λ 0 ), J(λ 0 )) = ch M(∅, J(λ 0 )) = 1.
(2) For i > 0, ch M(R(λ i ), J(λ i )) = e λ (i) · ch M(R(λ i−1 ), J(λ i )) (56) = e λ (i) · π 1 · · · π i−1 ch M(R(λ i−1 ), J(λ i−1 )).  Theorem 6.23. Let λ be a parittion sequence of length r, and let R = R(λ) and D = D(λ). Then the stable coefficients of R and D coincide, i.e. we have c µ R = c µ D for all partitions µ, and furthermore for any n such that R(λ) lives over I n (equivalently, D(λ) has columns of length at most n), we have ch M(GL r , R) = ch S D (C n ) and hence the monomial crystal M(GL r , R) is the crystal of the generalised Schur module S D (C n ).
Proof. Since the recurrences Lemma 6.17 and Lemma 6.22 are identical, letting J = J(λ) we have the equality of characters ch S flag D (V • ) = ch M(GL r , R, J). Applying the Demazure operator π w• to both sides yields (by Corollary 5.14 and [RS98, Theorem 21]) the equality of characters ch S D (C r ) = ch M(GL r , R). Since both D and R are stable for GL r we have c µ R = c µ D for all partitions µ. Comparing the two restriction rules given in Corollary 6.7 and Section 6.3 completes the proof.
Remark 6.24. Theorem 6.23 applies to all finite multisets R since we may assume that R is contained in down({(1, −1)}) after performing a vertical shift on the whole of R, which does not change the isomorphism class of M(R). We may always write the shifted R as R(λ) for some partition sequence λ, then M(GL n , R) is the crystal of S D(λ) (C n ). Conversely, every columnconvex diagram is D(λ) for some λ, and hence Theorem 6.23 gives a positive combinatorial formula for ch S D (C n ) in terms of a sum over elements of the corresponding product monomial crystal.
Remark 6.25. By [Kam+19b], the category O of truncated shifted Yangians provides categorifications of g-modules whose associated crystal is the product monomial crystal. Hence, by Theorem 6.23, we deduce that in type A these are categorifications of generalised Schur modules, in the column-convex case. In particular, the truncated shifted Yangians produce (the first known) categorifications of skew Schur modules.