Principal specialization of dual characters of flagged Weyl modules

Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald famously expressing the principal specialization of any Schubert polynomial in terms of reduced words. We study the principal specialization of dual characters of flagged Weyl~modules. Our result yields an alternative proof of a conjecture of Stanley about the principal specialization of Schubert polynomials, originally proved by Weigandt.


Introduction
Schubert polynomials S w were introduced by Lascoux and Schützenberger in [14] as distinguished polynomial representatives for the cohomology classes of Schubert cycles in the flag variety. Schubert polynomials generalize Schur polynomials, a classical basis of the ring of symmetric functions.
The principal specialization of Schur polynomials has a long history: s λ (1, . . . , 1) counts the number of semistandard Young tableaux of shape λ, a number famously enumerated by the hook-content formula, see for instance [21]. Macdonald [16,Eq. 6.11] famously expressed the principal specialization S w (1, . . . , 1) of the Schubert polynomial S w in terms of the reduced words of the permutation w. Fomin and Kirillov [9] placed this expression in the context of plane partitions for dominant permutations, while after two decades Billey et al. [2] provided a combinatorial proof. Principal specialization of Schubert polynomials has inspired a flurry of recent interest [10,18,20,22,23]. A major catalyst for the current line of study into S w (1, . . .  Weigandt's proof of Theorem 1.1 works by exploiting the structure of pipe dreams, one of the earliest combinatorial models for Schubert polynomials [1,8]. We give an alternative proof of Theorem 1.1 by generalizing its statement to the setting of dual characters of flagged Weyl modules of diagrams: For any diagram D, the dual character χ D of the flagged Weyl module of D satisfies We show in Corollary 3.4 that Theorem 1.2 specializes to Theorem 1.1. Additionally, Theorem 1.2 implies an analogous result for key polynomials (Corollary 3.5).

Background
We first define flagged Weyl modules and their dual characters. We then recall the definition of Schubert polynomials and the connection between Schubert polynomials and dual characters. The exposition of this section follows that of [7].
By a diagram, we mean a sequence D = (C 1 , C 2 , . . . , C n ) of finite subsets of [n], called the columns of D. We interchangeably think of D ⊆ [n] × [n] as a collection of boxes (i, j) in a grid, viewing an element i ∈ C j as a box in row i and column j of the grid. When we draw diagrams, we read the indices as in a matrix: i increases top-to-bottom and j increases left-to-right.
Let G = GL(n, C) be the group of n × n invertible matrices over C and B be the subgroup of G consisting of the n × n upper-triangular matrices. The flagged Weyl module is a representation of B associated to a diagram D. The flagged Weyl module of D will be denoted by M D . We will use the construction of M D in terms of determinants given in [17].
Denote by Y the n×n matrix with indeterminates y ij in the upper-triangular positions and zeros elsewhere. Let C[Y ] be the polynomial ring in the indeterminates {y ij } i≤j . Note that B acts on C[Y ] on the right via S be the submatrix of Y obtained by restricting to rows R and columns S.
Note that since Y is upper-triangular, the condition C ≤ D is technically unnecessary since det Y Cj Dj = 0 For any B-module N , the character of N is defined by char(N )(x 1 , . . . , x n ) = tr (X : N → N ) where X is the diagonal matrix with diagonal entries x 1 , . . . , x n , and X is viewed as a linear map from N to N via the B-action. Define the dual character of N to be the character of the dual module N * : . , x n ) be the dual character Then the diagrams C with C ≤ D are The corresponding determinants are y 11 y 22 y 2 33 , y 11 y 12 y 23 y 33 − y 11 y 13 y 22 y 33 y 11 y 12 y 2 33 y 11 y 22 y 23 y 33 y 11 y 12 y 2 23 − y 11 y 13 y 22 y 23 y 11 y 12 y 23 y 33 These determinants are all linearly independent eigenvectors of X, so The following two easy results describe the supports and coefficients of dual characters of diagrams. x D C ≤ D .
Definition 2.7. The Schubert polynomial S w of w ∈ S n is defined recursively on the weak Bruhat order. Let w 0 = n n − 1 · · · 2 1 ∈ S n , the longest permutation in S n . If w = w 0 then there is j ∈ [n − 1] with w(j) < w(j + 1) (called an ascent of w). The polynomial S w is defined by The diagram D(w) can be visualized as the set of boxes left in the n × n grid after you cross out all boxes weakly below (i, w(i)) in the same column, or weakly right of (i, w(i)) in the same row for each i ∈ [n]. The Schubert polynomial of w is computed by . Via Rothe diagrams, Schubert polynomials occur as special cases of dual characters of flagged Weyl modules: Theorem 2.10 ( [13]). Let w be a permutation with Rothe diagram D(w). Then, S w = χ D(w) .

Key Polynomials.
Key polynomials were first introduced by Demazure for Weyl groups [3], and studied in the context of the symmetric group by Lascoux and Schützenberger in [14,15]. Recall the key polynomial κ α of a composition α = (α 1 , α 2 , . . .) is defined as follows. When α is a partition, κ α = x α . Otherwise, suppose α i < α i+1 for some i. Then Definition 2.11 ( [11,19]). Fix a composition α, and set l = max{i : α i = 0} and n = max{l, α 1 , . . . , α l }. The key polynomial of α is computed by  In this section, we prove a lower bound for the principal specialization of the dual character of any diagram. We then specialize this bound to Schubert and key polynomials. The rank of D is rank(D) = (i,j)∈D rank D (i, j).
Proof. If r = 0 then the chain consists of just D and there is nothing to prove. Assume r > 0. We begin with the case that D has a single nonempty column. Without loss of generality, we may write D = (D 1 ) = ({a 1 , . . . , a m }). Since rank(D) > 0, D 1 = {1, . . . , m}. Let k be the largest integer less than a m such that k / ∈ D 1 . Choose i so that a i = k + 1 (which must exist by definition of k). Define Then C 1 < D 1 , and rank(C 1 ) = rank(D 1 ) − 1. By induction, the result follows whenever D has a single nonempty column. Since rank((D 1 , . . . , D n )) =

j∈[n]
rank((D j )), the general case follows from the single column case by performing the above construction to one column at a time.
Recall the inverse lexicographic order on monomials: x a < invlex x b if there exists 1 ≤ i ≤ n such that a j = b j for i + 1 ≤ j ≤ n, and a i < b i .
Proof. Let C = (C 1 , . . . , C n ) and D = (D 1 , . . . , D n ), so C j ≤ D j for all j ∈ [n]. Fix a column j. Then C j ≤ D j means we can write C j = {a 1 , . . . , a m } and Since C < D, we know C j < D j for at least one j ∈ [n]. For any such j, we have i∈Cj In particular x C = x D . Proof. By Theorem 2.5, By Lemma 3.2, there exists a chain of r = rank(D) + 1 diagrams C 0 < C 1 < · · · < C r−1 < D. Thus, by Lemma 3.3, . . , C r−1 , D} = r + 1 = rank(D) + 1.
By specializing Theorem 1.2 to Rothe diagrams, we obtain a new proof of Theorem 1.1: Proof. It is enough to show that p 132 (w) = rank(D(w)). By viewing 132-patterns of w graphically in D(w), one easily observes that 132-patterns are in transparent bijection with tuples (i, j, k) such that (i, j) ∈ D(w), 1 ≤ k < i, and (k, j) / ∈ D(w). The quantity rank(D) exactly counts these tuples.
By specializing Theorem 1.2 to skyline diagrams, we obtain an analogous result for key polynomials. For a composition α, let rinv(α) denote the set of right inversions of α, the pairs i < j such that α i < α j .
Corollary 3.5. For any composition α, We now characterize the case of equality in Theorem 1.2.
Definition 3.6. Let D be any diagram. A pair of boxes (i, j), (i , j ) ∈ D is called an unstable pair if Proof. Suppose D contains an unstable pair {(i, j), (i , j )}. A simple case analysis shows one can move boxes in D upwards to create diagrams C, C ≤ D of the same rank with x C = x C . This implies χ D (1, . . . , 1) = rank(D) + 1.
Assume D contains no unstable pair. If rank(D) = 0, then the result follows. Pick (i, j) ∈ D with rank D (i, j) ≥ 1. If rank D (i, j) > 1, then any other positive rank box would form an unstable pair with (i, j). Hence (i, j) is the only positive rank box of D, and the result follows easily.
Suppose rank D (i, j) = 1. To avoid unstable pairs, all other positive rank boxes of D either lie in row i, or they all lie in column j. In either case, they must all have rank exactly 1. If all positive rank boxes of D lie in column i, then one observes there is a unique diagram C ≤ D with rank k for each k = 0, 1, . . . , rank(D), implying the result.
If all positive rank boxes of D lie in row j, then one observes that all diagrams C ≤ D of a fixed rank have the same monomial x C , and their determinants span an eigenspace of dimension one in the flagged Weyl module.
We now relate equality in Theorem 1.2 with the question of when χ D is zero-one. Recall a polynomial f is called zero-one if all nonzero coefficients in f equal 1. Proof. In order for χ D (1, . . . , 1) = rank(D) + 1, it must happen that all diagrams C ≤ D with a fixed rank induce the same monomial x C and have dependent determinants in the flagged Weyl module. Since all diagrams C, C ≤ D with x C = x C must have the same rank, it follows that all eigenspaces in the flagged Weyl module of D have dimension one.
We now provide a conjectural characterization of diagrams D such that χ D is zero-one. Consider the six box configurations shown in Figure 1. In each configuration, an × (red) indicates the absence of a box; a shaded square (gray) indicates the presence of a box; and an unshaded square (white) indicates no restriction on the presence or absence of a box. Definition 3.9. Let D be any diagram. We say D contains a multiplicitous configuration if there are r 1 < r 2 < r 3 < r 4 and c 1 < c 2 so that D restricted to rows {r 1 , r 2 , r 3 , r 4 } and columns {c 1 , c 2 } equals one of the configurations shown in Figure 1, up to possibly swapping the order of the columns.  Proposition 3.11. If a diagram D contains a multiplicitous configuration, then χ D is not zero-one.
Proof. It follows immediately from [7,Theorem 5.8] that if the restriction of a diagram D to rows {i 1 , . . . i p } and columns {j 1 , . . . , j q } equals a diagram D , then the largest coefficient appearing in χ D is bounded below by the largest coefficient appearing in χ D . One easily checks that the dual characters of each of the multiplicitous configurations are not zero-one. In this final section, we recall a trivial upper bound for the principal specialization of the dual character of any diagram. We make a conjecture for the case of equality. From Corollary 2.6, it follows immediately that if c α is the coefficient of x α in χ D , then In particular, χ D (1, . . . , 1) ≤ #{C | C ≤ D}. Fan and Guo gave the following characterization for equality when the diagram D is northwest. Recall a diagram D is northwest if whenever (i, j), (i , j ) ∈ D with i > i and j < j , one has (i , j) ∈ D.