k-POSITIVITY OF DUAL CANONICAL BASIS ELEMENTS FROM 1324- AND 2143-AVOIDING KAZHDAN-LUSZTIG IMMANANTS

In this note, we show that certain dual canonical basis elements of C[SLm] are positive when evaluated on k-positive matrices, matrices whose minors of size k × k and smaller are positive. Skandera showed that all dual canonical basis elements of C[SLm] can be written in terms ofKazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by 1324and 2143-avoiding permutations. This extends previous work of the authors on KazhdanLusztig immanants and uses similar tools, namely Lewis Carroll’s identity (also known as the Desnanot-Jacobi identity).


Introduction
Given a function f : S n → C, the immanant associated to f , Imm f X : Mat n×n (C) → C, is the function Imm f X := w∈Sn f (w) x 1,w(1) · · · x n,w(n) , (1.1) where the x i,j are indeterminates. We evaluate Imm f X on a matrix M = (m i,j ) by specializing x i,j to m i,j for all i, j.
Immanants are a generalization of the determinant, where f (w) = (−1) (w) , and the permanent, where f (w) = 1. Positivity properties of immanants have been studied since the early 1990's [11][12][13]20]. One of the main results in this area is that when f is an irreducible character of S n , then Imm f (X) is nonnegative on totally nonnegative matrices, that is, matrices with all nonnegative minors [19]. In this note, we will investigate positivity properties of functions closely related to Kazhdan-Lusztig immanants, introduced by Rhoades and Skandera [16]. Definition 1.1. Let v ∈ S n . The Kazhdan-Lusztig immanant Imm v X : Mat n×n (C) → C is given by Imm v X := w∈Sn (−1) (w)− (v) P w0w,w0v (1) x 1,w1 · · · x n,wn (1.2) where P x,y (q) is the Kazhdan-Lusztig polynomial associated to x, y ∈ S n , w 0 ∈ S n is the longest permutation, and we write permutations w = w 1 w 2 . . . w n in one-line notation. (For the definition of P x,y (q) and their basic properties, see e.g. [1].) Our interest in Kazhdan-Lusztig immanants stems from their connection to the dual canonical basis of C[SL m ]. Using work of Du [9], Skandera [18] showed that the dual canonical basis elements of C[SL m ] are exactly Kazhdan-Lusztig immanants evaluated on matrices of indeterminates with repeated rows and columns.
Let X = (x ij ) be the m × m matrix of variables x ij and let [m] n denote the set of n-element multisets of [m] := {1, . . . , m}. For R, C ∈ [m] n with R = {r 1 ≤ · · · ≤ r n } and C = {c 1 ≤ · · · ≤ c n }, we write X(R, C) to denote the matrix (x ri,cj ) n i,j=1 (see Definition 2.8 Imm v X(R, C) : v ∈ S n for some n ∈ N and R, C ∈ [m] n .
The positivity properties of dual canonical basis elements have been of interest essentially since their definition, and are closely related to the study of total positivity. In 1994, Lusztig [14] defined the totally positive part G >0 of any reductive group G. He also showed that all elements of the dual canonical basis of O(G) are positive on G >0 . Fomin and Zelevinsky [10] later proved that for semisimple groups, G >0 is precisely the subset of G where all generalized minors are positive. Generalized minors are dual canonical basis elements corresponding to the fundamental weights of G and their images under Weyl group action.
Here, we study signs of dual canonical basis elements on a natural generalization of G >0 . Let S be some subset of generalized minors and G S >0 the subset of G where all elements of S are positive. Which dual canonical basis elements are positive on all elements of G S >0 ? In this note, we consider the case where G = SL m and S consists of the generalized minors corresponding to the first k fundamental weights and their images under the Weyl group action. In this situation, G S >0 is the set of k-positive matrices, matrices where all minors of size k and smaller are positive. Cluster algebra structures, topology, and variation diminishing properties of these matrices have been previously studied in [2,4,7,8].
We call a matrix functional k-positive if it is positive when evaluated on all k-positive matrices. Our main result is as follows: Let v ∈ S n be 1324-, 2143-avoiding and suppose that for n . Then Imm v X(R, C) is identically zero or it is k-positive.
We also characterize precisely when the functions Imm v X(R, C) appearing in Theorem 1.3 are identically zero (see Theorem 3.1). Theorem 1.3 extends the results of [5], in which we showed the function Imm v X([m], [m]) is k-positive under the assumptions of Theorem 1.3. Our techniques here are similar to [5]. Note that Theorem 1.3 does not follow from [5,Theorem 1.4] because for M k-positive, M (R, C) is k-nonnegative rather than k-positive.
Rephrasing Theorem 1.3 in terms of dual canonical basis elements, we have the following corollary.
Suppose v is 1324-, 2143-avoiding and for all The paper is organized as follows. Section 2 gives background on the objects we will be using to prove Theorem 1.3. It includes several useful lemmas proven in [5]. Section 3 contains the proof of Theorem 1.3. We conclude with a few thoughts on future directions in Section 4.

Background
In an abuse of notation, we frequently drop curly braces around sets appearing in subscripts and superscripts.
2.1. Background on 1324 and 2143-avoiding Kazhdan-Lusztig immanants. For integers i ≤ j, let [i, j] := {i, i + 1, . . . , j − 1, j}. We abbreviate [1, n] as [n]. For v ∈ S n , we write v i or v(i) for the image of i under v. We use the notation < for both the usual order on [n] and the Bruhat order on S n ; it is clear from context which is meant. To discuss non-inversions of a permutation v, we'll write i, j to avoid confusion with a matrix index or point in the plane. In the notation i, j , we always assume i < j. We use the notation [m] n for the collection of n-element multi-sets of [m]. We always list the elements of a multiset in increasing order.
We are concerned with two notions of positivity, one for matrices and one for immanants. Our results on k-positivity of Kazhdan-Lusztig immanants involve pattern avoidance.
Let v ∈ S n , and let w ∈ S m . Suppose v = v 1 · · · v n and w = w 1 · · · w m in one-line notation. The pattern w 1 · · · w m occurs in v if there exists 1 ≤ i 1 < · · · < i m ≤ n such that v i1 · · · v im are in the same relative order as w 1 · · · w m . Additionally, v avoids the pattern w 1 · · · w m if it does not occur in v.
Certain Kazdhan-Lusztig immanants have a very simple determinantal formula, which involves the graph of an interval. We think of an element (i, j) ∈ Γ[v, w] as a point in row i and column j of an n × n grid, indexed so that row indices increase going down and column indices increase going right (see Example 2.6). A square or square region in Γ[v, w] is a subset of Γ[v, w] which forms a square when drawn in the grid.
We will also need the following notion on matrices.  Note that v avoids patterns 1324 and 2143.
We can now state a simple determinantal formula for certain Kazhdan-Lusztig elements. This follows from results of [17]. (2.1) Using Proposition 2.7, we can similarly obtain a simple determinantal formula for certain dual canonical basis elements of C[SL m ]. Recall from Proposition 1.2 that every dual canonical basis element can be expressed as a Kazhdan-Lusztig immanant evaluated on a matrix of indeterminants with repeated rows and columns.
Definition 2.8. Let R = {r 1 ≤ r 2 ≤ · · · ≤ r n } and C = {c 1 ≤ c 2 ≤ · · · ≤ c n } be elements of [m] n and let M = (m ij ) be an m × m matrix. We denote by M (R, C) the matrix with (i, j)-entry equal to m ri,cj . We call r i the label of row i; similarly, c j is the label of column j. We view X(R, C) as a function from Mat m×m (C) to Mat n×n (C), which takes M to M (R, C).
Note that our convention is always to list multisets in weakly increasing order, so the row and column labels of X(R, C) are weakly increasing.
If M is the matrix from Example 2.2, then We will focus on the dual canonical basis elements Imm v X(R, C) where v is 1324-and 2143avoiding. Proposition 2.7 immediately gives a determinantal formula for these immanants.
n and let v ∈ S n be 1324-and 2143-avoiding. Then We are interested in the sign of Imm v X(R, C) on k-positive matrices, so long as Imm v X(R, C) is not identically zero. Clearly, the function in (2.2) is identically zero when the matrix X(R, C)| Γ[v,w0] has two identical rows or columns. We make the following definitions to discuss this situation.
The support of row r of P is the set of columns c ∈ [n] such that (r, c) ∈ P . The support of a column is defined analogously.
n . Then P is (R, C)-admissible if no two rows or columns with the same labels have the same support. For v avoiding 1324 and 2143, In the subsequent sections, we will show the converse holds as well (see Theorem 3.10).
Finally, we introduce some notation that will be useful in proofs.
The labels of rows and columns are preserved under deletions; to be more precise, if R = {r 1 ≤ · · · ≤ r n } is the multiset of row labels of P , the multiset of row labels of P J I is Combinatorics of graphs of upper intervals. We will now take a closer look at the graphs Γ[v, w 0 ] that appear in Lemma 2.10. We begin by giving an alternate definition for Γ[v, w 0 ].
We also say k, l sandwiches (i, j).
In other words, (i, j) is sandwiched by k, l if and only if (i, j) ∈ [n] 2 lies inside the rectangle with diagonal corners (k, v k ) and (l, v l ).
Using this alternate characterization, one can translate the assumptions of Theorem 1.3 into a condition on Γ[v, w 0 ].
We now introduce some notation and a proposition that we will need to prove our main result.
The name "bounding boxes" comes from the following lemma.
We also color the bounding boxes.
is on the antidiagonal, and blue if (i, v i ) is above the antidiagonal. If B i,vi and B n−vi+1,n−i+1 are both bounding boxes, then B i,vi = B n−vi+1,n−i+1 is both red and blue. We say such a box is purple. (See Figure 1 for an example.) Proposition 2.21 ( [5, Proposition 4.14]). Suppose v ∈ S n avoids 2143 and w 0 v is not contained in a maximal parabolic subgroup of S n . Order the bounding boxes of Γ[v, w 0 ] by the row of the northwest corner. If Γ[v, w 0 ] has more than one bounding box, then they alternate between blue and red and there are no purple bounding boxes.

Positivity of basis elements
In this section, we prove our main result.   3.1. Young diagram case. We first consider the case where Γ[v, w 0 ] is a Young diagram or the complement of a Young diagram (using English notation). Recall that the Durfee square of a Young diagram λ is the largest square contained in λ.
Let a = max{i | λ i = n} and b = λ n = max{j | λ j = n} where λ denotes the transpose of λ. In other words, a is the last row in λ with n boxes and b is the last column in λ with n boxes. From Lewis Carroll's identity, we have that Let's see what we know about the signs of these determinants using our inductive hypothesis. Say I := {i 1 < · · · < i k } and J := {j 1 < · · · < j k }, and let λ J I denote the Young diagram obtained from λ by removing rows indexed by I and columns indexed by J. Note that . . , c j k })| λ J I . Also, λ J I has Durfee square of size at most k. So we can use the inductive hypothesis to compute the signs of all of the determinants in (3.1) other than det(A).
Let's consider which determinants in (3.1) are zero. The shape λ b,n a,n contains the staircase (n − 2, . . . , 1) and the shapes λ n n , λ n a , and λ b n contain the staircase (n − 1, . . . , 1). However, λ b a may not contain the staircase (n − 1, . . . , 1) (e.g. consider λ = (3, 3, 1)), so det A b a may be zero. Now we need to determine when λ J I is (R \ {r i1 , . . . , r i k }, C \ {c j1 , . . . , c j k })-admissible. Consider A b,n a,n and pick two row indices p, q / ∈ {a, n} with p < q and r p = r q . Because λ is (R, C)-admissible, rows p, q have different support, so λ p > λ q . Further, because R is listed in weakly increasing order, p > a. We would like to argue that rows p := δ a,n (p) and q := δ a,n (q) of A b,n a,n have distinct support. Since p > a, we have (λ b,n a,n ) p = λ p − 1 and (λ b,n a,n ) q = λ q − 1, so (λ b,n a,n ) p > (λ b,n a,n ) q . An analogous argument shows that columns of A b,n a,n with the same index have different support. Proof. Letẇ 0 denote the matrix with ones on the antidiagonal and zeros elsewhere. For a multiset J = {j 1 ≤ · · · ≤ j n }, let J := {j 1 ≤ · · · ≤ j n } where j i := n + 1 − j n+1−i .
Let M be the antidiagonal transpose of M ; in symbols, M =ẇ 0 M Tẇ 0 . Taking antidiagonal transpose does not effect the determinant, so M is also k-positive.
If we transpose M (R, C)| µ across the antidiagonal, we obtain the matrix where ν is the Young diagram obtained from the skew-shape µ by reflecting across the antidiagonal. Applying Proposition 3.3, we have that det N has sign |λ| if ν is (C, R)-admissible and is zero otherwise. It is not hard to check that ν is (C, R)-admissible if and only if µ is (R, C)-admissible.
We can use Proposition 2.7 to rewrite Proposition 3.3 and Corollary 3.4 in terms of immanants.   3.2. General Case. The following proposition will allow us to restrict to permutations that are not elements of a maximal parabolic subgroup of S n . To state the lemma we temporarily denote the longest permutation in S j by w (j) . See Figure 2 for an example illustrating a block-antidiagonal Γ[v, w 0 ] and the notation of Proposition 3.7.
To analyze the determinants appearing in Lewis Carroll's identity for det X(R, C)| Γ[v,w0] , we will use the following two propositions. . Let x ∈ S n−1 be the permutation x : δ i (j) → δ vi (v j ) (that is, x is obtained from v by deleting v i from v in one-line notation and shifting the remaining numbers appropriately). Then Proof. Statement (1) follows from Lemma 2.16. Statements (2) and (3) are Proposition 4.17 from [5]. Proposition 3.9. Let v ∈ S n be 1324-and 2143-avoiding such that the last bounding box of Γ[v, w 0 ] is B n,vn , and the second to last box is B a,va for some a < n with ) d a is nonzero and has sign σ. Then Proof. This follows from the proof of [5,Theorem 4.18].
We can now determine the sign of det X(R, C)| Γ[v,w0] on k-positive matrices.
We follow the proof of [5,Theorem 4.18], and proceed by induction on n. If Γ[v, w 0 ] is a partition, a complement of a partition, or block-antidiagonal, we are done by Corollary 3.5, Corollary 3.6, or Proposition 3.7, respectively.
So we may assume that v has at least 2 bounding boxes and that adjacent bounding boxes have nonempty intersection (where bounding boxes are ordered as usual by the row of their northeast corner). Because v avoids 1324 and 2143, the final two bounding boxes of Γ[v, w 0 ] are of opposite color by Proposition 2.21. Without loss of generality, we assume the final box is red and the second to last box is blue. Otherwise, we can consider the antidiagonal transpose of M (R, C)| Γ [v,w0] . This is equal to (ẇ 0 M Tẇ 0 )(C, R)| Γ[w0v −1 w0,w0] (using the notation in the proof of Corollary 3.4) and has the same determinant as This means the final box is B n,vn , and the second to last box is B a,va for some a < n with 1 < v a < v n . We analyze the sign of det M (R, C)| Γ[v,w0] using Lewis Carroll's identity on rows a, b := v −1 (1) and columns 1, d := v a . Note that a < b and 1 < d.
The proof of [5,Theorem 4.18] shows that each of the 5 known determinants in this Lewis Carroll's identity is equal to det M (R , C )| Γ[v ,w0] for an appropriate choice of multisets R , C and permutation v . We first show that two of these determinants, forming a single term on the right-hand side of the identity, are non-zero. have the same support, we must have r b−1 ≤ r b < r b+1 < · · · < r n . So, letting r i denote the elements of R , indexed in increasing order, we have r b−1 < r b < · · · < r n−1 .
We  ]. An analogous statement holds for column q in Γ[v, w 0 ] d a . Consider rows p < p of Γ[v, w 0 ] with r p = r p and p, p = a. Because R is listed in weakly increasing order, r p = r p+1 = · · · = r p . By assumption, the support of these rows in Γ[v, w 0 ] must be different. Suppose rows s = δ a (p), s = δ a (p ) have the same support in Γ[z, w 0 ]; say is the largest number in their support. The reasoning in the above paragraph implies that δ −1 d ( ) is the largest number in the support of rows p, p in Γ[v, w 0 ], and thus also in rows p, p + 1, . . . , p − 1, p . So the smallest number in the support of rows p, p + 1, . . . , p must be different. On the other hand, after deleting column d and the elements of Q, the supports should be the same. These deletions remove the first element of a row only if that first element is in column d. Putting these together, we must have p = a − 1, p = a + 1, and row a + 1 has support starting at d; otherwise we obtain rows of Γ[v, w 0 ] with the same label and same support. But now row a is among rows p, p + 1, p , and rows a and p = a + 1 have support starting at d, a contradiction.
An identical argument with columns in place of rows shows that no two columns of Γ[z, w 0 ] have the same support and the same label. So Γ[z, w 0 ] is (R , C )-admissible. So by the inductive hypothesis, one term on the right-hand side of the identity is nonzero. Let σ denote the sign of this term. By Proposition 3.9, the other term on the right-hand side has sign −σ if it is nonzero. In either case, the right-hand side has sign σ, and in particular is nonzero. Thus, both determinants on the left-hand side are non-zero. By Proposition 3.9, the determinant det(M (R, C)| Γ[v,w0] ) 1,d a,b has sign σ · (−1) (v) , so dividing through by that determinant shows that det M (R, C)| Γ[v,w0] has sign (v).
Taking this theorem with Lemma 2.10, we can now prove Theorem 3.1.
Proof of Theorem 3.1. By Lemma 2.10, Let k ≤ k be the size of the largest square in Γ[v, w 0 ]. By Theorem 3.10, for M k -positive, the right hand side of this expression is positive. Any k-positive matrix is also k -positive, so we are done.

Future Directions
The results in [5] and this paper were inspired by the following conjecture of Pylyavskyy.  15]). Let 0 < k < n be an integer and let v ∈ S n avoid the pattern 12 · · · (k + 1). Then Imm v X is k-positive.
This conjecture remains open. The relation between pattern avoidance and k-positivity of immanants is an interesting direction of further inquiry.
The results of this paper showcase an interesting phenomenon: the behavior of the dual canonical basis element Imm v X(R, C) on k-positive matrices is the same as the behavior of the usual Kazhdan-Lusztig immanant Imm v X. Based on this, we make the following conjecture.

Conjecture 4.2. Suppose
Imm v X is k-positive. Then as long as Imm v X(R, C) is not identically zero, Imm v X(R, C) is k-positive.
We also make a related conjecture based on the same phenomenon, which is something of an intermediate conjecture; it would imply Conjecture 4.1 and would be implied by Conjectures 4.1 and 4.2 together. Conjecture 4.3. Let 0 < k < n ≤ m be integers and let v ∈ S n avoid the pattern 12 · · · (k + 1). Let R, C ∈ [m] n . If Imm v X(R, C) is not identically zero, then Imm v X(R, C) is k-positive.
The compact determinantal formulas we give for certain dual canonical basis elements may be useful to understand the relationship between the dual canonical basis of C[SL m ] and its cluster algebra structure. Technically, the cluster algebra in question is the coordinate ring of G w0,w0 , the open double Bruhat cell in SL m ; C[G w0,w0 ] differs from C[SL m ] by localization at certain principal minors. The cluster monomials of C[G w0,w0 ] are expected to be dual canonical basis elements. One natural question is: do the cluster monomials include the functions Imm v X(R, C), where v avoids 2143 and 1324? If so, can the k-positivity of these immanants be explained from a cluster algebraic viewpoint?
Work related to these questions appeared in the manuscript [6]; the connection to Kazhdan-Lusztig immanants is explained in [3,Section 3.3]. The results of [6] show that Imm v X(R, C) is a cluster variable for v avoiding 123, 2143, 1432, and 3214. The immanants occurring in [6] have a determinantal form given by Lemma 2.10; they further conjecture that all cluster variables of C[G w0,w0 ] can be written as ± det X(R, C)| P for some P ⊂ [n 2 ]. Conjecturally, the Kazhdan-Lusztig immanants that can be written as ± det X(R, C)| P are the exactly Imm v X(R, C) where v is 2143 and 1324 avoiding. This leads to the following conjecture.