Padded Schubert polynomials and weighted enumeration of Bruhat chains

TL;DR

This paper generalizes the enumeration of maximal chains in the strong Bruhat order on symmetric groups, introducing weights that connect to Macdonald's identities and Schubert polynomials, offering new combinatorial insights.

Contribution

It provides a unified framework for weighted enumeration of Bruhat chains, extending known formulas and introducing a one-parameter family of weights related to Schubert polynomials.

Findings

Weighted counts of Bruhat chains generalize factorial formulas

Introduction of weights leading to Macdonald's identities

New combinatorial interpretations for Schubert polynomials

Abstract

We prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ${n \choose 2}!$ for both the code weights and the Chevalley weights. We also define weights which give a one-parameter family of strong order analogues of Macdonald's reduced word identity for Schubert polynomials.

Equations

$$\mathrm{wt}(c):=\prod _{i=1}^{k-1}\mathrm{wt}(u_i⋖u_{i+1}).$$

$$m_{\mathrm{wt}}(v,w):=\sum _{c:v\to w}\mathrm{wt}(c)$$

$$a_{v⋖w}$$

$$b_{v⋖w}$$

$$c_{v⋖w}$$

$$d_{v⋖w}$$

$$m_{Chev}(e,w_0)({\alpha }_1,...,{\alpha }_{n-1})=\left(\genfrac{}{}{0ex}{}{n}{2}\right)!\cdot \prod _{1\le k<\ell \le n-1}\frac{{\alpha }_k+\cdots +{\alpha }_{\ell -1}}{\ell -k}$$

$$m_{Chev}(e,w_0)(1,...,1)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)!.$$

$$m_{code}(w,w_0)=\left(\left(\genfrac{}{}{0ex}{}{n}{2}\right)-\ell (w)\right)!\cdot {\mathfrak{S}}_w(1,...,1)$$

$$m_{code}(e,w_0)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)!.$$

$$f(v⋖w):=1+a_{v⋖w}{z}_A+b_{v⋖w}{z}_B+c_{v⋖w}{z}_C+d_{v⋖w}{z}_D.$$

$$m_{\mathrm{wt}}(e,w_0)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)!.$$

$$\mathrm{wt}(v⋖v{t}_{ij})={\alpha }_i+{\alpha }_{i+1}+\cdots +{\alpha }_{j-1}+(b_{v⋖v{t}_{ij}}-d_{v⋖v{t}_{ij}})z.$$

$$m_{\mathrm{wt}}(e,w_0)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)!\cdot \prod _{k<\ell }\frac{{\alpha }_k+\cdots +{\alpha }_{\ell -1}}{\ell -k}.$$

$$\mathrm{wt}(v⋖w)=1+b_{v⋖w}(2-z)+c_{v⋖w}z.$$

$$m_{\mathrm{wt}}(w,w_0)=\left(\left(\genfrac{}{}{0ex}{}{n}{2}\right)-\ell (w)\right)!\cdot {\mathfrak{S}}_w(1,...,1).$$

$$\mathrm{wt}(v⋖w=v{t}_{ij})$$

$$=(b_{v⋖w}+d_{v⋖w}+1)+(b_{v⋖w}-d_{v⋖w})$$

$$=1+2b_{v⋖w}.$$

$$N_i\cdot g(x_1,...,x_n):=\frac{g(x_1,...,x_n)-g(x_1,...,x_{i+1},x_i,...,x_n)}{x_i-x_{i+1}}.$$

$$\Delta =\sum _{i=1}^n{x}_i\frac{\partial }{\partial y_i}.$$

$$\Delta {\text{SS}}_w=\sum _{u:w⋖u}(1+2b_{w⋖u}){\text{SS}}_u.$$

$${\mathfrak{S}}_w(1,...,1)=\frac{1}{\ell (w)!}\sum _{s_{a_1}\cdots s_{a_{\ell (w)}}=w}{a}_1\cdots a_{\ell (w)}.$$

$$\Delta \left(\sum _{j=1}^n{\left(\frac{x_j}{y_j}\right)}^k{y}^{\rho }\right)=$$

$$=$$

$$(x_1+\cdots +x_m){\mathfrak{S}}_w=\sum _{\begin{array}{c}j\le m<k\\ \ell (w{t}_{jk})=\ell (w)+1\end{array}}{\mathfrak{S}}_{w{t}_{jk}}$$

$${\alpha }_1{x}_1+{\alpha }_2(x_1+x_2)+{\alpha }_3(x_1+x_2+x_3)+\cdots +{\alpha }_{n-1}(x_1+\cdots +x_{n-1})$$

$$=$$

$$M{\mathfrak{S}}_w=\sum _{w⋖w{t}_{ij}}({\alpha }_i+\cdots +{\alpha }_{j-1}){\mathfrak{S}}_{w{t}_{ij}}.$$

$$R{\mathfrak{S}}_w=\sum _{u:w⋖u}(b_{w⋖u}-d_{w⋖u}){\mathfrak{S}}_u.$$

$$R{\mathfrak{S}}_w=$$

$$=$$

$$x^{\gamma }$$

Full text

C.G. was supported by a National Science Foundation Graduate Research Fellowship under Grant No. 1122374

1. Introduction

Let $\bm{S}_{n}$ denote the strong Bruhat order on the symmetric group $S_{n}$ (see Section 2 for background and definitions). Given a function $\operatorname{wt}:\operatorname{Cov}(\bm{S}_{n})\to R$ from the set of covering relations of $\bm{S}_{n}$ to a ring $R$, and a saturated chain $c=(u_{1}\lessdot u_{2}\lessdot\cdots\lessdot u_{k})$, we define the weight of $c$ mutliplicatively:

[TABLE]

For $v\leq w$ in $\bm{S}_{n}$ we let

[TABLE]

denote the total weighted number of chains over all saturated chains $c$ from $v$ to $w$.

In this paper, we study several classes of weights which generalize the recently introduced code weights [2] and the more classical Chevalley weights, studied, for example, by Postnikov and Stanley [8] and by Stembridge [10]. Some building blocks for these new weights are given in Definition 1.1.

Definition 1.1**.**

For $v\lessdot w=vt_{ij}$ a covering relation in $\bm{S}_{n}$ with $i<j$, let $a_{v\lessdot w},b_{v\lessdot w},c_{v\lessdot w}$, and $d_{v\lessdot w}$ denote the number of dots in the regions $A,B,C$, and $D$ respectively in Figure 1. That is,

[TABLE]

Note that we always have $b_{v\lessdot w}+d_{v\lessdot w}=j-i-1$ and $a_{v\lessdot w}+c_{v\lessdot w}=v_{j}-v_{i}-1$.

The Chevalley weights $\operatorname{wt}_{Chev}(v\lessdot w):\operatorname{Cov}(\bm{S}_{n})\to\mathbb{Z}[\alpha_{1},...,\alpha_{n-1}]$ assign weight $\alpha_{i}+\cdots+\alpha_{j-1}$ to the covering relation $v\lessdot w=vt_{ij}$, where $t_{ij}=(i\>j)$ is a transposition. It was shown by Stembridge [10] that:

[TABLE]

where $w_{0}=n(n-1)\cdots 21$ denotes the longest permutation. Specializing all $\alpha_{i}=1$ recovers a classical fact, due essentially to Chevalley:

[TABLE]

Recently, a new set of weights, the code weights $\operatorname{wt}_{code}:\operatorname{Cov}(\bm{S}_{n})\to\mathbb{N}$ were defined in the course of proving the Sperner property for the weak Bruhat order [3]. In the notation of Definition 1.1, the code weights are defined by $\operatorname{wt}_{code}(v\lessdot w)=1+2b_{v\lessdot w}$. In [2], it was shown that

[TABLE]

where $\mathfrak{S}_{w}$ is the Schubert polynomial (see Section 2), providing a strong Bruhat order analogue of Macdonald’s well known identity for $\mathfrak{S}_{w}(1,...,1)$ as a weighted enumeration of chains in the weak Bruhat order (see Proposition 2.3). Letting $w=e$ in (3) gives:

[TABLE]

One motivation of this work is to understand and generalize the coincidence between (2) and (4); this is done in Theorem 1.2. This theorem, together with the dual weak and strong order identities of [4] and [2], also gives the first explanation of why the weighted path counts $e\to w_{0}$ for Macdonald’s weak order weights and Chevalley’s strong order weights should agree.

Theorem 1.2**.**

Let $f:\operatorname{Cov}(\bm{S}_{n})\to\mathbb{Z}[z_{A},z_{B},z_{C},z_{D}]$ be the weight function defined by

[TABLE]

Let $\operatorname{wt}:\operatorname{Cov}(\bm{S}_{n})\to\mathbb{Z}[z]$ be any weight function obtained from $f$ by specializing the variables so that $\{z_{A},z_{B},z_{C},z_{D}\}=\{0,0,z,2-z\}$ as multisets, then:

[TABLE]

In particular, $m_{\operatorname{wt}}(e,w_{0})$ does not depend on $z$.

Theorem 1.3 provides a common generalization of (1) and (4); see Example 1.5.

Theorem 1.3**.**

Let $\operatorname{wt}:\operatorname{Cov}(\bm{S}_{n})\to\mathbb{Z}[\alpha_{1},...,\alpha_{n-1},z]$ be defined by

[TABLE]

Then

[TABLE]

In particular, $m_{\operatorname{wt}}(e,w_{0})$ does not depend on $z$.

Theorem 1.4 extends (3) to a one-parameter family of strong Bruhat analogues of Macdonald’s identity.

Theorem 1.4**.**

Let $\operatorname{wt}:\operatorname{Cov}(\bm{S}_{n})\to\mathbb{Z}[z]$ be defined by

[TABLE]

Then for any $w\in S_{n}$ we have

[TABLE]

In particular, $m_{\operatorname{wt}}(w,w_{0})$ does not depend on $z$.

Example 1.5**.**

Various specializations of the above Theorems give previously known results:

1. (1)

Letting $z_{B}=2$ and $z_{A}=z_{C}=z_{D}=0$ in Theorem 1.2 recovers (4), while letting $z_{B}=z_{D}=1$ and $z_{A}=z_{C}=0$ recovers (2). 2. (2)

Letting all $z=\alpha_{1}=\cdots=\alpha_{n-1}=1$ in Theorem 1.3 the weight becomes:

[TABLE]

This recovers the identity (4) for the code weights. 3. (3)

Letting $z=0$ in Theorem 1.3 recovers Stembridge’s identity (1) for the Chevalley weights. 4. (4)

Letting $z=0$ in Theorem 1.4 recovers the strong order Macdonald identity (3).

Section 2 covers background and definitions. Theorem 1.3 is proven in Section 3. Section 4 proves Theorem 1.4 and deduces Theorem 1.2 from Theorems 1.3 and 1.4.

2. Background and definitions

2.1. Bruhat order

Let $s_{1},...,s_{n-1}$ denote the adjacent transpositions in the symmetric group $S_{n}$. For any permutation $w\in S_{n}$, its length $\ell(w)$ is the minimal number of simple transpositions needed to write $w=s_{i_{1}}\cdots s_{i_{\ell}}$ as a product.

The (strong) Bruhat order $\bm{S}_{n}=(S_{n},\leq)$ is defined by its covering relations: $v\lessdot w$ whenever $w=vt_{ij}$ for some $i,j$ and $\ell(w)=\ell(v)+1$; this order relation encodes the containment of Schubert varieties in the flag variety. The Bruhat order has unique minimal element the identity permutation $e$, and unique maximal element $w_{0}=n(n-1)...21$ of length ${n\choose 2}$, called the longest element. The Hasse diagram of $\bm{S}_{3}$ is shown in Figure 2.

It is well known that the maps $v\mapsto w_{0}v$ and $v\mapsto vw_{0}$ are antiautomorphisms of the Bruhat order $\bm{S}_{n}$ and that $v\mapsto v^{-1}$ is an automorphism [1]; Proposition 2.1 determines the effect of these maps on the quantities $a,b,c,$ and $d$ from Definition 1.1.

Proposition 2.1**.**

Let $v\lessdot w$ be a covering relation in $\bm{S}_{n}$.

1. (1)

$a_{v\lessdot w}=d_{v^{-1}\lessdot w^{-1}}$* and $b_{v\lessdot w}=c_{v^{-1}\lessdot w^{-1}}$,* 2. (2)

$b_{v\lessdot w}=d_{w_{0}w\lessdot w_{0}v}$, and 3. (3)

$a_{v\lessdot w}=c_{ww_{0}\lessdot vw_{0}}$.

Proof.

These are clear from Figure 1 after observing that inversion corresponds to reflecting the permutation matrix across the main (top-left to bottom-right) diagonal, that left multiplication by $w_{0}$ corresponds to reflecting across the vertical axis, and that right multiplication by $w_{0}$ corresponds to reflecting across the horizontal axis. ∎

2.2. Schubert polynomials and padded Schubert polynomials

For $w\in S_{n}$ the Schubert polynomials $\mathfrak{S}_{w}(x_{1},...,x_{n})$, introduced by Lascoux and Schützenberger [5], represent the classes of Schubert varieties in the cohomology $H^{*}(G/B)$ of the flag variety. They can be defined recursively as follows:

• •

$\mathfrak{S}_{w_{0}}(x_{1},...,x_{n})=x_{1}^{n-1}x_{2}^{n-1}\cdots x_{n-2}^{2}x_{1}=\bm{x}^{\rho}$, where $\rho=(n-1,n-2,...,1)$ denotes the staircase composition, and

• •

$\mathfrak{S}_{ws_{i}}=N_{i}\cdot\mathfrak{S}_{w}$ when $\ell(ws_{i})<\ell(w)$.

Here $N_{i}$ denotes the $i$-th Newton divided difference operator:

[TABLE]

The Schubert polynomials $\{\mathfrak{S}_{w}\}_{w\in S_{n}}$ form a basis for the vector space $V_{n}=\operatorname{span}_{\mathbb{Q}}\{\bm{x}^{\gamma}\>|\>\gamma\leq\rho\}$, where here $\leq$ denotes component-wise comparison. Let $\widetilde{V}_{n}=\operatorname{span}_{\mathbb{Q}}\{\bm{x}^{\gamma}\bm{y}^{\rho-\gamma}\>|\>\gamma\leq\rho\}$, then the padded Schubert polynomials $\widetilde{\mathfrak{S}}_{w}$, introduced in [2], are defined as the images of the $\mathfrak{S}_{w}$ under the natural map $\bm{x}^{\gamma}\mapsto\bm{x}^{\gamma}\bm{y}^{\rho-\gamma}$ from $V_{n}\to\widetilde{V}_{n}$. Define a differential operator $\Delta:\widetilde{V}_{n}\to\widetilde{V}_{n}$ by

[TABLE]

Proposition 2.2** ([2]).**

For any $w\in S_{n}$ we have:

[TABLE]

We will also need Macdonald’s well-known reduced word identity for the number of monomials in $\mathfrak{S}_{w}$:

Proposition 2.3** ([6]).**

[TABLE]

3. Proof of Theorem 1.3

We will modify a proof idea for (1) due to Stanley [9]. Let us define some linear operators on the coinvariant algebra $\mathbb{C}[x_{1},\ldots,x_{n}]/I$, where $I$ is the ideal generated by all symmetric polynomials in $x_{1},\ldots,x_{n}$ with vanishing constant terms. The core of the argument comes from interpreting the operator $\Delta$ with respect to two different bases of $\mathbb{C}[x_{1},\ldots,x_{n}]/I$: one is $\{\mathfrak{S}_{w}|w\in S_{n}\}$ and the other one is $\{\bm{x}^{\gamma}|\gamma\leq\rho\}$.

Recall that we have defined $\Delta$ on $\widetilde{V}_{n}$. We can define it naturally on $V_{n}$ since $V_{n}\rightarrow\widetilde{V}_{n}$ is an isomorphism. Namely, it can be seen from the definition that \Delta x^{\gamma}=\big{(}\sum_{i=1}^{n}(n-i-\gamma_{i})x_{i}\big{)}x^{\gamma} for $\gamma\leq\rho$ (this operator $\Delta$ appears implicitly in [4]). Moreover, we can extend this definition of $\Delta$ to $\mathbb{C}[x_{1},\ldots,x_{n}]$ by the same formula. We claim that such definition is in fact well-defined on the quotient $\mathbb{C}[x_{1},\ldots,x_{n}]/I$. This is formulated in the following technical lemma, which is necessary for the correctness of the main proof but is not related to the key idea of the proof.

Lemma 3.1**.**

The linear operator \Delta:x^{\gamma}\mapsto\big{(}\sum_{i=1}^{n}(n-i-\gamma_{i})x_{i}\big{)}x^{\gamma} is well-defined on $\mathbb{C}[x_{1},\ldots,x_{n}]/I$ and coincides with $\sum_{i=1}^{n}x_{i}\frac{\partial}{\partial y_{i}}$ on $\widetilde{V}_{n}$.

Proof.

We need to check that if $f\in I$, then $\Delta f\in I$. For convenience, we will first pad every monomial $x^{\gamma}$ to $x^{\gamma}y^{\rho-\gamma}$, allowing negative exponents on $y$-variables, so that we can use $\Delta=\sum_{i}\frac{\partial}{\partial y_{i}}x_{i}$, and then specialize $y_{i}$’s to 1. This is compatible with the definition as in the statement of the lemma. This means $\Delta(fg)=f\Delta(g)+g\Delta(f)$. As a result, it suffices to check if $f$ is a generator of $I$, then $\Delta f\in I$.

Let us pick the power sum symmetric functions $f=x_{1}^{k}+\cdots+x_{n}^{k}$ as generators, for $k\geq 1$. After padding, we get $\sum_{j}(\frac{x_{j}}{y_{j}})^{k}y^{\rho}.$ Then

[TABLE]

It is clear that both terms belong to $I$ after specializing $y_{i}$’s to 1. So we are done. ∎

We will also make use of the following classical identity, called Monk’s rule (or Monk’s formula).

Proposition 3.2** (Monk’s rule, see e.g. [7]).**

For any $1\leq m<n$, the identity

[TABLE]

holds in $\mathbb{C}[x_{1},\ldots,x_{n}]/I$.

Now let $\alpha_{1},\ldots,\alpha_{n-1}$ be as in Theorem 1.3 and define a linear operator $M$ as multiplication by

[TABLE]

where $\beta_{i}=\alpha_{i}+\cdots+\alpha_{n-1}$. By Monk’s rule,

[TABLE]

Note that Monk’s rule only holds modulo the ideal $I$, and not as an identity of polynomials. Define another linear operator $R$ by

[TABLE]

Write $M_{1}=M$ and define $M_{k+1}=[M_{k},R]:=M_{k}R-RM_{k}$ for $k\geq 1$. Here, $[,]$ is the standard Lie bracket.

Lemma 3.3**.**

The operator $M_{k}$ is the same as multiplication by the element $(k-1)!(\beta_{1}x_{1}^{k}+\beta_{2}x_{2}^{k}+\cdots+\beta_{n-1}x_{n-1}^{k}).$

Proof.

Let’s analyze $R$ a bit more. We have

[TABLE]

where the last equality follows from Proposition 2.2 and Monk’s rule (as a special case of $M$ by assigning $\alpha_{1}=\cdots=\alpha_{n-1}=1$).

We use induction on $k$. Since multiplications by polynomials commute with each other, we have $M_{k}R-RM_{k}=M_{k}\Delta-\Delta M_{k}$. Let’s compute what it does on monomials $\bf{x}^{\gamma}$:

[TABLE]

while on the other hand,

[TABLE]

Here, the calculation of $\Delta Mx^{\gamma}$ uses the fact that $\Delta$ is defined on all of $\mathbb{C}[x_{1},\ldots,x_{n}]/I$ (Lemma 3.1), since the coefficient of $x_{j}$ may exceed $n-j$. As a result, we see that $(M_{k}\Delta-\Delta M_{k})x^{\gamma}=k!(\sum_{i=1}^{n-1}\beta_{i}x_{i}^{k+1})x^{\gamma}$. So the induction step goes through. ∎

Remark*.*

In fact, the operator $R$ can be more elegantly written as

[TABLE]

when $f\in\widetilde{V}_{n}$ is already padded.

Lemma 3.4**.**

View $M_{k}=(k-1)!\sum_{i=1}^{n-1}\beta_{i}x_{i}^{k}$ as polynomials. If $\sum kp_{k}={n\choose 2}$ and $p_{1}<{n\choose 2}$, then $\prod_{k}M_{k}^{p_{k}}$ lies in the ideal $I$ .

Proof.

Write $M_{k}=(k-1)!\sum_{i=1}^{n}\beta_{i}x_{i}^{k}$ with $\beta_{n}=0$. As $\prod_{k}M_{k}^{p_{k}}$ is homogeneous of degree ${n\choose 2}$, we can write it as $f\mathfrak{S}_{w_{0}}$ modulo $I$, where $f$ depends only on $\beta_{i}$’s. In fact, we can obtain $f\mathfrak{S}_{w_{0}}$ by first multiplying out $\prod_{k}M_{k}^{p_{k}}$ and then performing subtraction with respect to the homogeneous part of degree ${n\choose 2}$ in $\mathbb{C}[x_{1},\ldots,x_{n}]/I$. This shows that $f$ is a polynomial of degree at most ${n\choose 2}-2\ell+\ell={n\choose 2}-\ell$.

On the other hand, if $\beta_{i}=\beta_{i+1}$, then $\prod_{k\geq 1}M_{k}^{p_{k}}$ is symmetric in $x_{i}$ and $x_{i+1}$. Consequently, $N_{i}(\prod_{k}M_{k}^{p_{k}})=0$, where $N_{i}$ is the $i$-th divided difference operator introduced in Section 2. But $0=N_{i}(f\mathfrak{S}_{w_{0}})=f(N_{i}\mathfrak{S}_{w_{0}})=f\mathfrak{S}_{w_{0}s_{i}}$. As $\mathfrak{S}_{w_{0}s_{i}}\neq 0$, we deduce that $(\beta_{i}-\beta_{i+1})|f$. By symmetry, $(\beta_{i}-\beta_{j})|f$ for all $1\leq i<j\leq n$. As $\deg_{\beta}f=\sum p_{k}<{n\choose 2}$, we conclude that $f=0$. ∎

Lemma 3.5**.**

With $M,R$ as above, $(M+zR)^{n\choose 2}\cdot 1=M^{n\choose 2}\cdot 1.$

Proof.

Notice that $R\cdot 1=R\cdot\mathfrak{S}_{e}=0$ as $b_{w\lessdot s_{i}}=d_{w\lessdot s_{i}}=0$. The rest is a simple consequence of Lemma 3.3 and Lemma 3.4. Namely, expand $(M+zR)^{n\choose 2}$ and move $R$’s towards the right such that in each step, we replace $\cdots RM_{k}\cdots$ by $\cdots M_{k}R\cdots-\cdots M_{k+1}\cdots$, keeping the total degree. In the end when no such moves are possible, either $R$ appears on the right side, resulting in a term equal to 0 (since $R\cdot 1=0$), or $\prod_{k\geq 1}M_{k}^{p_{k}}$ appears with $\sum kp_{k}={n\choose 2}$, which is also 0 except the single term $M^{n\choose 2}$. ∎

Theorem 1.3 now follows easily.

Proof of Theorem 1.3.

The weights in Theorem 1.3 are given by:

[TABLE]

Recall that we have

[TABLE]

so putting them together,

[TABLE]

An iteration (or induction) immediately gives

[TABLE]

Taking $w=e$ and $\ell={n\choose 2}$ in the above setting, we see that $m_{\operatorname{wt}}(e,w_{0})$ is the coefficient of $\mathfrak{S}_{w_{0}}$ in $(M+zR)^{n\choose 2}$, modulo $I$. By Lemma 3.5, such coefficient does not depend on $z$. When $z=0$, our result is given by Stembridge [10] (see also Stanley [9]). ∎

4. Proof of Theorem 1.4

We first note a simple fact about the specialization of $\SS_{w}$: since $\SS_{w}$ has total $x$-degree $\ell(w)$ and total $y$-degree ${n\choose 2}-\ell(w)$, we have

[TABLE]

We then have the following lemma.

Lemma 4.1**.**

Fix $w\in S_{n}$. Then

[TABLE]

Proof.

By Proposition 2.3, we have

[TABLE]

Thus $\mathfrak{S}_{u}$ and $\mathfrak{S}_{u^{-1}}$ have the same number of monomials, so we have $\SS_{u}(1,\ldots,1)=\SS_{u^{-1}}(1,\ldots,1)$. Therefore, as $\ell(u)=\ell(u^{-1})$, $(\Delta\SS_{w})(1,\ldots,1)=(\Delta\SS_{w^{-1}})(1,\ldots,1)$. In addition, notice that $w\lessdot u$ if and only if $w^{-1}\lessdot u^{-1}$ and that $b_{w\lessdot u}=c_{w^{-1}\lessdot u^{-1}}$ by Proposition 2.1.

Apply Proposition 2.2 to $w$ and $w^{-1}$ separately. We have

[TABLE]

Now take the principal specialization and subtract these two equations. The left-hand side becomes zero as explained above. Recalling from above that $\SS_{u^{-1}}(1,\ldots,1)=\SS_{u}(1,\ldots,1)$, we obtain the desired equality. ∎

Now we are ready to prove Theorem 1.4.

Proof of Theorem 1.4.

We proceed by induction on ${n\choose 2}-\ell(w)$. The base case $w=w_{0}$ is trivial as both sides equal 1. Now fix $w$ and assume that the statement is true for all $u$ with $\ell(u)>\ell(w)$. The following calculation is straightforward:

[TABLE]

By Lemma 4.1, the second term in the above expression becomes 0. And by the principal specialization of Proposition 2.2, we have that

[TABLE]

Thus the first term in the above expression becomes $({n\choose 2}-\ell(w))!\mathfrak{S}_{w}(1,\ldots,1)$, which is what we want. ∎

We can now complete the proof of Theorem 1.2.

Proof of Theorem 1.2.

There are six cases to consider, depending on which pair of $z_{A},z_{B},z_{C},$ and $z_{D}$ are equal to $z$ and $2-z$ (the others being zero); which element of the pair is sent to $z$ or $2-z$ does not matter, since the claimed result is independent of $z$.

For the pair $\{z_{B},z_{C}\}$, letting $w=e$ in Theorem 1.4 proves the result. For $\{z_{B},z_{D}\}$, letting $\alpha_{1}=\cdots=\alpha_{n-1}$ in Theorem 1.3 gives weights

[TABLE]

which clearly give all of the desired linear combinations of $b_{v\lessdot w}$ and $d_{v\lessdot w}$.

Applying the symmetries from Proposition 2.1 then yields the remaining pairs. ∎

Acknowledgements

The authors wish to thank Alex Postnikov and John Stembridge for helpful suggestions.