A generalization of Edelman--Greene insertion for Schubert polynomials

Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well.


Introduction
Schur functions, the ubiquitous basis for symmetric functions with deep connections to representation theory and geometry, may be regarded as the generating functions for standard Young tableaux. In an analogous way, Stanley [21] defined a generating function for reduced words that he proved was symmetric and conjectured was Schur positive. Edelman and Greene [10] established a bijective correspondence between reduced words and ordered pairs of Young tableaux of the same partition shape such that the left is increasing with reduced reading word and the right is standard. Thus through this correspondence they proved Stanley's conjecture and, moreover, gave an explicit formula for the Schur expansion as the number of such left tableaux that can appear in the correspondence. Schubert polynomials were introduced by Lascoux and Schützenberger [14] as polynomial representatives of Schubert classes for the cohomology of the flag manifold with nice algebraic and combinatorial properties. They can be defined as the generating polynomials of reduced words [5,7], and in the stable limit, they become the Stanley symmetric functions [16]. Parallel to this, Demazure characters for the general linear group [8] can be regarded as the generating polynomials for standard key tableaux [4,6], and in the stable limit, the key tableaux become Young tableaux and the Demazure characters become Schur functions [15]. Lascoux and Schützenberger [15] noticed that the Schubert polynomials expand nonnegatively into Demazure characters parallel to the nonnegative expansion of Stanley symmetric functions into Schur functions. The proof [15,18] uses the same structure of partitioning reduced words into equivalence classes based on the Edelman-Greene right tableau, yet there is no direct formula for the coefficients given.
In this paper, we complete the analogy between the function and polynomial settings by providing a new bijective correspondence between reduced words and ordered pairs of key tableaux of the same weak composition shape such that the left is Yamanouchi with reduced reading word and the right is standard. Thus through this correspondence we prove the Demazure positivity of Schubert polynomials and, moreover, give an explicit formula for the Demazure expansion as the number of such left tableaux that can appear in the correspondence.
Our real purpose, in addition to this explicit result, is to provide a framework by which one can extract explicit Schur expansions of symmetric functions through the machinery of dual equivalence and explicit Demazure expansions of polynomials through the machinery of weak dual equivalence. Dual equivalence and its weak variant give universal methods for proving positivity results, but they do so indirectly without giving tractable formulas. Our techniques in this paper utilize dual equivalence to show how it can be manipulated to give the desired formulas.
This paper is structured as follows. We begin in Section 2 with a review of definitions for reduced words for permutations. We develop parallel theories of the generating functions for reduced words, reviewing Stanley symmetric functions [21] in Section 2.1 and generating polynomials for reduced words, reviewing Schubert polynomials [14] in Section 2.2.
Questions of positivity arise in Section 3, where we consider the Coxeter-Knuth equivalence relations [10] on reduced words. Maintaining our parallel study, in Section 3.1, we review the machinery of dual equivalence [2] to see that the generating function of a Coxeter-Knuth equivalence class on reduced words is a Schur function, thus recovering the Schur positivity result of Edelman and Greene for Stanley symmetric functions [10]. In Section 3.2, we review the machinery of weak dual equivalence [4] to see that the generating polynomial of a Coxeter-Knuth equivalence class on reduced words is a Demazure character, thus recovering the Demazure positivity result of Lascoux and Schützenberger for Schubert polynomials [15,18].
In Section 4, we embark on the quest to extract explicit formulas for these expansions by finding canonical representatives for the Coxeter-Knuth equivalence classes. In Section 4.1, we recover the explicit formula of Edelman and Greene for the Schur expansion Stanley symmetric functions [10] using simple techniques that avoid the subtlety of their insertion algorithm. In Section 4.2 we use similar techniques to arrive at our main result: an explicit combinatorial formula for the Demazure expansion of Schubert polynomials.
Finally, we return to the inspiration of this work in Section 5, where we present explicit insertion algorithms. In Section 5.1, we review the Edelman-Greene correspondence that associates to each reduced word a pair of Young tableaux, and then in Section 5.2 we use results from Section 4 to modify this correspondence to associate to each reduced word a pair of key tableaux. In this way, we complete the parallel stories with satisfactory formulas for both cases.

Generating functions for reduced words
The symmetric group S n has generators s i for 1 i < n, the simple transpositions interchanging i and i + 1, and relations s 2 i is the identity, s i s j = s j s i for |i − j| 2, and s i s i+1 s i = s i+1 s i s i+1 for 1 i n − 2.
An expression for a permutation w ∈ S n is a way of writing w in terms of these simple generators, i.e. w = s ρ · · · s ρ1 . Notice the reversal of indexing. The length of w, denoted by (w), is the number of pairs (i < j) such that w i > w j . If an expression for w has exactly (w) terms, then it is reduced. In this case, the sequence of indices ρ = (ρ (w) , . . . , ρ 1 ) such that w = s ρ (w) · · · s ρ1 is called a reduced word for w.
For example, there are two reduced expressions for the permutation 321, namely s 1 s 2 s 1 and s 2 s 1 s 2 , both of which have length 3 since there are 3 inversions in 321. Therefore we say that (1, 2, 1) and (2, 1, 2) are reduced words for 321. For a more elaborate example, Fig. 1 shows the 11 reduced words for the permutation 153264.
Gessel introduced the fundamental quasisymmetric functions [11], indexed by compositions, that form an important basis for quasisymmetric functions.
Definition 2.1 ( [11]). For α a composition, the fundamental quasisymmetric function F α is where the sum is over weak compositions b for which the composition flat(b) obtained by removing all parts equal to 0 refines α.
For example, restricting to three variables to make the expansion finite, we have [21] defined a family of symmetric functions indexed by permutations that are the fundamental quasisymmetric generating functions for reduced words. To define this, we associate a composition to each reduced word.
Definition 2.3. The descent tableau of a reduced word ρ, denoted by D(ρ), is the filling of unit cells in the first quadrant constructed as follows. Place ρ (w) into the first column of row | Des(ρ)|. For i = (w) − 1, . . . , 2, 1, place ρ i immediately right of ρ i+1 if ρ i+1 < ρ i ; otherwise in the first column of the next row down.
For example, ρ = (3, 6,4,7,5,2,4) is inserted as shown on the left side of Figure 2, and σ = (6, 7, 3, 4, 5, 2, 4) inserts as shown on the right side of Figure 2.  By construction, rows of D(ρ) are increasing and the descent composition for ρ is given by the lengths of the rows of the descent tableau for ρ, read bottom to top.
2.2. Schubert polynomials. Lascoux and Schützenberger [14] introduced Schubert polynomials as a basis for the polynomial ring that gives polynomial representatives of Schubert classes for the cohomology of the flag manifold with nice algebraic and combinatorial properties. Assaf and Searles [5] introduced fundamental slide generating polynomials as a generalization of the fundamental quasisymmetric functions that form a basis for the full polynomial ring. They showed that Schubert polynomials are the fundamental slide generating polynomials for reduced words.
where flat(a) denotes the composition obtained by removing all zero parts.
Proposition 2.6 ( [5]). For a weak composition a, we have (4) lim where 0 m × a is the weak composition obtained by prepending m 0's to a.
We generalize the descent composition of a reduced word to a weak composition as defined in [3, Definition 3.2].
We may visualize Definition 2.7 via a simple insertion algorithm as follows.   We say that ρ is virtual if des(ρ) = ∅. To facilitate virtual objects, set where the sum may be taken over non-virtual reduced words ρ.
Proposition 2.10 ( [16]). For w a permutation, we have where 1 m ×w is the permutation obtained by adding m to each w i and then prepending 12 · · · m.

Equivalence relations
We consider simple involutions based on the Coxeter relations for the simple transpositions that generate the symmetric group. Given ρ ∈ R(w), for 1 j < (w), let c j denote the commutation relation that acts by exchanging ρ j and ρ j+1 if |ρ j − ρ j+1 | > 1 and the identity otherwise.
Any reduced words in the same equivalence class under {c j , b j } are called Coxeter equivalent. A classical result of Tits [22]   Knuth [12] considered relations on permutations that characterize when two permutations give rise to the same Schensted insertion tableau [20]. Analogously, Edelman and Greene [10] characterize when two reduced words give rise to the same Edelman-Greene insertion tableau using elementary Coxeter-Knuth relations.
Algebraic Combinatorics, Vol. 4 #2 (2021) Definition 3.1. For 1 < i < (w), the elementary Coxeter-Knuth relation d i acts on a reduced word ρ ∈ R(w) by where c j denotes a commutation relation and b j denotes a braid relation.
We partition R(w) by stating any reduced words in the same equivalence class under {d i } are Coxeter-Knuth equivalent. For example, see  Inverting history, a natural question to ask is whether this partitioning can be realized on the level of symmetric functions by decomposing the Stanley symmetric functions or on the level of polynomials by decomposing Schubert polynomials.
3.1. Dual equivalence. Based on the explicit elementary dual equivalence involutions on standard Young tableaux, Assaf [1,2] defined an abstract notion of dual equivalence that can be used to prove that a given fundamental quasisymmetric generating function is symmetric and Schur positive.
A Young diagram is the set of unit cells in the first quadrant with λ i cells in row i for some partition λ. A Young tableau is a filling of a Young diagram with positive integers. A Young tableau is increasing if it has strictly increasing rows (left to right) and columns (bottom to top). A Young tableau is standard if it is increasing and uses each integer 1, 2, . . . , n exactly once. For example, Figure 6 shows the standard Young tableaux of shape (3, 2).  Schur functions may be defined combinatorially as the fundamental quasisymmetric generating functions for standard Young tableaux. This follows from the classical definition (see [17]) by results of Gessel [11].
Definition 3.2. For λ a partition, the Schur function s λ is For example, from Figure 6 we have A dual equivalence for a set of objects endowed with a descent statistic is a family  Edelman and Greene [10] proved this by generalizing the Robinson-Schensted-Knuth insertion algorithm [12,19,20] on permutations.
Theorem 3.4 ( [10]). For w a permutation, we have where row(T ) is the row reading word (left to right along rows from the top) of T .
The Edelman-Greene correspondence is an elegant solution to the Schur positivity conjecture, but the arguments involved in the proof require intricate analysis of bumping paths with many separate cases. Thus one can hope to find a simpler proof that avoids much of this subtlety.
Edelman and Greene [10, Corollary 6.15] relate Coxeter-Knuth equivalence with dual equivalence through the Edelman-Greene recording tableaux. Implicit in their work and explicit in [4, Theorem 2.10], the Coxeter-Knuth involutions give a dual equivalence on reduced words. That is, the Coxeter-Knuth relations d i partition reduced words for a given permutation into dual equivalence classes, each of which has fundamental quasisymmetric generating function equal to a single Schur function. However, while the proof of Theorem 3.5 is simple, the resulting formula requires computing each equivalence class in its entirety, falling short of the explicit formula in Theorem 3.4.
Algebraic Combinatorics, Vol. 4 #2 (2021) 366 Generalized Edelman-Greene insertion 3.2. Weak dual equivalence. The Demazure characters, introduced by Demazure [9], originally arose as characters of Demazure modules for the general linear group [8]. These polynomials were studied combinatorially by Lascoux and Schützenberger [15] who call them standard bases and more extensively by Reiner and Shimozono [18] who call them key polynomials. We use the key tableaux model [4] based on ideas of Kohnert [13] developed further by Assaf and Searles [6].
A key diagram is a collection of left-justified unit cells in the right half place with a i cells in row i for some weak composition a. A key tableau is a filling of a key diagram with positive integers. The definition for standard key tableaux [4, Definition 3.10] is more subtle than for standard Young tableaux.
Definition 3.6 ([4]). A standard key tableau is a bijective filling of a key diagram with {1, 2, . . . , n} such that rows decrease (left to right) and if some entry i is above and in the same column as an entry k with i < k, then there is an entry right of k, say j, such that i < j.
For a standard key tableau T , say i is a descent of T if i + 1 lies weakly right of i in T . Note that this is the reverse of the concept of descents for standard Young tableaux. Next we define a weak descent composition [4, Definition 3.12].
Definition 3.7 ( [4]). For a standard key tableau T , define the weak descent composition of T , denoted by des(T ), as follows. Let (τ (k) | · · · |τ (1) ) be the run decomposition of n · · · 21 based on descents of T , that is, each τ (i) has no descents between adjacent letters and is as long as possible. Set t i = min(τ (i) ) for i = 1, . . . , k. Sett k = t k , and for i < k, sett i = min(t i ,t i+1 − 1). Ift 1 0, then define des(ρ) = ∅; otherwise, set thet i th part of des(T ) to be des(T )t i = |τ (i) | and set all other parts to 0. For example, the standard key tableaux of shape (0, 3, 0, 2) shown in Figure 7 have weak descent compositions given beneath.
Using this notion, we have the following reformulation of Demazure characters given in [4,Corollary 3.16] that we take as our definition.
Definition 3.8. Given a weak composition a, the Demazure character is For example, from Figure 7 we compute Implicit in the work of Lascoux and Schützenberger [15] and explicit in that of Assaf and Searles [6, Corollary 4.9], we have the following analog of Proposition 2.10 for Demazure characters.
Generalizing dual equivalence, a weak dual equivalence is a family of involutions d 2 , . . . , d n−1 that give a dual equivalence when weak descent compositions are flattened to descent compositions by removing parts equal to 0 and for which the fundamental slide generating polynomial of each restricted equivalence class under d i , . . . , d j for j − i 3 is a single Demazure character. Parallel to the symmetric case, the main theorem for weak dual equivalence [4,Theorem 3.33] states that this local Demazure positivity implies global Demazure positivity. One might now anticipate that Schubert polynomials expand nonnegatively into Demazure characters, parallel to (10), and indeed, we have, Lascoux and Schützenberger [15] give a formula for the key polynomial expansion of a Schubert polynomial as a sum over increasing Young tableau whose row reading word is a reduced word for w, where for each such ρ one computes the left nil key by considering all reduced words of w that are Coxeter-Knuth equivalent to ρ. For details that fill the gaps in [15], see [18,Theorem 4].
Theorem 3.11 ( [15,18]). For w a permutation, we have where K 0 − (ρ) is the left nil key of ρ. While theoretically interesting for the nonnegativity, this result does not provide a direct formula as one is required to compute each Coxeter-Knuth class, and so the computation is effectively equivalent to computing the fundamental slide expansion. A simplified proof comes as an immediate application of weak dual equivalence. That is, the Coxeter-Knuth relations d i partition reduced words for a given permutation into weak dual equivalence classes, each of which has fundamental slide generating polynomial equal to a single Demazure character. This gives a simplified proof of Theorem 3.11, though the resulting formula is no more tractable.

Positive expansions
By Theorem 3.5, each Coxeter-Knuth equivalence class corresponds to a term in the Schur expansion of a Stanley symmetric function. Similarly, by Theorem 3.12, each Coxeter-Knuth equivalence class corresponds to a term in the Demazure expansion of a Schubert polynomial. To make these positivity results more compelling, we wish to have canonical representatives from each Coxeter-Knuth equivalence class from which an exact formula can be easily computed.
Edelman and Greene [10] resolved this for the Schur expansion of the Stanley symmetric functions, but we wish to give a simple, self-contained proof of their formula Algebraic Combinatorics, Vol. 4 #2 (2021) that avoids the subtleties of their insertion algorithms. The end result will be the same, however, namely that each Coxeter-Knuth equivalence class contains a unique reduced word whose descent tableau is an increasing Young tableau. Then the shape of these tableaux determines the Schur expansion.  For example, we compute the Schur expansion of Stanley symmetric function S 13625847 by constructing the five increasing Young tableaux in Figure 8, giving However, in the Schubert case, these are not the correct Coxeter-Knuth equivalence class representatives for giving the Demazure expansion. In light of Proposition 3.9, there are many different candidates for which weak composition should index each class, even knowing the correct partition. Using the same techniques with which we prove the Edelman-Greene formula below, we also give an explicit algorithm to construct the correct Coxeter-Knuth equivalence class representatives for the polynomial case.  Figure 9. The set of Yamanouchi key tableaux whose row reading words are reduced words for w = 13625847.

4.1.
Increasing Young tableaux. We begin by considering the descent tableaux for reduced words, and, more generally, any tableau with weakly increasing rows for which the reading word is reduced.
Visually, begin with σ left justified under τ , and from left to right, for each not strict column, slide entries of σ from that column onward right by one position. For example, Fig. 10 shows the drop alignments for two pairs of increasing words.
If j 1 = t, then τ j1 = σ j1 are adjacent, contradicting the fact that the word is reduced. If j 1 < t, then τ j1+1 > τ j1 + 1 = σ j1 + 1, and so the above word is Coxeter equivalent to the word In this case as well, τ j1 = σ j1 are adjacent, contradicting the fact that the word is reduced. Thus condition (3) must hold whenever τ σ is reduced.
We call this factorization the drop alignment because, as we show below, we may drop the unsupported cells x 1 , . . . , x k from τ down to σ without changing the Coxeter-Knuth equivalence class.
The following elementary lemma will be useful in proving that τ σ is Coxeter-Knuth equivalent to drop(τ σ).
For k > 1, the above case shows τ σ is Coxeter-Knuth equivalent to However, notice that this new pair factors uniquely by combining τ (k−1) and τ (k) , as well as σ (k−1) , x, and σ (k) 1 · · · σ (k) k −1 , then also combining σ (k) k and σ (k+1) . Therefore there are fewer factors in the result, so by induction, we may drop the remaining x 1 , . . . , x k−1 as well, completing the proof Extending Definition 4.3, we may define the drop of any reduced word based on the rows of its descent tableau or more generally based on the blocks of any increasing decomposition, that is, any partitioning ρ = (ρ (k) | · · · |ρ (1) ) such that each block ρ (i) is increasing.
Definition 4.6. Let ρ be a reduced word, and let (ρ (k) | · · · |ρ (1) ) be an increasing decomposition of ρ. Define drop i (ρ) by replacing ρ (i+1) ρ (i) with drop(ρ (i+1) ρ (i) ). We visualize increasing factorizations as tableaux with strictly increasing rows, and then the drop maps can be visualized as dropping cells in the tableaux, as shown in Fig. 13. Proof. If D(ρ) is increasing, then in Definition 4.1 we will always left justify ρ (j) with respect to ρ (j+1) resulting in no x i 's in the unique factorization of Proposition 4.2, and so drop j (ρ) = ρ for all j. Moreover, this is the only case in which there are no x i 's, and so the only case when drop j (ρ) = ρ for all j.
We apply the maps drop i until reaching this terminal state, in which case we have an increasing Young tableau. To see that this is independent of the choice of which rows to drop when, we observe that these maps satisfy the nil-Hecke relations on increasing factorizations of reduced words (with empty blocks allowed).
Proof. By the Drop Lemma, ρ is Coxeter-Knuth equivalent to drop i (ρ). In particular, when ρ is reduced, so is drop i (ρ), so we may iterate the maps. For relation (i), notice that since x i < σ (i) 1 for i = 1, . . . , k, with notation as in Proposition 4.2, the columns of D(drop(τ σ)) are strict when the two rows are left justified. Therefore, by uniqueness of the factorization in Proposition 4.2, there will be no x i 's for drop(τ σ). In particular, drop will act trivially.
Relation (ii) follows from the fact that drop k considers only rows k and k + 1, so for |i − j| > 1, the sets of indices {i, i + 1} and {j, j + 1} are disjoint.
Finally, for relation (iii) it is enough to consider a three term factorization, say τ σρ with each of τ, σ, ρ increasing and the concatenation reduced. We must show that drop 1 drop 2 drop 1 (τ σρ) is equal to drop 2 drop 1 drop 2 (τ σρ). Align ρ below σ with letters of unsupported letters of σ denoted x 1 , . . . , x k , and then, maintaining that alignment, align σ below τ with letters of unsupported letters of τ denoted y 1 , . . . , y l ; call this the initial alignment. For a generic example, see Fig. 14. We consider blocks of the initial alignment with respect to the factorization of τ that put each y i at the end of a block as indicated.
Consider drop 1 drop 2 (τ σρ). Applying drop 2 (τ σρ) = drop(τ σ)ρ results in all y i 's moving down into σ. Consider the drop alignment of σ ∪ {y j } above ρ. The letters x i remain unsupported and, in addition to this, we have unsupported letters say w 1 , . . . , w l such that each w j lies weakly right of y j and strictly left of the nearest x i to the right of y j . When applying drop 1 to drop(τ σ)ρ, the x i 's and w j 's drop into ρ in row 1. This gives the following.
Working from (15), the letters z i now in row 2 will again align above the letters x i now in row 1, thus the original letters of ρ still drop align with respect to the letters Algebraic Combinatorics, Vol. 4 #2 (2021) of σ ∪ {y j } the same as before, ensuring the letter w j drop to row 1. Thus applying drop 1 to the righthand side of (15) results in the following.
Similarly, working from (16) and applying drop 2 to the righthand side of (16) drops the same unsupported cells z i from row 3, and so the end results agree. For a reduced word ρ, the drop of ρ, denoted by drop(ρ), is defined as follows: Let ρ (k) denote the kth row of D(ρ). Choose any k such that either (ρ (k+1) ) > (ρ (k) ) or ρ

Repeat until the result is an increasing Young tableau.
By Theorem 4.8, the definition of drop(ρ) is independent of the order in which rows are consolidated, and so we have the following.

4.2.
Yamanouchi key tableaux. In order to give an explicit, direct formula for the Demazure expansion of a Schubert polynomial, we begin by characterizing the analogs of increasing Young tableaux that will give our canonical representatives for each Coxeter-Knuth equivalence class in the polynomial setting.
Proof. We construct T a by filling the key diagram for a with entries n, n − 1, . . . , 2, 1 left to right along rows beginning with the top. Then rows decrease left to right and columns decrease top to bottom, showing T a ∈ SKT(a). Definition 3.7 will havet i = t i for all i, so T a will indeed have des(T a ) = a. Uniqueness follows since any alternative filling T ∈ SKT(a) necessarily has flat(des(T )) = flat(des(T a )) = flat(a). Finally, the latter condition follows from the upper unitriangularity of Demazure characters with respect to monomials.
Moreover, this expansion holds in general, giving the following formula.
Theorem 4.14. Given a permutation w, we have Proof. By [4,Theorem 3.33], the fundamental slide generating polynomial of a Coxeter-Knuth equivalence class is a single Demazure character. Therefore by Lemma 4.12, each Coxeter-Knuth equivalence class has a unique element whose weak descent composition is dominated by every other element of the class, and so every Coxeter-Knuth equivalence class contains a unique Yamanouchi reduced word. By Lemma 4.12 again, the weak descent composition of the Yamanouchi reduced word indexes the Demazure character corresponding to the class.
The formula in Theorem 4.14 is still indirect since the definition of Yamanouchi requires consideration of the entire Coxeter-Knuth equivalence class. In order to avoid searching entire classes to find the Yamanouchi reduced words, we present an algorithm by which they can be constructed from the increasing Young tableaux with reduced reading words. Beginning with an increasing Young tableau, we raise letters from lower rows while staying within the same Coxeter-Knuth class by inverting the drop map from Definition 4.9. To begin, we must align.
Visually, right justify τ above σ and, from right to left, for each not weakly strict column, slide entries of τ from that column onward left by one position. For example, Fig. 12 shows the lift alignments for two pairs of increasing words.
Parallel to the drop case, if there are k instances in the lift alignment of τ above σ where a cell of σ has no cell below it, then we denote these cells of σ as x 1 , . . . , x k and factor σ = σ (1) x 1 σ (2) · · · x k σ (k+1) and, correspondingly, τ = τ (0) τ (1) · · · τ (k+1) as shown in Fig. 15. When the concatenation τ σ is a reduced word, then this factorization has the following properties. Figure 15. An illustration of the lift alignment of τ above σ and corresponding factorizations of σ and τ .
The following analog of Proposition 4.2 has a completely analogous proof.
Definition 4.17. Given two increasing words τ, σ such that τ σ is reduced, define where the factorization is the unique one in Proposition 4.16 and for 1 j ǩ For example, lifting the aligned word on the right side of Fig. 12 results in the word on the right side of Fig. 10, though the labeling of letters is not the same for the two procedures. Lifting the left word in Fig. 12 lifts both the 3 and the 7, so we see that lifting and dropping are not inverse operations in general.
Finally, suppose 2 > 0 and τ 16(3) and the increasing property of τ , we have σ Thus applying Lemma 4.4 shows Eq. (19) is Coxeter-Knuth equivalent to (20) 2 . Based on the assumptions on b, we have the elementary relations For k > 1, the above case shows τ σ is Coxeter-Knuth equivalent to However, notice that this new pair factors uniquely by combining τ (0) and τ lift 3 Figure 16. The lifting algorithm applied to an increasing Young tableau.
Reversing the symmetric situation, we can define an increasing key tableau with the corresponding row condition and such that the result is lift-invariant. Unlike the case for drop i , we do not wish to apply the maps lift i indiscriminately until reaching some increasing key tableau. Foremost among the reasons is that these maps do not, in general, satisfy the nil-Hecke relations. For example, Fig. 16 shows all nontrivial lift i operators applied to the leftmost tableau P , which is the unique increasing Young tableau in the Coxeter-Knuth class. Notice Furthermore, both the third and fourth tableaux (from the left) in the middle row of Fig. 16 are increasing, though only the fourth is Yamanouchi.
Nevertheless, we do have a canonical lifting path from the unique increasing Young tableau to the unique Yamanouchi tableau. To define this path, we say that lift i acts faithfully on a tableau T if lift i (T ) = T , and we extend this notion to a sequence of lifting maps in the obvious way.

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Sami H. Assaf Lemma 4.21. Let P be an increasing Young tableau with reduced reading word, and suppose each operator in lift i k • · · · • lift i1 (P ) acts faithfully. Then where s i acts on compositions by interchanging parts in positions i and i + 1.
Proof. In the notation of Definition 4.15, since (τ (i) ) = (σ (i) ) for all i > 0, raising the x i 's from σ to τ in lift(τ σ) precisely exchanges the two parts of Des(τ σ) provided τ (0) = ∅. Moreover, lift acts nontrivially only if (σ) > (τ ), and so each application of lift i will increase the coinversion number of the descent composition. In particular, this ensures the expression s i k · · · s i1 is reduced. Thus it remains only to show τ (0) = ∅ at each lift.
Say a tableau T has property P if for every pair of rows i < j such that row i is strictly longer than row j and for every entry z in row i, there are at least as many x < z in row i as there are y < z in row j > i. Observe for T with property P, aligning rows i < j with row i strictly longer than row j will result in τ (0) = ∅. Since columns of P strictly increase bottom to top, P has property P. Furthermore, if T has property P, then a pair of rows i < j of lift k (T ) violating property P must have either i < k and j = k, k + 1 or i = k, k + 1 and j > k + 1 since row k will be strictly shorter than row k + 1 in lift k (T ). However, since entries that lift upward maintain their columns and only larger entries move left, if i < k and j = k (resp. j = k + 1), then (i, k + 1) (resp. (i, k)) violated property P in T , and similarly for the case j > k + 1. Thus lift i k • · · · • lift i1 (P ) has property P and, in particular, has τ (0) = ∅ at each step. The result follows.
For i j, define the lifting sequence lift [i,j] by (21) lift We say that a lifting sequence lift [i,j] acts faithfully on a tableau T if lift i acts faithfully on T and lift k acts faithfully on lift [i,k−1] (T ) for all i < k j.
Definition 4.22. For P an increasing Young tableau whose row reading word is reduced, define the lift of P , denoted by lift(P ), to be the tableau of key shape constructed as follows. Set T 0 = P , and for k > 0, where (a) j k is the maximum j for which there exists i j such that lift [i,j] acts faithfully on T k−1 , and (b) i k is the minimum i j k for which lift [i,j k ] acts faithfully on T k−1 .
For example, with P the leftmost tableau in Fig. 16, we have lift(P ) = lift [1,1] • lift [2,3] which is the rightmost tableau in Fig. 16 and is Yamanouchi. Proof. By Theorem 3.12, there is a des-preserving isomorphism, say θ, from the Coxeter-Knuth equivalence class of ρ to SKT(a) for some weak composition a. Moreover, by Theorem 4.10 and Proposition 3.9, we must have sort(a) = Des(ρ).
Given any T ∈ SKT(a) for which des(T ) rearranges the parts of a, if S ∈ SKT(a) has des(S) = des(T ), then S = T , which is to say that for each weak composition b that rearranges a, there is at most one element of SKT(a) with weak descent composition b. In particular, θ(ρ) is determined as is θ(σ) for any σ obtained from ρ by a sequence Algebraic Combinatorics, Vol. 4 #2 (2021) of lifts by Lemma 4.21. Moreover, by Lemma 4.21, we may extend the maps lift i to those SKT(a) whose weak descent compositions rearrange a so that they intertwine with the isomorphism θ. Thus it suffices to show that Definition 4.22 applied via θ to SKT(a) gives the Yamanouchi standard key tableau whose row reading word is the reverse of the identity.
Let P = θ(ρ) ∈ SKT(a), and set λ = Des(P ) say with length . Let α be the composition obtained by removing zero parts of a. After the first pass of Definition 4.22 applied to P , we have T 1 = lift [i1,j1] (P ) ∈ SKT(a) where j 1 is the largest index j for which λ j = α j but λ k = α k for all k > j, and i 1 is the largest index i < j 1 for which λ i = α j1 . Therefore Des(T 1 ) = s j1 · · · s i1 · λ agrees with α in all positions j j 1 . Furthermore, reading the rows of T 1 left to right from the top down to j 1 , we precisely have the reverse of the identity. Thus we may proceed by induction on the first j 1 − 1 < rows of T 1 .
For an example of the maps lift i induced on elements of SKT(a) whose weak descent compositions rearrange a, see For example, lifting the increasing Young tableaux in Figure 8 we arrive at the Yamanouchi key tableaux in Figure 9, as promised.

Insertion algorithms
Edelman and Greene [10] define an insertion algorithm mapping reduced words to pairs of Young tableaux where the left is increasing and the right is standard. In this context, the left tableau gives the canonical Coxeter-Knuth equivalence class representative for obtaining the Schur expansion of a Stanley symmetric function, Algebraic Combinatorics, Vol. 4 #2 (2021) and the right tableau gives an explicit bijection between elements of the Coxeter-Knuth equivalence class and standard Young tableaux of fixed shape. We recall their definitions and main results for the purpose of generalizing them to the polynomial setting. In the generalization, the left tableau will be a Yamanouchi key tableau, and the right tableau will be a standard key tableau. Thus the left tableau gives the canonical Coxeter-Knuth equivalence class representative for obtaining the Demazure expansion of a Schubert polynomial, and the right tableau gives an explicit bijection between elements of the Coxeter-Knuth equivalence class and standard key tableaux of fixed shape. 5.1. Edelman-Greene insertion. Edelman and Greene [10, Definition 6.20] defined the following procedure for inserting a letter into an increasing tableau.
Definition 5.1 ( [10]). Let P be an increasing Young tableau, and let x be a positive integer. Let P i be the ith lowest row of P . Define the Edelman-Greene insertion of x into P , denoted by P x ←, as follows. Set x 0 = x and for i 0, insert x i into P i+1 as follows: if x i z for all z ∈ P i+1 , place x i at the end of P i+1 and stop; otherwise, let This algorithm generalizes the insertion algorithm of Schensted [20], building on work of Robinson [19], later generalized by Knuth [12]. Robinson-Schensted insertion becomes a bijective correspondence between permutations and pairs of standard Young tableaux by constructing a second tableau to track the order in which new cells are added. The pair is typically denoted by (P, Q), where P is called the insertion tableau, and Q is called the recording tableau.
Edelman and Greene derived many properties of this generalized insertion algorithm, including that the insertion tableau P (ρ) defined by inserting ρ k , . . . , ρ 1 into Algebraic Combinatorics, Vol. 4 #2 (2021) the empty tableau is a well-defined increasing tableau whose row reading word is a reduced word for w. They proved the following [10, Theorem 6.24] relating their insertion to Coxeter-Knuth equivalence. Further, Edelman and Greene characterize how the recording tableaux differ for two reduced words that differ by an elementary Coxeter-Knuth equivalence. Refining [10, Definition 6.14], we have the following definition from [2].   Theorem 5.4 follows from Theorem 3.5, proving that Edelman-Greene insertion establishes a Des-preserving bijection between elements of a Coxeter-Knuth equivalence class and standard Young tableaux of fixed shape.
Edelman and Greene use their insertion and recording tableaux to establish the following bijective correspondence [10, Theorem 6.25].
Taking fundamental quasisymmetric generating functions gives Theorem 3.4.

Weak insertion.
We generalize Edelman-Greene insertion to an algorithm on reduced words that outputs a pair of tableaux of key shape such that the insertion tableau is a Yamanouchi key tableau (in particular, it is increasing) and the recording tableau is a standard key tableau. Leveraging Definition 5.1 along with Definitions 4.9 and 4.22, we have the following.
Definition 5.6. For P a Yamanouchi key tableau and x a positive integer, define the weak insertion of x into P , denoted by P x , to be lift(drop(P ) x ←).
Construct the weak correspondence of a reduced word ρ = (ρ k , . . . , ρ 1 ) by successively inserting the letters of ρ from k to 1 into the empty tableau to create the weak insertion tableau of ρ, denoted by P (ρ). For example, Figure 21 shows the weak insertion tableau for the reduced word ρ = (3, 6, 4, 7, 5, 2, 4).  Proof. By Theorem 4.10, dropping a word so that the result is an increasing Young tableau maintains the Coxeter-Knuth equivalence class. By Theorem 4.23, the Yamanouchi words are constructed by lifting the increasing Young tableaux, and by Theorem 4.14 they are the canonical representatives for each weak dual equivalence class. By Theorem 3.12, Coxeter-Knuth equivalence classes are weak dual equivalence classes, and so the result follows.
In order to define a weak recording tableau, we must show that the successive shapes when inserting a word are nested. To this end, we have the following.
Lemma 5.8. Let ρ be a Yamanouchi reduced word and x a letter such that the concatenation ρx is reduced, and set σ = lift(drop(ρx)). Then des(ρ) i des(σ) i for all i, and if j is the unique index such that des(σ) j = des(ρ) j + 1, then des(σ) j = des(ρ) i for all i < j.
Proof. By Lemma 4.21, the locations of the nonempty rows of P (ρ) are determined by the entries in the first column of P (ρ), and these are the nonzero entries of des(lift(drop(ρ))). Thus inserting an additional letter will increase the length of one (possibly empty) row. Therefore des(ρ) i des(σ) i for all i.
Let j denote the unique index for which des(σ) j = des(ρ) j + 1. Suppose, for contradiction, des(σ) j = 1 and des(ρ) i = 1 for some i < j. Note the entries in P (σ) in rows i, j must be i, j, respectively. Furthermore, in P (σ), both i and j are singleton cells, so the Edelman-Greene insertion of x into ρ must have caused i to bump j. In particular, j must have been a singleton cell of P (ρ), but this contradicts that des(σ) j = des(ρ) j + 1. Similarly, if des(σ) j = des(ρ) i = c > 1 for some i < j, then the entry in row j column c must be strictly larger than the entry in row i column c, and the same bumping argument applies, presenting the same contradiction. Thus we must have des(σ) j = des(ρ) i for all i < j as desired.
Proof. By Lemma 5.8, the successive shapes created during the weak insertion of ρ are nested, making the recording tableau well-defined. Entries are added to the recording tableau in decreasing order, ensuring that rows decrease left to right. The latter condition of Lemma 5.8 ensures that if i is added above an entry k, then the length of the lower row containing k must be longer than that length of the upper row containing i. In particular, there must be an entry j right of k added before i, thus j > i. Therefore Q(ρ) is indeed a standard key tableau.
We can characterize how the recording tableaux differ for two reduced words that differ by an elementary Coxeter-Knuth equivalence using elementary weak dual equivalences [4, Definition 3.21]. where b j cycles j − 1, j, j + 1 so that j shares a row with j ± 1.
The elementary weak dual equivalence involutions on standard key tableaux of shape (0, 3, 0, 2) are shown in Fig. 23. We relate Coxeter-Knuth equivalence with weak dual equivalence through the weak recording tableaux as follows. Proof. By [4,Theorem 3.24], the bijection Φ : SKT(a) → SYT(sort(a)) that drops entries in a key tableau to partition shape, replaces i with n − i + 1, and sorts columns to increase bottom to top intertwines the elementary weak dual equivalence involutions on standard key tableaux with the elementary dual equivalence involutions on standard Young tableaux by Φ( d i (T )) = d n−i+1 (Φ(T )). The result now follows from Theorem 5.4.
The weak insertion and recording tableaux establish the following bijective correspondence, parallel to Corollary 5.5.
Corollary 5.12. The weak correspondence ρ → P (ρ), Q(ρ) establishes a bijection where Y R a (w) is the set of Yamanouchi reduced words σ for w such that des(σ) = a.
Taking fundamental slide generating polynomials gives Corollary 4.24.