Cyclic Demazure modules and positroid varieties

A positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties. Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which we call the cyclic Demazure module. In this note, we show that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.


Introduction
The classical Borel-Weil theorem identifies the global sections Γ(G/B, L λ ) of a line bundle on a flag variety with the irreducible highest weight representation V (λ). When the same line bundle is restricted to a Schubert variety X w , the global sections Γ(X w , L λ ) can be identified with the Demazure module V w (λ). In this paper, we study the global sections Γ(Π f , O(d)) of a line bundle on a positroid subvariety Π f of the Grassmannian Gr(k, n).
Positroid varieties (see Section 4) are certain intersections of cyclically rotated Schubert varieties in the Grassmannian. They were introduced in Postnikov's work [Pos] on the totally nonnegative Grassmannian, and subsequently studied in algebro-geometric terms by . Via [KLS13], the work of Lakshmibai and Littelmann [LaLi] gives a description of the vector space Γ(Π f , O(d)) in terms of standard monomials. In the present work, we give a new description of Γ(Π f , O(d)) that is compatible with the cyclic symmetry of the Grassmannian and its positroid varieties.
We define in Section 5 the cyclic Demazure module V f (dω k ) as the intersection of cyclically rotated Demazure modules. We show in Theorem 5.5 that a graded piece of the homogeneous coordinate ring of a positroid variety can be identified with the (dual of the) cyclic Demazure module.
Our approach is based on the key observation (Theorem 2.1(4)) that the dual canonical basis of the Grassmannian is invariant under signed cyclic rotation. This relies heavily on the work of Rhoades [Rho]. In Theorem 2.1(2), we show that dual canonical basis in degree two is identical to the Temperley-Lieb invariants of [Lam14], which are defined in a combinatorially explicit manner.
We define the cyclic Demazure crystal B f (dω k ) as the intersection of cyclically rotated (via promotion) Demazure crystals. We show in Theorem 5.4 that V f (dω k ) has a basis given by the canonical basis elements indexed by B f (dω k ). We obtain the following dichotomy (Theorem 5.13): a dual canonical basis element either (1) vanishes on Π f (if it is outside B f (dω k )), or (2) it takes strictly positive values on the totally positive part Π f,>0 (if it is inside B f (dω k )).
Our work was initially motivated by the budding theory of Grassmann polytopes, and many of the results here were announced initially in [Lam16]. However, the results herein T.L. was supported by NSF grants DMS-1160726 and DMS-1464693, and by a Von Neumann fellowship at the Institute of Advanced Study.
were so simple and clean, we felt that they deserved a short and separate exposition. We plan to pursue our intended applications in other work. In Section 6, we also indicate some further directions of study.
We also note that R(k, n) is a unique factorization domain. We let R(k, n) d denote the d-th graded piece of R(k, n), spanned by monomials ∆ I 1 ∆ I 2 · · · ∆ I d . We begin by reviewing the classical description of R(k, n) d in representation theoretic terms.
For a partition λ with at most n parts, we have an irreducible, finite-dimensional representation V (λ) of GL(n) with highest weight λ. We state some basic facts concerning V (λ).
The Young diagram of λ is the collection of boxes in the plane with λ 1 boxes in the 1st row, λ 2 boxes in the 2nd row, and so on, where all boxes are upper-left justified. A semistandard tableaux T of shape λ is a filling of the Young diagram of λ by the numbers 1, 2, . . . , n so that each row is weakly-increasing, and each column is strictly increasing. The weight wt(T ) of a tableau T is the composition (α 1 , α 2 , . . . , α n ) where α i is equal to the number of i-s in T . For example, 1 1 3 4 4 2 3 4 5 4 4 is a semistandard tableau with shape (5, 4, 2) with weight (2, 1, 2, 5, 1). Let B(λ) denote the set of semistandard tableaux of shape λ. (Note that this set depends on n, which is suppressed from the notation.) The dimension dim(V (λ)) is equal to the cardinality of B(λ). A vector v in a GL(n)-representation V is called a weight vector with weight α = (α 1 , α 2 , . . . , α n ) if the diagonal matrix diag(x 1 , x 2 , . . . , x n ) sends v to (x α 1 1 x α 2 2 · · · x αn n )v. Lusztig [Lus94] and Kashiwara [Kas93a] have constructed a canonical basis, or global basis of the U q (sl n )-module V q (λ), which is a quantization of V (λ). We shall only use the evaluation of this basis at q = 1. After picking a highest weight vector v + for V (λ), there exists a distinguished basis {G(T ) | T ∈ B(λ)} of V (λ) such that each G(T ) is a weight vector with weight wt(T ). We shall also let {H(T ) | T ∈ B(λ)} denote the dual basis of V (λ) * , called the dual canonical basis. Let (·, ·) denote the unique nondegenerate symmetric bilinear form on V (λ) satisfying (v + , v + ) = 1, and (x·v, u) = (v, x T ·u) where x ∈ gl n and x T denotes the transpose. We may identify V (λ) with V (λ) * via (·, ·), and H(T ) becomes a basis of V (λ).
2.3. Crystals. The set B(λ) has the structure of a crystal graph. We will only need the operationsẽ for i = 1, 2, . . . , n − 1. Let T ∈ B(λ). The rowword row(T ) of T is obtained by reading the rows of T from left to right, starting from the bottom row.
For a fixed i ∈ {1, 2, . . . , n − 1}, we think of each occurrence of i in row(T ) to be a closed parentheses ")" and each occurrence of i + 1 in row(T ) to be an open parentheses "(". We then pair these parentheses as usual until no pairing can be done. We are left with a sequence that looks like "))))((". The operationf i changes the i corresponding to the rightmost unpaired ")" into a i + 1. The operationẽ i changes the i + 1 corresponding to the leftmost unpaired "(" into a i. The result will be the rowword of a unique tableauẽ i (T ) or f i (T ) of shape λ. If there is no such ")" (resp. "(") thenf i (T ) (resp.ẽ i (T )) is defined to be 0.

Kirillov-Reshetikhin crystals.
Let ω k = (1, 1, . . . , 1) be the partition with k 1's. Then V (ω k ) is isomorphic to the k-exterior power Λ k (C n ) of the standard representation C n of GL(n) and the canonical basis of V (ω k ) is simply the basis {e i 1 ∧ e i 2 ∧ · · · ∧ e i k }. For an integer d ≥ 1, the representation V (dω k ) for a rectangular partition has very special properties. The set B(dω k ) is the set of semistandard Young tableaux with k rows and d columns. For example, 1 1 3 4 4 2 3 4 5 5 4 4 6 6 6 belongs to B(5ω 3 ).
The crystal B(dω k ) has an additional operation called promotion, which is a bijection χ : B(dω k ) → B(dω k ). We have χ n = 1. Promotion is defined as follows: first remove all occurrences of the letter n in T . Then slide the boxes to the bottom right of the rectangle, always keeping the rows weakly-increasing and columns strictly-increasing. Once all slides are complete, we add one to all letters, and fill the empty boxes with the letter 1 to obtain χ(T ). For example, 1 1 3 4 4 2 3 4 5 5 4 4 6 6 6 → 1 1 3 4 4 2 3 4 5 5 4 4 → 1 1 2 3 3 4 4 4 4 4 5 5 → 1 1 1 2 2 3 4 4 5 5 5 5 5 6 6 We have , and this defines extra operationsẽ 0 and f 0 on B(dω k ). Together these structures form part of the affine crystal structure of B(dω k ), which in this case is a Kirillov-Reshetikhin crystal.
2.5. The dual canonical basis of the Grassmannian. By the classical Borel-Weil theorem, the degree d component R(k, n) d of the graded ring R(k, n) is canonically isomorphic, as a GL(n)-representation, to the dual V (dω k ) * of the highest weight representation V (dω k ).
Let χ : C n → C n denote the (signed) cyclic rotation linear map given by sending e i to e i+1 for i = 1, 2, . . . , n − 1 and sending e n to (−1) k−1 e 1 . It also induces a rotation map χ : Gr(k, n) → Gr(k, n). Since χ ∈ GL(n), we obtain a cyclic rotation map χ : Theorem 2.1. The vector space R(k, n) d has a dual canonical basis {H(T ) | T ∈ B(dω k )} with the following properties: (1) For d = 1, we have H(T ) = ∆ I , where I is the set of entries in the one-column tableau T .
Already for d = 3, the canonical basis of V (3ω k ) is combinatorially obscure to us. In [Lam14], we studied the closely related web basis in combinatorial terms.
Proof. Letχ denote the unsigned cyclic rotation map that sends e i to e i+1 mod n . We first show thatχ(G(T )) = ±G(χ(T )).
Let A(n) denote the coordinate ring of n × n matrices, so that ]. The ring A(n) has a dual canonical basis b P,Q labeled by pairs of semistandard tableaux P, Q of the same shape and entries bounded by n. The cyclic rotationχ acts on A(n) by sending the matrix entry x ij to x i+1,j where indices are taken modulo n. Equivalently, thinking of A(n) as the space of polynomial functions on End(C n ), we have for f ∈ A(n) and g ∈ End(C n ). In [Rho,Proposition 5.5], Rhoades shows that when P, Q have rectangular shape, we have where the other terms belong to the span of the dual canonical basis indexed by shapes different to the shape of P, Q.
The basis studied in [Rho] is connected to the canonical bases of the highest weight representations V (dω k ) via the works of Skandera [Ska] and Du [Du92]. Specifically, Du shows that the comodule map τ : is a subset of the dual canonical basis of A(n) and G(P ) belongs to the canonical basis of V (dω k ). (The coincidence of Du's basis with Lusztig's is shown in [Du95].) A computation from the definitions shows that (χ ⊗χ)τ (v + ) = τ (v + ). It follows that The set T (d, k) consists of all b P,Q where Q is some fixed semistandard tableaux of rectangular shape d k . It follows thatχ · b P,Q ∈ T (d, k), and in (2) all the "other terms" from (1) cancel out. We conclude thatχ(G(P )) = ±G(χ(P )). It follows easily from the fact that G(P ) is a weight-vector that χ(G(P )) = ±G(χ(P )) and by duality we have χ * (H(P )) = ±H(χ(P )). Now χ(Gr(k, n) ≥0 ) = Gr(k, n) ≥0 , so it follows from Theorem 2.1(3) that we must have χ * (H(P )) = H(χ(P )). (Note that H(P ) cannot be identically 0 on Gr(k, n) ≥0 because the latter is Zariski-dense in Gr(k, n).)
Define a bijection θ : A k,n → B(2ω k ) as follows. Given (τ, T ), the tableau θ(τ, T ) has columns I 1 , I 2 , where I 1 ∩ I 2 = T , and for each strand (a, b) ∈ τ with a < b, we have a ∈ I 1 and b ∈ I 2 .
Proposition 3.2. We have ∆ (τ,T ) = H(θ(τ, T )). Thus Temperley-Lieb immanants are the dual canonical basis of V (2ω k ): Proof. We deduce the proposition from setting q = 1 in work of Brundan [Bru] and Cheng-Wang-Zhang [CWZ]. In [CWZ,Section 4], the dual canonical basis for V (ω k ) ⊗ V (ω k ) is constructed for any k. The dual canonical basis elements are denoted L f in [CWZ]; we shall write them as L I,J where I, J are k-element subsets of [n] (note that [CWZ] are working with n = ∞). The standard basis of V (ω k ) ⊗ V (ω k ) will be denoted by K I,J . By [Bru,Theorem 26], there is a linear map ξ : are the two columns of a semistandard tableau T of shape 2 k , and to 0 otherwise. In our notation, the map ξ also sends the standard basis element K I,J to the monomial ∆ I ∆ J . By Theorem 3.1, it thus suffices to show that for a 2-column tableau T with columns A, B, that the coefficient of L T := L A,B in K I,J is equal to 1 or 0 depending on whether θ −1 (T ) is compatible with (I, J) or not.
Let A, B be two k-element subsets of [n]. Cheng-Wang-Zhang [CWZ] define a set of pairs It is then shown in [CWZ,Corollary 4.18] 1 that the coefficient of L A,B in K I,J is equal to 1 if (I, J) can be obtained from (A, B) by swapping the pairs in some subset Σ ⊆ Σ − A,B , and equal to 0 otherwise. Now, suppose that A, B are the two columns of a semistandard tableaux T of shape 2 k . It is then easy to check that Σ − A,B is exactly the set of strands of the non-crossing matching of θ −1 (A, B). This completes the proof. We shall need to consider pairs (I, J) where I is an ordered sequence (i 1 , i 2 , . . . , i k ) of distinct integers in [n]. LetĪ ∈ [n] k denote the k-element subset consisting of the elements of the sequence I. We say that (I, J) is standard if (Ī, J) is. We also have a matching τ (I, J) := τ (Ī, J). We write (I = (i 1 , . . . , i a , . . . , i k ), J) → a (I ′ = (i 1 , . . . , j, . . . , i k ), J ′ = J \ {j} ∪ {i a }) if (i a , j) ∈ τ (I, J) and (I ′ , J ′ ) is a standard pair.
A legal path P of length |P | = r between (I, J) and (K, L) is a sequence where I 0 is equal to I arranged in order,Ī r = K, and a 1 ≥ a 2 ≥ · · · ≥ a r . Note that I ∪ J = K ∪ L as multisets whenever a legal path exists.
The following result can be deduced from [CWZ]. We give an independent proof.
Theorem 3.3. We have where the first summation is over all standard pairs (I, J) and the second summation is over legal paths from (I, J) to θ(τ, T ). is not legal, because the sequence 3, 2, 3 is not weakly decreasing.
If (i < j), (r < s) ∈ τ are two strands of a noncrossing matching, we say that (i, j) is nested under (r, s) if (r < i < j < s). We say that (i, j) is nested immediately under (r, s) if, in addition, there is no strand (p, q) such that (i, j) is nested under (p, q), and (p, q) is nested under (r, s).
Lemma 3.5. Suppose that (I, J) → a (I ′ , J ′ ), where (i a , j) is the strand swapped. Then there is a unique (i b , j ′ ) ∈ τ (I, J) such that (i a , j) is nested immediately under (i b , j ′ ). Furthermore, τ (I ′ , J ′ ) is obtained from τ (I, J) by replacing the two strands (i a , j) and (i b , j ′ ) by the two strands (i b , i a ) and (j, j ′ ).
Proof. If (i a , j) is not nested under any other strand, then swapping i a with j in (Ī, J) cannot give a standard pair. This gives the first statement. It is easy to see that replacing (i a , j) and (i b , j ′ ) by the two strands (i b , i a ) and (j, j ′ ) does indeed give a noncrossing matching, and the second statement follows.
Lemma 3.6. Suppose that we have a legal path ending at (I = (i 1 , i 2 , . . . , i k ), J). Suppose that a < b and that (i a , j) and (i b , j ′ ) are both in τ (I, J). Then (i a , j) is never nested under (i b , j ′ ).
Proof. Let P = (I 0 , J) → a 1 (I 1 , J 1 ) → a 2 · · · → ar (I r , J r ) be a legal path ending at (I r , J r ) = (I = (i 1 , i 2 , . . . , i k ), J), and suppose a < b. We proceed by induction on r. If r = 0, the claim is clear. If i a < i b , the claim is clear. Thus we may assume that i a > i b and a r ≤ a. If a r < a, then by induction and Lemma 3.5, the last swap → ar (I r , J r ) does not affect the strands (i a , j) and (i b , j ′ ) incident to i a and i b .
Finally, suppose that a r = a, and let (I ′ , J ′ ) = (I r−1 , J r−1 ). There are two cases: In the first case, the claim follows from Lemma 3.5, and in the second case, the claim follows from the inductive assumption and Lemma 3.5.
Proof of Theorem 3.3. Any i ∈ T is present in both I and J for all terms on the RHS. Thus it suffices to prove the statement assuming that T = ∅ and τ is a complete noncrossing matching on [n] = [2k]. Henceforth, we make this assumption; thus we restrict to standard pairs (I, J) using each element in [2k] exactly once. Restricting Theorem 3.1 to these standard pairs, we must show that (3) and (4) give inverse matrices.
Define a partial order ≤ on standard pairs by (I, J) ≤ (C, D) if i r ≤ c r for r = 1, 2, . . . , k, where I = {i 1 < i 2 < · · · < i k } and C = {c 1 < c 2 < · · · < c k }. where the first summation is over all standard pairs (C, D) such that θ −1 (C, D) ∈ C(K, L), and the second summation is over all legal paths P from (I, J) to (C, D). The statement is clear when (I, J) = (K, L). Suppose (I, J) < (K, L). We provide a sign-reversing involution ι on the terms in (5). If τ (C, D) is compatible with (K, L) we can obtain (K, L) from (C, D) by swapping some (uniquely determined) subset S(C, D) of the strands in τ (C, D). Let a legal path P = · · · (C ′′ , D ′′ ) → x (C, D) from (I, J) to (C, D) be given, where C = (c 1 , c 2 , . . . , c k ). With respect to (C, D), the minimum strand (c i , d) in S(C, D) is the strand where i is minimal. We define ι(P ) by splitting into two cases. Case (1): If |S(C, D)| > 0 and (c i , d) is the minimal strand, and either (1) P is empty, or (2) P is nonempty and i ≤ x, then ι(P ) = P → i (C ′ , D ′ ) is the path obtained by concatenating to P the swap (C, D) → i (C ′ , D ′ ). Note that τ (C ′ , D ′ ) is still compatible with (K, L): if (c i , d) is nested under a strand (c s , d ′ ), then by Lemmas 3.5 and 3.6, we cannot have (c s , d ′ ) ∈ S(C, D). Thus S(C ′ , D ′ ) = S(C, D) \ (c i , d).

Schubert varieties and positroid varieties
The Schubert variety X I (F • ) is given by Here and elsewhere, we always mean complex (co)dimension when referring to complex subvarieties.
Let E • be the standard flag defined by E i = span(e n , e n−1 , . . . , e n−i+1 ). Then we set the standard Schubert varieties to be X I := X I (E • ). Suppose v 1 , v 2 , . . . , v n are the columns of a k × n matrix (with respect to the basis e 1 , e 2 , . . . , e n ) representing X ∈ Gr(k, n). Then the condition dim(X ∩E j ) = d is equivalent to the condition dim span(v 1 , . . . , v n−j ) = k−d. Thus the Schubert variety X I (E • ) is cut out by rank conditions on initial sequences of columns of X.

4.2.
Bounded affine permutations, Grassmann necklaces, and positroids. A (k, n)bounded affine permutation is a bijection f : Z → Z satisfying conditions: ( The set B(k, n) of (k, n)-bounded affine permutations forms a lower order ideal in the Bruhat order of the affine symmetric group ( [KLS13]).
Let I = {i 1 < i 2 < · · · < i k } and J = {j 1 < j 2 < · · · < j k } be two k-element subsets of [n]. We define a partial order ≤ on [n] k by I ≤ J if i r ≤ j r for r = 1, 2, . . . , k. We write ≤ a for the cyclically rotated ordering a < a a + 1 < a · · · < a n < a 1 < a · · · < a a − 1 on [n]. Replacing ≤ by ≤ a , we also have the cyclically rotated version partial order I ≤ a J on [n] k . A (k, n)-Grassmann necklace [Pos] is a collection of k-element subsets I = (I 1 , I 2 , . . . , I n ) satisfying the following property: for each a ∈ [n]: (1) There is a partial order on the set of (k, n)-Grassmann necklaces, given by I ≤ J if I a ≤ a J a for all a = 1, 2, . . . , n.
Proposition 4.1. The map f → I(f ) is a bijection between (k, n)-bounded affine permutations and (k, n)-Grassmann necklaces.
The inverse map I → f (I) is given as follows. Suppose a / ∈ I a . Then define f (a) = a. Suppose a ∈ I a and I a+1 = I a − {a} ∪ {a ′ }. Then define f (a) = b where b ≡ a ′ mod n and a < b ≤ a + n.
By [KLS13,KLS14], the restriction map Γ (Gr(k, n), is the line bundle on Gr(k, n) associated to the Plücker embedding (and in particular, Γ(Π f , O(d)) = 0 for d < 0). Thus the homogeneous coordinate ring R(Π f ) := d≥0 Γ(Π f , O(d)) of Π f is a quotient of the homogenous coordinate ring R(k, n). We write I(Π f ) for the homogeneous ideal of Π f and denote byΠ f := Spec(R(Π f )) ⊂Ĝr(k, n) the affine cone over the positroid variety Π f .
Recall that for X ∈ Gr(k, n), the matroid M X of X is defined as . Define Π f,>0 :=Π f ∩ Gr(k, n) ≥0 . The following result of Oh characterizes the matroids of totally nonnegative points.

The cyclic Demazure module
5.1. Demazure modules and Demazure crystals. Let I(X I ) ⊂ R(k, n) denote the homogeneous ideal of the Schubert variety X I (see Section 4.1) and let I(X I ) d ⊂ R(k, n) d denote the degree d component. Let R(X I ) d = Γ(X I , O(d)) denote the degree d part of the homogeneous coordinate ring of X I . The restriction map Γ (Gr(k, n), O(d)) → Γ(X I , O(d)) is known to be surjective, and thus the space R(X I ) d is naturally a quotient of R(k, n) d = V (dω k ) * .
For I ∈ [n] k , we have an extremal weight vector G(T I ) ∈ V (dω k ). The vector G(T I ) spans the weight space of V (dω k ) with weight α = (α 1 , . . . , α n ) given by α i = d if i ∈ I and α i = 0 otherwise. The Demazure module V I (dω k ) is defined to be the B − -submodule of V (dω k ) generated by the vector G(T I ). It is a classical result that I(X I ) d can be identifed with V I (dω k ) ⊥ ⊂ V (dω k ) * (see for example [Kum,Chapter 8]).
k , we have a tableau T I ∈ B(dω k ) with all entries in the r-th row equal to i r , where I = {i 1 , i 2 , . . . , i k }. The canonical basis vector G(T I ) is an extremal weight vector of V (dω k ). Define the Demazure crystal B I (dω k ) ⊂ B(dω k ) to be the subset of B(dω k ) obtained by repeatedly applying the operatorsf 1 ,f 2 , . . . ,f n−1 to T I .
The following result is due to Kashiwara [Kas93b].
By Theorem 5.1, we obtain: Let us give a more explicit description of B I (dω k ).
Proposition 5.3. The set B I (dω k ) consists of tableaux T which are entry-wise greater than or equal to T I .
Proof. Let S denote the set of tableaux T ∈ B(dω k ) that are entry-wise greater than or equal to T I . Since the operatorsf i decreases a single entry of a tableau, it is clear that B I (dω k ) is contained in S. Also, it is known that the set S indexes a basis for R(X I ) d known as the standard monomial basis, see for example [LaLi]. Thus |B I (dω k )| = |S|, so B I (dω k ) = S.

If we identify B(ω k ) with the set [n]
k of k-element subsets of [n], then B f (ω k ) is simply the positroid M(f ) (8). Also, define the cyclic Demazure module V f (dω k ) to be intersection Let R(Π f ) denote the homogeneous coordinate ring of the positroid variety Π f .
(1) I(Π f ) d is isomorphic to V f (dω k ) ⊥ and has a basis given by Theorem 5.5 reduces in the case d = 1 to Theorem 4.2.
Indeed, for any a, the function ∆ Ia is non-zero onΠ f , and so is ∆ d Ia .
Remark 5.7. It follows from Theorem 5.5 that the vectors H(T ) ∈ V (dω k ) * that do not restrict to identically zero on Π f form a basis for R(Π f ) d . This is not the case for the standard monomial basis (cf. [LaLi]). Example 5.10. Consider k = 2 and n = 5. Let us consider the positroid variety Π f where f = [63547] ∈ B(2, 5) and compute B f (2ω 2 ). The Grassmann necklace is I(f ) = (12,12,13,15,15). Since B 12 (2ω 2 ) = B(2ω 2 ), we have The set B 15 (2ω 2 ) consists of all tableaux of the form a b 5 5 with 1 ≤ a ≤ b ≤ 4 and thus χ(B 15 (2ω 2 )) consists of all tableaux of the form 1 1 a b with 2 ≤ a ≤ b ≤ 5. In particular, every tableau in B f (2ω 2 ) has exactly two 1-s. Intersecting with χ 2 (B 14 (2ω 2 )) imposes no additional restriction. On the other hand, looking at tableaux in B 23 (2ω 2 ) with two 3-s, we get the six tableaux and thus B f (2ω 2 ) consists of the tableaux 1 1 5 5 1 1 2 5 1 1 3 5 1 1 2 2 1 1 2 3 1 1 3 3 We give an example of a Schubert variety whose ideal does not have a basis given by a subset of the dual canonical basis.
Example 5.11. Let X ⊂ Gr(2, 4) be given by the single equation {∆ 13 = 0}. This is a permutation of a standard Schubert variety that is not a positroid variety. Then the degree two part of I(X) has a one-dimensional weight space for the weight (1, 1, 1, 1). It is spanned by the vector ∆ 13 ∆ 24 . This vector is a sum of two elements of the dual canonical basis by Theorem 3.1.

Positivity.
Theorem 5.13. For f ∈ B(k, n) and T ∈ B(dω k ), if H(T ) is not identically zero on Π f , then it takes strictly positive values everywhere on Π f,>0 .
Fix a lift of e I toĜr(k, n). Then the value of the dual canonical basis element H(T ) ∈ V (dω k ) * on the point X = x i 1 (t 1 ) · · · x i d (t d ) · e I ∈Ĝr(k, n) is given by H(T ), x i 1 (t 1 ) · · · x i d (t d ) · G(T I ) where ·, · denotes the natural pairing between V (dω k ) * and V (dω k ), and G(T I ) is the canonical basis element of extremal weight indexed by T I .
Thus it suffices to show that for any T ∈ B(dω k ), the coefficient of G(T ) in x i 1 (t 1 ) · · · x i d (t d )· G(T I ) is equal to a (possibly zero) polynomial in t 1 , t 2 , . . . , t d with nonnegative coefficients. For i = 1, 2, . . . , n − 1, it follows from the proof of [Lus94,Proposition 3.2] that the matrix coefficients of x i (a) on the canonical basis of V (dω k ) are polynomials in t 1 , t 2 , . . . , t d with nonnegative coefficients. By Theorem 2.1(4), the same holds for i = 0. The claim follows.
6. Future directions 6.1. The character of the cyclic Demazure module. The character of highest weight representation V (dω k ) is given by the celebrated Weyl character formula. The character of the Demazure module V I (dω k ) is given by the Demazure character formula [Dem, And]. Problem 6.9. Give an explicit combinatorial formula for the expansion of η d,d ′ k (G(T )) in the canonical basis.