2.1 Permutations

We write $[n]$ for the set $\{1, 2, \dots , n\}$ .

Let $S_{\mathbb {Z}}$ denote the group of permutations of $\mathbb {Z}$ that fix all but finitely many elements. For $i \in \mathbb {Z}$ , the is the involution that switches i and $i+1$ . Note that $S_{\mathbb {Z}}$ is generated by $\{s_i\}_{i \in \mathbb {Z}}$ . The of $w \in S_{\mathbb {Z}}$ is the length of a minimal expression for w as a product of simple transpositions.

We write $S_+$ for the subgroup generated by $\{s_i\}_{i> 0}$ and write $S_n$ for the subgroup generated by $\{s_i\}_{0 < i < n}$ . For a permutation $w \in S_n$ , we often write w in one-line notation as $w(1)w(2) \dots w(n)$ . If we write w in one-line notation, then $ws_i$ is obtained by swapping the entries in positions i and $i+1$ ; $s_iw$ is obtained by swapping the letters i and $i+1$ . The is $n (n-1) \dots 1$ . The inclusion map $\iota : S_n \to S_{n+1}$ sends w to $w(1)w(2)\cdots w(n)(n+1)$ .

on permutations is defined by $u \leq _L w$ if $w = vu$ for some permutation v with $ \ell (u) + \ell (v) = \ell (w)$ . Similarly, in this case, we write $v \leq _R w$ and call this the . Let $t_{i,j} \in S_{\mathbb {Z}}$ denote the involution swapping i and j. is the transitive closure of the covering relations $wt_{i,j} \lessdot w$ for $\ell (wt_{i,j}) = \ell (w) - 1$ . We write Bruhat order comparisons as $u \leq w$ , without subscripts. The weak orders are weak in the sense that the corresponding relations are subsets of the Bruhat order relation.

For a permutation $w \in S_{\mathbb {Z}}$ , say that i is a of w if $w(i)> w(i+1)$ , equivalently if $ws_i < w$ . Say that i is a of $w \in S_{\mathbb {Z}}$ if $w^{-1}(i)> w^{-1}(i+1)$ , equivalently if $s_i w < w$ . The of a permutation $w \in S_{\mathbb {Z}}$ is the function $c(w) : \mathbb {Z} \to \mathbb {Z}_{\geq 0}$ such that $c(w)(i)$ equals the number of $j> i$ such that $w(j) < w(i)$ ; as a shorthand, we often write $c_i(w) = c(w)(i)$ .

A permutation $w \in S_{\mathbb {Z}}$ is if k is its unique descent or if it has no descent. Note that the identity permutation is k-Grassmannian for all k. We say w is if it is k-Grassmannian for some k. We say that w is (resp. ) if $w^{-1}$ is k-Grassmannian (resp. Grassmannian).

The has generators $T_i$ for $i \in \mathbb {Z}$ satisfying

(2.1) $$ \begin{align} T_i^2 &= T_i, \nonumber\\ T_i T_j &= T_j T_i \quad \text{(if}\ |i-j|> 1\text{), and} \\ T_i T_{i+1} T_i &= T_{i+1} T_i T_{i+1}.\nonumber \end{align} $$

For every $w \in S_{\mathbb {Z}}$ of length k, there is a corresponding element $T_w \in \mathcal {H}_{\mathbb {Z}}$ obtained by taking any reduced decomposition $w = s_{i_1} \cdots s_{i_k}$ and setting $T_w = T_{i_1} \cdots T_{i_k}$ . The elements $T_w$ for $w \in S_{\mathbb {Z}}$ are a linear basis of $\mathcal {H}_{\mathbb {Z}}$ .

2.2 Schubert polynomials

are defined recursively as follows. For $w_0^{(n)} \in S_n$ , set the Schubert polynomial $\mathfrak {S}_{w_0^{(n)}} = x_1^{n-1} x_2^{n-2} \cdots x_n^0$ . For w such that $ws_i < w$ , set

$$\begin{align*}\mathfrak{S}_{ws_i} = N_i \mathfrak{S}_w, \end{align*}$$

where $N_i$ is the that acts on $f \in \mathbb {Z}[x_1, \dots , x_n]$ by

$$\begin{align*}N_i(f)= \frac{f - s_i \cdot f}{x_i - x_{i+1}}. \end{align*}$$

Here, $s_i$ acts on a polynomial by swapping variables $x_i$ and $x_{i+1}$ .

We have $\mathfrak {S}_w = \mathfrak {S}_{\iota (w)}$ , so we may treat Schubert polynomials as indexed by the elements of $S_+$ . The set of Schubert polynomials $\{\mathfrak {S}_w \}_{w \in S_+}$ is a linear basis of the free $\mathbb {Z}$ -module $\mathbb {Z}[x_1, x_2, x_3, \dots ]$ . In particular, there are structure coefficients defined by

$$\begin{align*}\mathfrak{S}_u \cdot \mathfrak{S}_v = \sum_w d_{u,v}^w \mathfrak{S}_w. \end{align*}$$

For $u,v,w \in S_n$ , these structure coefficients agree with the Schubert structure coefficients defined in Section 1, that is, $c_{u,v}^w = d_{u,v}^w$ . Hence, we can study the structure coefficients $c_{u,v}^w$ by using Schubert polynomials in place of Schubert cycles. References for basic facts about Schubert polynomials include [Reference MacdonaldMac91, Reference Manivel and SwallowMan01].

2.3 Backstable clans

A is a partial matching of the integers such that there exist $i,j \in \mathbb {Z}$ such that $i - k$ is paired with $j+k$ for all $k> 0$ , together with an assignment of labels from to the unmatched integers. We say that such a backstable clan $\gamma $ is on $[i,j]$ and call $\gamma $ an . (Note that if $\gamma $ is supported on $[i,j]$ , then $\gamma $ is also supported on $[i-a,j+a]$ for any $a \geq 0$ .) When we draw diagrams to illustrate an $[i,j]$ -clan $\gamma $ , we often restrict to the interval $[i,j]$ since all information about $\gamma $ can be extracted from this finite region. For examples of backstable clans, see Figures 1, 2 and 3.

Figure 1

The rainbow clans $\Omega _{5,2}$ (left) and $\Omega _{0,1}$ (right).

Figure 2

Some examples of the Hecke action on clans.

Figure 3

Some representative almost rainbow clans.

In the previous literature, ‘clans’ are restricted to the interval $[1,n]$ ; we identify these objects with $[1,n]$ -clans. Clans were introduced in [Reference Matsuki and ŌshimaMO90, Reference YamamotoYam97] in the context of K-orbits. For more recent work using clans in a related K-orbit context, see, for example, [Reference Wyser and YongWY14, Reference Woo and WyserWW15]. We believe that backstable clans will additionally be amenable to the study of backstabilized K-orbits, analogous to the backstable Schubert calculus of [Reference Lam, Lee and ShimozonoLLS21]; however, we do not pursue that application here. In this paper, backstable clans are a tool for explicating the Schubert calculus of $\mathsf {Flags}_n$ .

For a backstable clan $\gamma $ , let $\zeta (\gamma )$ denote the number of labels minus the number of labels. If $\gamma $ is supported on $[i,j]$ and $\zeta (\gamma ) = \zeta $ , we say that $\gamma $ is a , where $p = \frac {1}{2} (i+j+\zeta -1)$ and $q = \frac {1}{2}(i+j - \zeta -1)$ . Note that $\zeta = p-q$ and $i+j-1 = p+q$ ; moreover, a backstable clan $\gamma $ supported on $[i,j]$ with $\zeta (\gamma ) = p-q$ and $i+j-1 = p+q$ is a $(p,q)$ -clan.

For a backstable clan $\gamma $ , we write $\gamma (i) = j$ if i is matched with j in $\gamma $ . Write to denote an unspecified element of . We write if i is unmatched in $\gamma $ and labeled with . If $i \in \mathbb {Z}$ is matched, we say that i is if $\gamma (i)> i$ and if $\gamma (i) < i$ . A backstable clan $\gamma $ is if we never have $a<b<c<d \in \mathbb {Z}$ with $\gamma (a) = c$ and $\gamma (b) = d$ .

For each pair of integers $p,q \in \mathbb {Z}$ , the is the $(p,q)$ -clan such that

See Figure 1 for some examples. Note that the rainbow clan is always noncrossing and never has both and appearing.

For each generator $T_i$ of the $0$ -Hecke algebra $\mathcal {H}_{\mathbb {Z}}$ , we define an action of $T_i$ on $(p,q)$ -clans. This action is defined through various cases; however, all have the flavor of acting locally at the numbers i and $i+1$ and of transforming the $(p,q)$ -clan to more closely resemble the rainbow clan $\Omega _{p,q}$ . Precisely, for a clan $\gamma $ , we have:

• • if and $i+1$ is initial, then

  

• • if i is final and , then

  

• • if i and $i+1$ are both initial with $\gamma (i) < \gamma (i+1)$ , then

  $$\begin{align*}(T_i \cdot \gamma)(i) = \gamma(i +1), (T_i \cdot \gamma)(\gamma(i +1)) = i, (T_i \cdot \gamma)(i+1) = \gamma(i), \text{ and } (T_i \cdot \gamma)(\gamma(i)) = i+1; \end{align*}$$

• • if i and $i+1$ are both final with $\gamma (i) < \gamma (i+1)$ , then

  $$\begin{align*}(T_i \cdot \gamma)(i) = \gamma(i +1), (T_i \cdot \gamma)(\gamma(i +1)) = i, (T_i \cdot \gamma)(i+1) = \gamma(i), \text{ and } (T_i \cdot \gamma)(\gamma(i)) = i+1; \end{align*}$$

• • if i is final and $i+1$ is initial, then

  $$\begin{align*}(T_i \cdot \gamma)(i) = \gamma(i +1), (T_i \cdot \gamma)(\gamma(i +1)) = i, (T_i \cdot \gamma)(i+1) = \gamma(i), \text{ and } (T_i \cdot \gamma)(\gamma(i)) = i+1; \end{align*}$$

• • if and , then $(T_i \cdot \gamma )(i) = i+1$ and $(T_i \cdot \gamma )(i+1) = i$ ;

in all other cases, $T_i$ acts trivially. Since this action respects the braid relations of Equation (2.1) by [Reference WyserWys13, p. 839], we obtain an action of each $0$ -Hecke element $T_w$ . (Wyser only considers $(p,q)$ -clans supported on $[1,p+q]$ ; however, by translating $[i,j]$ -clans to be supported on $[1,j-i+1]$ , the general result is immediate.) Examples of the $0$ -Hecke action on backstable clans are shown in Figure 2.

We say that a $(p,q)$ -clan $\gamma $ is an if $T_i \cdot \gamma = \Omega _{p,q}$ for at least one $i \in \mathbb {Z}$ . Examples of almost rainbow clans are depicted in Figure 3. Write $\omega _{p}$ for the almost rainbow $(p,p)$ -clan with and . For example, $\omega _3$ is illustrated in the center of Figure 3.

Suppose that $u \in S_{\mathbb {Z}}$ is p-inverse Grassmannian and $v \in S_{\mathbb {Z}}$ is q-inverse Grassmannian. We define a noncrossing backstable clan $\gamma _{u,v}$ associated to the pair $(u,v)$ . Write $\hat {p} = p + \frac {1}{2}$ and $\hat {q}=q + \frac {1}{2}$ . Then $\gamma _{u,v}$ is the unique noncrossing $(p,q)$ -clan such that

1. 1. if $u(i) < \hat {p}$ and $v(i) < \hat {q}$ , then i is initial;

2. 2. if $u(i)> \hat {p}$ and $v(i)> \hat {q}$ , then i is final;

3. 3. if $u(i) < \hat {p}$ and $v(i)> \hat {q}$ , then ; and

4. 4. if $u(i)> \hat {p}$ and $v(i) < \hat {q}$ , then .

See Figure 4 for an example of this construction.

Figure 4

The clan $\gamma _{u,v}$ for $u = 12435$ and $v = 13425$ . Here, $p = 3$ , $q = 2$ , $\hat {p} = 3.5$ , and $\hat {q} = 2.5$ .

Proof. By assumption, there exists $N>0$ such that $u(i) = i$ and $v(i) = i$ for $i \geq N$ and for $i \leq -N$ . Therefore, all $i<-N$ are initial and all $i>N$ are final. Since these sets are then both countably infinite, there is a unique noncrossing way to pair them up. Thus, $\gamma _{u,v}$ is a backstable clan; it remains to show that it is a $(p,q)$ -clan.

Choose an interval $[i,j]$ on which $\gamma _{u,v}$ is supported. Suppose $p \geq q$ . By expanding the interval $[i,j]$ as necessary, assume that $i \leq q$ and $p \leq j$ .

On the interval $[i,j]$ , define

and define

$$\begin{align*}a = |A|, b = |B|, c =|C|, \text{ and } c = |D|. \end{align*}$$

On the interval $[i,j]$ , both u and v take all of the values in $[i,j]$ . We see that the $z \in [i,j]$ with $u(z) \in [i,p]$ are those $z \in A \cup C$ , while those z with $u(z) \in [p+1, j]$ are those $z \in B \cup D$ . Therefore, $a+c = p-i+1$ and $b+d = j-p$ . Similarly, the $z \in [i,j]$ with $v(z) \in [i,q]$ are those $z \in A \cup D$ , while those z with $v(z) \in [q+1,j]$ are those $z \in B \cup C$ . Therefore, $a+d = q-i+1$ and $b+c = j-q$ .

By the definition of ‘supported’, the interval $[i,j]$ contains equal numbers of initial and of final elements, so $a=b$ . Thus, $\zeta (\gamma _{u,v}) = c-d = j-q-(j-p) = p-q$ , as desired for a $(p,q)$ -clan. Moreover, we have $p-i+1 - (j-q) = 0$ , so $p+q = i+j-1$ , and so $\gamma _{u,v}$ is a $(p,q)$ -clan.

The case $p < q$ is entirely analogous; we omit the details.

In the case that p and q are positive, $u,v \in S_{p+q}$ , and $u \leq w_0^{(p+q)}v$ , Lemma 2.1 was previously established by Wyser [Reference WyserWys13, p. 840].

For any positive integers $p,q \in \mathbb {Z}_{>0}$ , we define a set $\Psi _{p,q}$ of almost rainbow $(p,q)$ -clans. Let

$$ \begin{align*} \Psi_{p,q} = \begin{cases} \{ \gamma : \gamma \text{ is almost rainbow and } T_i \cdot \gamma = \Omega_{p,q} \text{ for some } i \in \mathbb{Z}_{>0} \text{ with } i \neq q \}, & \!\!\!\text{if } p \neq q; \\ \{ \gamma : \gamma \text{ is almost rainbow and } T_i \cdot \gamma = \Omega_{p,q} \text{ for some } i \in \mathbb{Z}_{>0} \text{ with } i \neq q \} \cup \{\omega_p \}, & \!\!\! \text{if } p = q. \\ \end{cases} \end{align*} $$

Note that in general $\Psi _{p,q} \neq \Psi _{q,p}$ . If we relax the conditions $i \in \mathbb {Z}_{>0}$ to $i \in \mathbb {Z}$ , then the enlarged set of almost rainbow $(p,q)$ -clans can be used to compute backstable Schubert structure coefficients (introduced in [Reference Lam, Lee and ShimozonoLLS21]).

4.1 Cohomology

We will need the differential operator $\nabla : \mathbb {Z}[x_1, x_2, \dots ] \to \mathbb {Z}[x_1, x_2, \dots ]$ defined by

$$\begin{align*}\nabla = \sum_{i=1}^\infty \frac{\partial}{\partial x_i}. \end{align*}$$

Our key tool will be the following, developed earlier in our joint work with Z. Hamaker and D. Speyer.

Proposition 4.1 [Reference Hamaker, Pechenik, Speyer and WeigandtHPSW20, Proposition 1.1].

For $w \in S_+$ , we have

$$\begin{align*}\nabla \mathfrak{S}_w = \sum_{s_k w < w} k \mathfrak{S}_{s_k w}. \end{align*}$$

From this proposition, we can establish our primary family of linear relations.

Proposition 4.2. Let $u,v,w \in S_+$ . Then

(4.1) $$ \begin{align} \sum_{s_iu < u} i c_{s_iu,v}^w + \sum_{s_jv < v} j c_{u,s_jv}^w = \sum_{s_kw> w} k c_{u,v}^{s_k w}. \end{align} $$

Proposition 4.2 enables one to discern properties of an unknown $c_{u,v}^w$ from properties of other Schubert structure coefficients. In particular, our relations yield some new vanishing and nonvanishing conditions, as well as congruence conditions. We do not know how to relate our linear relations to the the nonpositive recurrence of [Reference KnutsonKnu03].

Proof of Proposition 4.2.

Write

$$\begin{align*}\mathfrak{S}_u \cdot \mathfrak{S}_v = \sum_p c_{u,v}^p \mathfrak{S}_p, \end{align*}$$

and apply the differential operator $\nabla $ to both sides to obtain

(4.2) $$ \begin{align} \sum_{s_i u < u} i \mathfrak{S}_{s_i u} \mathfrak{S}_v + \sum_{s_j v < v} j \mathfrak{S}_u \mathfrak{S}_{s_j v} = \sum_p \sum_{s_k p < p} k c_{u,v}^p \mathfrak{S}_{s_k p} \end{align} $$

by Proposition 4.1 and the Leibniz formula. Now, extract the coefficient of $\mathfrak {S}_w$ from both sides of Equation (4.2) to obtain

$$\begin{align*}\sum_{s_i u < u} i c_{s_i u, v}^w + \sum_{s_jv < v} j c_{u,s_jv}^w = \sum_{s_k w> w} k c_{u,v}^{s_k w}, \end{align*}$$

as desired.

Proposition 4.2 has some surprising corollaries. The following result can be extracted straightforwardly from Monk’s formula [Reference MonkMon59], but we can alternatively derive it easily from Proposition 4.2.

Proof. Specialize Proposition 4.2 to the case $u = s_i$ and $w=v$ . Then we get

$$\begin{align*}i + \sum_{s_j v < v} j c_{s_i, s_jv}^w = \sum_{s_k v> v} k c_{s_i, v}^{s_k v}. \end{align*}$$

Since the sum on the left is nonnegative and $i>0$ , we obtain

$$\begin{align*}0 < \sum_{s_k v> v} k c_{s_i, v}^{s_k v}. \end{align*}$$

But then

$$\begin{align*}0 < \sum_{s_k v> v} c_{s_i, v}^{s_k v}, \end{align*}$$

so there is some k with $s_k v> v$ and $c_{s_i, v}^{s_k v}> 0$ . By dimension counting, the second of these conditions implies the first, so the corollary follows.

Proposition 4.2 also implies many congruence relations among Schubert structure coefficients.

Corollary 4.4. Suppose all left descents of u and v are multiples of $\alpha \in \mathbb Z_+$ . Then, for any $w \in S_+$ , we have

$$\begin{align*}\sum_{s_k w> w} k c_{u,v}^{s_k w} \equiv 0\quad \pmod{\alpha}. \end{align*}$$

If moreover $u=v$ , then

$$\begin{align*}\sum_{s_k w> w} k c_{u,u}^{s_k w} \equiv 0\quad \pmod {2\alpha}. \end{align*}$$

What is remarkable about Corollary 4.4 is that although our sum is congruent to $0$ modulo $\alpha $ , the individual terms of the sum generally are not. For this reason, knowing some of the relevant Schubert structure coefficients imposes strong conditions on the remaining ones. Before giving an example of the application of Corollary 4.4, we need the following easy lemma.

Example 4.6. Suppose $u=13254$ , and note that it only has left descents $2$ and $4$ . Let $w = 231645$ , and suppose we have correctly computed already that $c_{u,u}^w = 1$ .

Now, let $a = s_1w = 132645$ . Corollary 4.4 gives that

(4.3) $$ \begin{align} \sum_{s_k a> a} k c_{u,u}^{s_k a} \equiv 0\quad \pmod{4}. \end{align} $$

But we can expand this sum as

$$\begin{align*}\sum_{s_k a> a} k c_{u,u}^{s_k a} = 1 c_{u,u}^{s_1 a} +3 c_{u,u}^{s_3 a} +4 c_{u,u}^{s_4 a} +6 c_{u,u}^{s_6 a},\end{align*}$$

using Lemma 4.5 to see that the other potential terms vanish. Since we know the term with coefficient $1$ contributes $1$ , the sum of the other terms must be congruent to $3$ modulo $4$ . In particular, we learn for free that $c_{u,u}^{s_3 a} \neq 0$ . Even better, it is immediate without further computation that $c_{u,u}^{s_3 a}$ is odd. In fact, it turns out that $c_{u,u}^{s_3 a} = 1$ .

If we compute also that $c_{u,u}^{s_3 a} = 1$ , we learn then that $c_{u,u}^{s_6 a}$ must be even. In fact, it turns out that $c_{u,u}^{s_6 a} = 0$ .

4.2 K-theory

The structure sheaves of Schubert varieties $X_w \subset \mathsf {Flags}_n$ give classes $[\mathcal {O}_{X_w}]$ in the Grothendieck ring $K^0(\mathsf {Flags}_n)$ of algebraic vector bundles over $\mathsf {Flags}_n$ . These classes form an additive basis and give rise to K-theoretic Schubert structure coefficients $K_{u,v}^w$ defined by

$$\begin{align*}[\mathcal{O}_{X_u}] \cdot [\mathcal{O}_{X_v}] = \sum_w K_{u,v}^w [\mathcal{O}_{X_w}]. \end{align*}$$

When $\ell (w) = \ell (u) + \ell (v)$ , we have $K_{u,v}^w = c_{u,v}^w$ , but, unlike $c_{u,v}^w$ , $K_{u,v}^w$ can be nonzero when $\ell (w)> \ell (u) + \ell (v)$ .

represent Schubert structure sheaf classes analogously to how Schubert polynomials represent Schubert cycles. We may also define Grothendieck polynomials recursively. For $w_0^{(n)} \in S_n$ , we set $\mathfrak {G}_{w_0^{(n)}} = \mathfrak {S}_{w_0^{(n)}} = x_1^{n-1}x_2^{n-2}\cdots x_n^0$ . For w such that $ws_i < w$ , set

$$\begin{align*}\mathfrak{G}_{ws_i} = \overline{N}_i \mathfrak{G}_w, \end{align*}$$

where $\overline {N}_i(f) = N_i( (1-x_{i+1}) f)$ . We have $\mathfrak {G}_w = \mathfrak {G}_{\iota (w)}$ , so we think of Grothendieck polynomials as also being indexed by elements of $S_+$ . The set of Grothendieck polynomials $\{\mathfrak {G}_w \}_{w \in S_+}$ is another linear basis of $\mathbb {Z}[x_1, x_2, x_3, \dots ]$ . The structure coefficients defined by

$$\begin{align*}\mathfrak{G}_u \cdot \mathfrak{G}_v = \sum_w L_{u,v}^w \mathfrak{G}_w \end{align*}$$

agree with the K-theoretic Schubert structure coefficients $K_{u,v}^w$ provided $u,v,w \in S_n$ .

Let $\beta $ be an indeterminate. Define the by

$$\begin{align*}\mathfrak{G}_w^{(\beta)}(x_1, \dots, x_n) = (-\beta)^{-\ell(w)} \mathfrak{G}_w(-\beta x_1, \dots, -\beta x_n). \end{align*}$$

The $\beta $ -Grothendieck polynomials were introduced in [Reference Fomin and KirillovFK94] and represent classes in the connective K-theory of $\mathsf {Flags}_n$ [Reference HudsonHud14]. We will find the $\beta $ -Grothendieck polynomials slightly easier to work with in our context. Let the structure coefficients for $\beta $ -Grothendieck polynomials be $K_{u,v}^w(\beta )$ . We have $K_{u,v}^w(-1) = K_{u,v}^w$ .

Let $\mathrm {Des}(w)$ denote the set of descents of the permutation w. The of w is

$$\begin{align*}\mathrm{maj}(w) = \sum_{i \in \mathrm{Des}(w)} i. \end{align*}$$

We also need the following differential operators related to $\nabla $ :

$$\begin{align*}\nabla^\beta = \nabla + \beta^2 \frac{\partial}{\partial \beta} \quad \text{and} \quad E = \sum_{i=1}^\infty x_i \frac{\partial}{\partial x_i}. \end{align*}$$

We can now recall [Reference Pechenik, Speyer and WeigandtPSW24, Theorem A.1], as reformulated in [Reference Pechenik, Speyer and WeigandtPSW24, Corollary A.2], an analogue of Proposition 4.1 for Grothendieck polynomials and our key tool in this subsection.

Proposition 4.7 [Reference Pechenik, Speyer and WeigandtPSW24, Theorem A.1].

For $w \in S_+$ , we have

$$\begin{align*}\nabla^\beta \mathfrak{G}_w^{(\beta)} = \beta (\mathrm{maj}(w^{-1}) - \ell(w)) \mathfrak{G}_w^{(\beta)} + \sum_{s_kw < w} k \mathfrak{G}^{(\beta)}_{s_k w} \end{align*}$$

and

$$\begin{align*}(\mathrm{maj}(w^{-1}) + \nabla - E)\mathfrak{G}_w = \sum_{s_kw < w} k \mathfrak{G}_{s_k w}. \end{align*}$$

Using essentially the same proof as for Proposition 4.2, but with [Reference Pechenik, Speyer and WeigandtPSW24, Theorem A.1] in place of [Reference Hamaker, Pechenik, Speyer and WeigandtHPSW20, Proposition 1.1], we obtain the following analogue of Proposition 4.2, giving linear relations among K-theoretic Schubert structure coefficients.

Proposition 4.8. Let $u,v,w \in S_+$ . Then

$$ \begin{align*} \beta K_{u,v}^w(\beta) \bigg( &\mathrm{maj}(u^{-1}) + \mathrm{maj}(v^{-1}) - \mathrm{maj}(w^{-1}) - \ell(u) - \ell(v) + \ell(w) \bigg) \\ &+ \sum_{s_iu < u} i K_{s_iu,v}^w(\beta) + \sum_{s_jv < v} j K_{u,s_jv}^w(\beta) = \sum_{s_kw> w} k K_{u,v}^{s_k w}(\beta). \end{align*} $$

Proof. Write

$$\begin{align*}\sum_p K_{u,v}^p(\beta) \mathfrak{G}^{(\beta)}_p=\mathfrak{G}^{(\beta)}_u \cdot \mathfrak{G}^{(\beta)}_v, \end{align*}$$

and apply $\nabla ^\beta $ to both sides, using the first part of Proposition 4.7. Then on the left we have

$$ \begin{align*} \nabla^\beta \left( \sum_p K_{u,v}^p(\beta) \mathfrak{G}^{(\beta)}_p \right) &= \sum_p K_{u,v}^p(\beta) \nabla^\beta \mathfrak{G}^{(\beta)}_p \\ &= \sum_p K_{u,v}^p(\beta) \left( \beta \mathfrak{G}^{(\beta)}_p (\mathrm{maj}(p^{-1}) - \ell(p)) + \sum_{s_kp < p} k \mathfrak{G}^{(\beta)}_{s_k p} \right), \end{align*} $$

while on the right we have

$$ \begin{align*} \nabla^\beta ( \mathfrak{G}^{(\beta)}_u \cdot \mathfrak{G}^{(\beta)}_v) &= \nabla^\beta (\mathfrak{G}^{(\beta)}_u) \mathfrak{G}^{(\beta)}_v + \mathfrak{G}^{(\beta)}_u (\nabla^\beta \mathfrak{G}^{(\beta)}_v) \\&= \left( \beta \mathfrak{G}^{(\beta)}_u (\mathrm{maj}(u^{-1}) - \ell(u)) + \sum_{s_iu < u} i \mathfrak{G}^{(\beta)}_{s_i u} \right) \mathfrak{G}^{(\beta)}_v \\& \quad + \mathfrak{G}^{(\beta)}_u \left(\beta \mathfrak{G}^{(\beta)}_v (\mathrm{maj}(v^{-1}) - \ell(v)) + \sum_{s_j v < v} j \mathfrak{G}^{(\beta)}_{s_j v} \right). \end{align*} $$

Now, we can extract the coefficient of $\mathfrak {G}_w^{(\beta )}$ from both of these obtain

$$ \begin{align*} \sum_{s_k w> w} k K_{u,v}^{s_k w}(\beta) + \beta (&\mathrm{maj}(w^{-1}) - \ell(w)) K_{u,v}^w(\beta) = \beta(\mathrm{maj}(v^{-1}) - \ell(v)) K_{u,v}^w(\beta)\\ & + \beta (\mathrm{maj}(u^{-1}) - \ell(u)) K_{u,v}^w(\beta) +\sum_{s_i u < u} i K_{s_i u, v}^w(\beta)+ \sum_{s_j v < v} j K_{u, s_jv}^w(\beta). \end{align*} $$

The proposition follows by rearranging and collecting terms.

Let $\mathfrak {G}_w(\boldsymbol{1})$ denote the specialization of the Grothendieck polynomial $\mathfrak {G}_w$ obtained by setting all variables equal to $1$ . It is well known to experts that $\mathfrak G_w( \boldsymbol{1})=1$ (see [Reference Yu. Smirnov and TutubalinaST21] for an explicit proof and [Reference Mészáros, Setiabrata and DizierMSS24] for further discussion). We present a new short proof using the second part of Proposition 4.7.

Proof. We proceed by induction on Coxeter length. In the base case, $\mathfrak G_{\mathrm {id}}=1$ and there is nothing to show. Now, fix $w\in S_+$ with $\ell (w) \geq 1$ , and assume the statement holds for all $v\in S_+$ with $\ell (v)<\ell (w)$ .

First, note, for any $f\in \mathbb Z[x_1,x_2,\ldots ]$ , we have $(\nabla -E)(f)|_{\mathbf x=1}=0$ . Thus,

$$ \begin{align*} \mathrm{maj}(w^{-1})\mathfrak G_w(\boldsymbol{1})&=\sum_{s_kw<w}k\mathfrak G_{s_kw}(\boldsymbol{1}) &\text{(by Proposition}\ {4.7})\\ &=\sum_{s_kw<w}k &\text{(by induction)}\\ &=\mathrm{maj}(w^{-1}). \end{align*} $$

Since $\ell (w)\geq 1$ , $\mathrm {maj}(w^{-1})\neq 0$ which implies $\mathfrak G_w(\boldsymbol{1})=1$ .

4.3 Stabilization

Recall the stabilization operation from Section 3.

The following is an analogue of Proposition 4.2; we drop the coefficients on the linear relations at the expense of adding one extra term.

Proposition 4.10. Let $u,v,w \in S_+$ . Then

(4.4) $$ \begin{align} \sum_{s_iu < u} c_{s_iu,v}^w + \sum_{s_jv < v} c_{u,s_jv}^w = c_{1\times u,1\times v}^{s_1(1\times w)}+\sum_{s_kw> w} c_{u,v}^{s_k w}. \end{align} $$

Proof. It follows easily from the pipe dream formula for Schubert polynomials (e.g., [Reference Fomin and KirillovFK96, Reference Bergeron and BilleyBB93, Reference Knutson and MillerKM05]) that

$$\begin{align*}\mathfrak S_{1\times w}(0,x_1,x_2,\dots)=\mathfrak S_w(x_1, x_2,\dots). \end{align*}$$

(Indeed, we will prove a stronger version of this statement as Lemma 5.1.) By Lemma 3.2, $c_{u,v}^w=c_{1\times u,1\times v}^{1\times w}$ .

Apply Proposition 4.2 to $1 \times u$ , $1 \times v$ and $1\times w$ . Then we get

(4.5) $$ \begin{align} \tiny \displaystyle \sum_{s_{i+1}(1\times u) < 1\times u} (i+1) c_{s_{i+1}(1\times u),1\times v}^{1\times w} + \sum_{s_{j+1} (1\times v) < 1\times v} (j+1) c_{1\times u,s_{j+1} (1\times v)}^{1\times w} = \sum_{s_{k+1}(1\times w)> 1\times w} (k+1) c_{1\times u,1\times v}^{s_{k+1} (1\times w)}. \end{align} $$

Thus,

(4.6) $$ \begin{align} \sum_{s_{i} u < u} (i+1) c_{s_{i} u, v}^{ w} + \sum_{s_{j}v < v} (j+1) c_{ u,s_{j} v}^{w} = c_{1\times u,1\times v}^{s_1 (1\times w)}+ \sum_{s_{k} w> w} (k+1) c_{ u, v}^{s_{k} w}. \end{align} $$

Furthermore, from applying Proposition 4.2 to u, v and w, we have

(4.7) $$ \begin{align} \sum_{s_iu < u} i c_{s_iu,v}^w + \sum_{s_jv < v} j c_{u,s_jv}^w = \sum_{s_kw> w} k c_{u,v}^{s_k w}. \end{align} $$

Therefore, subtracting Equation (4.7) from Equation (4.6) yields

$$ \begin{align*} \sum_{s_iu < u} c_{s_iu,v}^w + \sum_{s_jv < v} c_{u,s_jv}^w & = c_{1\times u,1\times v}^{s_1(1\times w)}+\sum_{s_kw> w} c_{u,v}^{s_k w}.\\[-50pt]\end{align*} $$

In many cases, one can see that the extra term on the right side of Proposition 4.10 is in fact $0$ . In these situations, Proposition 4.10 becomes identical to Proposition 4.2, but with the coefficients dropped, yielding two linear relations among the same structure coefficients. In most cases, these relations are linearly independent.

One can also do analogous analysis in K-theory; we omit the details since we will not use the K-theoretic analogue in what follows.

4.4 Iterations of differential operators

Iterating the application of $\nabla $ allows us to obtain additional linear relations among Schubert structure coefficients. These linear relations are somewhat more complicated to state, but the proof is analogous to that of Proposition 4.2.

Let $\mathfrak {S}_w(\boldsymbol{1})$ denote the specialization of the Schubert polynomial $\mathfrak {S}_w$ obtained by setting all variables equal to $1$ . For w with $\ell (w) = k$ , a for w is a sequence $(a_1, a_2, \dots , a_k)$ of positive integers such that $w = s_{a_1} s_{a_2} \cdots s_{a_k}$ . Let $R(w)$ denote the set of all reduced words for w. The Macdonald reduced word identity [Reference MacdonaldMac91, Eq. (6.11)] is the following.

Proposition 4.12 [Reference MacdonaldMac91, Eq. (6.11)].

Let $w \in S_+$ have $\ell (w) = k$ . Then

$$\begin{align*}k! \mathfrak{S}_w(\boldsymbol{1}) = \sum_{a \in R(w)} a_1 a_2 \cdots a_k. \end{align*}$$

We can use the Macdonald reduced word identity to obtain more linear relations.

Proposition 4.13. Let $u,v\in S_+$ , and fix $1\leq k\leq \ell (u)+\ell (v)$ . Let $w\in S_+$ with

$$\begin{align*}\ell(w)=\ell(u)+\ell(v)-k. \end{align*}$$

Then,

(4.8) $$ \begin{align} \sum_{\substack{\hat{w}\geq_Lw \\ \ell(\hat{w})=\ell(w)+k}} \!\! c_{u,v}^{\hat{w}}\cdot \mathfrak S_{\hat{w}w^{-1}}(\boldsymbol{1})=\sum_{i=0}^{k} \sum_{\substack{\hat{u}\leq_L u \\ \ell(\hat{u})=\ell(u)-i}} \;\;\sum_{\substack{\hat{v} \leq_L v \\ \ell(\hat{v})=\ell(v)-(k-i)}} \!\!\!\! c_{\hat{u},\hat{v}}^{w}\cdot\mathfrak S_{u\hat{u}^{-1}}(\boldsymbol{1}) \cdot \mathfrak S_{v\hat{v}^{-1}}(\boldsymbol{1}). \end{align} $$

Proof. First, note that

$$ \begin{align*} \nabla^k(\mathfrak S_\pi)=\sum_{\substack{\hat{\pi}\leq_L \pi \\ \ell(\hat{\pi})=\ell(\pi)-k}} \left(\sum_{a \in R(\pi \hat{\pi}^{-1})} a_1a_2 \cdots a_k \right)\cdot \mathfrak S_{\hat{\pi}} \end{align*} $$

by Proposition 4.1. By Proposition 4.12, we can replace the inner summation to obtain

(4.9) $$ \begin{align} \nabla^k(\mathfrak S_\pi)=\sum_{\substack{\hat{\pi}\leq_L \pi \\ \ell(\hat{\pi})=\ell(\pi)-k}} k!\mathfrak S_{\pi \hat{\pi}^{-1}}(\boldsymbol{1})\cdot \mathfrak S_{\hat{\pi}}. \end{align} $$

Write

$$\begin{align*}\sum_{\hat{p}}c_{u,v}^{\hat{p}} \mathfrak S_{\hat{p}}=\mathfrak S_u \cdot \mathfrak S_v. \end{align*}$$

Applying $\nabla ^k$ to both sides, we obtain

$$ \begin{align*} \nabla^k\left(\sum_{\hat{p}} c_{u,v}^{\hat{p}} \mathfrak S_{\hat{p}} \right)=\nabla^k(\mathfrak S_u\cdot \mathfrak S_v) =\sum_{i=0}^{k} \binom{k}{i} \nabla^i(\mathfrak S_u)\nabla^{k-i}(\mathfrak S_v). \end{align*} $$

Thus, Equation (4.9) yields

$$ \begin{align*} \sum_{\hat{p}} c_{u,v}^{\hat{p}} & \sum_{\substack{p\leq_L \hat{p} \\ \ell(p)=\ell(\hat{p})-k}} k!\mathfrak S_{\hat{p}p^{-1}}(\boldsymbol{1}) \mathfrak S_p \\ &=\sum_{i=0}^{k} \binom{k}{i} \left(\sum_{\substack{\hat{u} \leq_Lu \\ \ell(\hat{u})=\ell(u)-i}} \! i!\mathfrak S_{u\hat{u}^{-1}}(\boldsymbol{1}) \mathfrak S_{\hat{u}}\right) \left (\sum_{\substack{\hat{v}\leq_L v \\ \ell(\hat{v})=\ell(v)-(k-i)}} \!\! \!(k-i)!\mathfrak S_{v\hat{v}^{-1}}(\boldsymbol{1}) \mathfrak S_{\hat{v}} \right )\\ &=\sum_{i=0}^{k} k! \sum_{\substack{\hat{u}\leq_L u \\ \ell(\hat{u})=\ell(u)-i}} \:\;\; \sum_{\substack{\hat{v} \leq_L v \\ \ell(\hat{v})= \ell(v)-(k-i)}} \mathfrak S_{u\hat{u}^{-1}}(\boldsymbol{1}) \mathfrak S_{v\hat{v}^{-1}}(\boldsymbol{1})\sum_q c_{\hat{u},\hat{v}}^q \mathfrak S_q. \end{align*} $$

Extract the coefficient of $\mathfrak S_w$ from each side to obtain

$$ \begin{align*} k! \!\!\!\!\sum_{\substack{\hat{w}\geq_Lw \\ \ell(\hat{w})=\ell(w)+k}} \!\!\! c_{u,v}^{\hat{w}} \mathfrak S_{\hat{w}w^{-1}}(\boldsymbol{1})= k! \sum_{i=0}^{k} \sum_{\substack{\hat{u}\leq_Lu \\ \ell(\hat{u})=\ell(u)-i}} \;\;\;\sum_{\substack{\hat{v}\leq_Lv \\ \ell(\hat{v})=\ell(v)-(k-i)}} \!\!\!\! c_{\hat{u},\hat{v}}^{w}\mathfrak S_{u\hat{u}^{-1}}(\boldsymbol{1}) \mathfrak S_{v\hat{v}^{-1}}(\boldsymbol{1}). \end{align*} $$

The proposition follows by dividing out $k!$ .

Proposition 4.2 may alternatively be proved as a corollary to Proposition 4.13 by setting $k=1$ . On the other extreme, we have the following corollary. Define

$$\begin{align*}\delta(w,i) = \begin{cases} 1, & \text{if}\ i \in \mathrm{Des}(w); \\ 0, & \text{if}\ i \notin \mathrm{Des}(w). \end{cases} \end{align*}$$

Corollary 4.14. Let $u,v\in S_+$ and fix $i \in \mathbb {Z}_{>0}$ . Then,

(4.10) $$ \begin{align} \sum_{p: i \in \mathrm{Des}(p)} c_{u,v}^{p}\mathfrak S_{ps_i}(\boldsymbol{1})=\delta(v,i)\mathfrak S_u(\boldsymbol{1})\mathfrak S_{vs_i}(\boldsymbol{1})+\delta(u,i)\mathfrak S_{us_i}(\boldsymbol{1})\mathfrak S_{v}(\boldsymbol{1}). \end{align} $$

The following was observed as [Reference KnutsonKnu01, Lemma 1.1], where the phenomenon was referred to as dc-triviality. We obtain another easy proof of this fact.