2.1 Notation

Throughout the paper, we set $\chi (\text {True})=1$ and $\chi (\text {False})=0$.

2.1.2 Partitions

Let ${\mathbb {Y}}$ denote the set of partitions or Young diagrams. We consider a partition $\lambda =(\lambda _1,\ldots ,\lambda _\ell )$ as an infinite sequence $(\lambda _1,\ldots ,\lambda _\ell ,0,0,\ldots )$ if necessary. Throughout the paper, Young diagrams are drawn in English notation: the boxes are top left justified in the plane. For a Young diagram $\lambda$, we let $\lambda '$ denote the conjugate (or transpose) Young diagram. The dominance order on partitions of the same size is given by $\lambda \leqslant \mu$ if $\sum _{i=1}^k \lambda _i \leqslant \sum _{i=1}^k \mu _i$ for all $k$.

There is a bijection between ${\mathbb {Y}}$ and $S_{\mathbb {Z}}^0$, given by $\lambda \mapsto w_\lambda$, where

(2.1)\begin{equation} w_\lambda(i) := i + \begin{cases} \lambda_{1-i} & \text{if $i\leqslant 0$}, \\ -\lambda'_i & \text{if $i > 0$.} \end{cases} \end{equation}

A reduced expression for $w_\lambda$ is obtained by labeling the box $(i,j)$ in the $i$th row and $j$th column of the diagram of $\lambda$ by $s_{j-i}$ and reading the rows from right to left starting with the bottom row.

If $\mu \subset \lambda$, we define

(2.2)\begin{equation} w_{\lambda/\mu} := w_\lambda w_\mu^{{-}1}. \end{equation}

We note that $w_\lambda \doteq w_{\lambda /\mu } w_\mu$. An element $w\in S_{\mathbb {Z}}$ is 321-avoiding if there is no triple of integers $i<j<k$ such that $w(i)>w(j)>w(k)$.

Lemma 2.1 [Reference Billey, Jockusch and StanleyBJS93, § 2]

An element $w\in S_{\mathbb {Z}}$ is $321$-avoiding if and only if $w=w_{\lambda /\mu }$ for some partitions $\mu \subset \lambda$.

Example 2.2 For $\lambda =(3,2)$, the values of $w_\lambda :{\mathbb {Z}}\to {\mathbb {Z}}$ are given. For $\mu =(1)$ we have $w_\mu =s_0$. Reduced decompositions for $w_\lambda$ and $w_{\lambda /\mu }$ are given.

2.2 Schubert polynomials

Following [Reference Lascoux and SchützenbergerLS82], we define Schubert polynomials using divided difference operators. Let ${\mathbb {Q}}[x_+]:= {\mathbb {Q}}[x_1,x_2,x_3,\ldots ]$ be the polynomial ring in infinitely many positively indexed variables and ${\mathbb {Q}}[x]:={\mathbb {Q}}[\dotsc ,x_{-1},x_0,x_1,\dotsc ]$ the polynomial ring in variables indexed by integers. Define the ${\mathbb {Q}}$-algebra automorphism $\gamma : {\mathbb {Q}}[x] \to {\mathbb {Q}}[x]$ given by $x_i \mapsto x_{i+1}$.

For $i\in {\mathbb {Z}}$ the divided difference operator $A_i: {\mathbb {Q}}[x]\to {\mathbb {Q}}[x]$ is defined by

(2.3)\begin{equation} A_i(f) := \dfrac{f - s_i(f)}{x_i - x_{i+1}}. \end{equation}

We have the operator identities

(2.4)\begin{align} A_i^2 &= 0, \end{align}

(2.5)\begin{align}A_iA_j &= A_j A_i \quad \text{for } |i-j|>1, \end{align}

(2.6)\begin{align}A_i A_{i+1} A_i &= A_{i+1} A_i A_{i+1}. \end{align}

For $w\in S_{\mathbb {Z}}$ this allows the definition of

(2.7)\begin{equation} A_w := A_{i_1} A_{i_2}\dotsm A_{i_\ell} \quad \text{where $(i_1,i_2,\dotsc,i_\ell)\in{\rm Red}(w)$.} \end{equation}

For $w\in S_n$, the Schubert polynomial ${\mathfrak {S}}_w\in {\mathbb {Q}}[x_+]$ is defined by

(2.8)\begin{align} {\mathfrak{S}}_{w_0^{(n)}}(x_+) &:= x_1^{n-1}x_2^{n-2}\dotsm x_{n-1}^1, \end{align}

(2.9)\begin{align}{\mathfrak{S}}_w(x_+) &:= A_i {\mathfrak{S}}_{ws_i}(x_+)\quad\text{for any $i$ with $ws_i>w$.} \end{align}

The polynomials ${\mathfrak {S}}_w(x_+)$ are well defined for $w\in S_n$ by (2.5) and (2.6).

We recall the monomial expansion of ${\mathfrak {S}}_w$ due to Billey, Jockusch, and Stanley.

Theorem 2.5 [Reference Billey, Jockusch and StanleyBJS93]

For $w \in S_+$, we have

(2.10)\begin{equation} {\mathfrak{S}}_w (x_+)= \sum_{a_1a_2 \cdots a_\ell \in {\rm Red}(w)} \sum_{\substack{1 \leqslant b_1 \leqslant b_2 \leqslant \cdots \leqslant b_\ell\\ a_i<a_{i+1} \implies b_i < b_{i+1} \\ b_i \leqslant a_i}} x_{b_1} x_{b_2} \cdots x_{b_\ell}. \end{equation}

Define the code $c(w) = (\dotsc ,c_{-1},c_0,c_1,\dotsc )$ of $w\in S_{\mathbb {Z}}$ by

(2.11)\begin{equation} c_i := |\{j>i \mid w(j)<w(i)\}|. \end{equation}

The support of an indexed collection of integers $(c_i\mid i\in J)$ is the set of $i\in J$ such that $c_i\ne 0$. The code gives a bijection from $S_{\mathbb {Z}}$ to finitely supported sequences of nonnegative integers $(\dotsc ,c_{-1},c_0,c_1,\dotsc )$. It restricts to a bijection from $S_+$ to finitely supported sequences of nonnegative integers $(c_1,c_2,\dotsc )$.

For a sequence $b=(b_1,b_2,b_3,\ldots )$ of integers, let $x^b$ denote $x_1^{b_1}x_2^{b_2}\cdots$. For two monomials $x^b$ and $x^c$ in ${\mathbb {Q}}[x]$, we say that $x^c>x^b$ in reverse-lex order if $b\ne c$ and for the maximum $i\in {\mathbb {Z}}$ such that $b_i\ne c_i$ we have $b_i<c_i$. The following triangularity of Schubert polynomials with monomials can be seen from Bergeron and Billey's rc-graph formula for Schubert polynomials [Reference Bergeron and BilleyBB93], and is also proven in [Reference Billey and HaimanBH95].

Proposition 2.6 The transition matrix between Schubert polynomials and monomials is unitriangular:

(2.12)\begin{equation} {\mathfrak{S}}_w (x_+)= x^{c(w)} + \text{reverse-lex lower terms.} \end{equation}

Theorem 2.7 The Schubert polynomials are the unique family of polynomials $\{{\mathfrak {S}}_w (x_+)\in {\mathbb {Q}}[x_+] \mid w \in S_+\}$ satisfying the following conditions:

(2.13)\begin{gather} {\mathfrak{S}}_\mathrm{id} (x_+)= 1, \end{gather}

(2.14)\begin{gather}{\mathfrak{S}}_w(x_+)\ \text{is homogeneous of degree $\ell(w)$}, \end{gather}

(2.15)\begin{gather}A_i {\mathfrak{S}}_w(x_+) = \begin{cases} {\mathfrak{S}}_{w s_i}(x_+) & \text{if $ws_i < w$}, \\ 0 & \text{otherwise.} \end{cases} \end{gather}

The elements $\{{\mathfrak {S}}_w (x_+)\mid w \in S_+\}$ form a basis of ${\mathbb {Q}}[x_+]$ over ${\mathbb {Q}}$.

2.3 Double Schubert polynomials

Let ${\mathbb {Q}}[x_+,a_+] := {\mathbb {Q}}[x_1,x_2,\ldots ,a_1,a_2,\ldots ]$. The divided difference operators $A_i$, $i>0$ act on ${\mathbb {Q}}[x_+,a_+]$ by acting on the $x$-variables only. Double Schubert polynomials [Reference Lascoux and SchützenbergerLS82] are defined by the action of divided difference operators on the expression in (2.19). We summarize the fundamental statements concerning double Schubert polynomials in the following theorem.

Theorem 2.9 There exists a unique family $\{{\mathfrak {S}}_w(x_+;a_+) \in {\mathbb {Q}}[x_+,a_+] \mid w \in S_+\}$ of polynomials satisfying the following conditions:

(2.16)\begin{align} {\mathfrak{S}}_\mathrm{id}(x_+;a_+) &= 1, \end{align}

(2.17)\begin{align}{\mathfrak{S}}_w(a_+;a_+) &= 0\quad\text{if } w \neq \mathrm{id} , \end{align}

(2.18)\begin{align}A_i {\mathfrak{S}}_w(x_+;a_+) &= \begin{cases}{\mathfrak{S}}_{w s_i}(x_+;a_+) & \text{if } ws_i < w , \\ 0 & \text{otherwise.} \end{cases} \end{align}

The elements $\{{\mathfrak {S}}_w(x_+;a_+) \mid w \in S_+\}$ form a basis of ${\mathbb {Q}}[x_+,a_+]$ over ${\mathbb {Q}}[a_+]$.

Proof. Uniqueness is proved as in Theorem 2.7. For existence, let

(2.19)\begin{equation} {\mathfrak{S}}_{w_0^{(n)}}(x_+;a_+) = \prod_{\substack{1\leqslant i,j\leqslant n \\ i+j\leqslant n}} (x_i-a_j). \end{equation}

This agrees with (2.16). It is straightforward to verify the double analogue of Lemma 2.4.

For (2.17) it suffices to prove the stronger vanishing property

(2.20)\begin{equation} {\mathfrak{S}}_v(w a_+;a_+) = 0 \quad\text{unless $v \leqslant w$.} \end{equation}

Here ${\mathfrak {S}}_v(wa_+;a_+):={\mathfrak {S}}_v(a_{w(1)},a_{w(2)},\dotsc ;a_+)$. Let $v,w \in S_n$ with $v\not \leqslant w$. If $v=w_0^{(n)}$, then by inspection ${\mathfrak {S}}_v(wa_+;a_+)=0$. So suppose $v<w_0^{(n)}$. Let $1\leqslant i\leqslant n-1$ be such that $vs_i>v$. Then $vs_i \not \leqslant w$ and also $vs_i \not \leqslant ws_i$. Substituting $x_k\mapsto a_{w(k)}$ into $A_i {\mathfrak {S}}_{vs_i}(x_+;a_+) ={\mathfrak {S}}_v(x_+;a_+)$ and using induction we have ${\mathfrak {S}}_v(wa_+;a_+) = (a_i-a_{i+1})^{-1}({\mathfrak {S}}_{vs_i}(wa_+;a_+) - {\mathfrak {S}}_{vs_i}(ws_ia_+;a_+)) = 0$, proving (2.20).

The basis property follows from the fact that ${\mathfrak {S}}_w(x_+;0)={\mathfrak {S}}_w(x_+)$ are a ${\mathbb {Q}}$-basis of ${\mathbb {Q}}[x_+]$.

2.4 Double Schubert polynomials into single

The following identity is proved in Appendix B.

Lemma 2.10 For $w \in S_+$, we have

(2.21)\begin{equation} \sum_{w \doteq uv} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}} (a_+){\mathfrak{S}}_v(a_+)= \delta_{w,\mathrm{id}}. \end{equation}

Proposition 2.11 ([Reference MacdonaldMac91, (6.1)], [Reference Fomin and StanleyFS94, Lemma 4.5])

Let $w \in S_+$. Then

(2.22)\begin{equation} {\mathfrak{S}}_w(x_+;a_+) = \sum_{w \doteq uv} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}}(a_+){\mathfrak{S}}_v(x_+). \end{equation}

Proof. It suffices to verify the conditions of Theorem 2.9. Equation (2.16) is clear. Equation (2.17) holds by Lemma 2.10. We prove (2.18) by induction on $\ell (w)$. The case $\ell (w) = 0$ is trivial. We have

\begin{align*} A_i \sum_{w \doteq uv} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}}(a_+){\mathfrak{S}}_v(x_+) & = \sum_{\substack{w \doteq uv \\ vs_i < v}} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}}(a_+){\mathfrak{S}}_{vs_i}(x_+) \\ &= \begin{cases} \sum_{\substack{ws_i \doteq uv' }} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}}(a_+){\mathfrak{S}}_{v'}(x_+) & \text{if } ws_i < w, \\ 0 & \text{otherwise.} \end{cases} \end{align*}

This establishes (2.18) by induction.

2.5 Left divided differences

Let $A_i^a$ be the divided difference operator acting on the $a$-variables.

Lemma 2.12 For $i>0$ and $w\in S_+$,

(2.23)\begin{equation} A_i^a {\mathfrak{S}}_w(x_+;a_+) = \begin{cases} - {\mathfrak{S}}_{s_iw}(x_+;a_+) & \text{if } s_iw<w , \\ 0 & \text{otherwise.} \end{cases} \end{equation}

3.1 Symmetric functions in nonpositive variables

For $b\in {\mathbb {Z}}$, let $\Lambda (x_{\leqslant b})$ be the ${\mathbb {Q}}$-algebra of symmetric functions in the variables $x_i$ for $i\in {\mathbb {Z}}$ with $i\leqslant b$. We write $\Lambda = \Lambda (x_{\leqslant 0})= \Lambda ( x_-)$, emphasizing that our symmetric functions are in variables with nonpositive indices. See Appendix A for the comparison with symmetric functions in variables with positive indices.

The tensor product $\Lambda \otimes \Lambda$ is isomorphic to the ${\mathbb {Q}}$-algebra of formal power series of bounded total degree in $x_-$ and $a_-$ which are separately symmetric in $x_-$ and $a_-$. Under this isomorphism, we have $g\otimes h\mapsto g( x_-)h(a_-)$. We use this alternate notation without further mention.

The ${\mathbb {Q}}$-algebra $\Lambda$ is a Hopf algebra over ${\mathbb {Q}}$, generated as a polynomial ${\mathbb {Q}}$-algebra by primitive elements

\[ p_k = \sum_{i\leqslant0} x_i^k. \]

That is, $\Delta (p_k) = 1 \otimes p_k + p_k \otimes 1$ (or $\Delta (p_k)=p_k( x_-)+p_k(a_-)$). Equivalently, for $f\in \Lambda$, $\Delta (f)$ is given by plugging both $x_-$ and $a_-$ variable sets into $f$. The counit takes the coefficient of the constant term, or equivalently, is the ${\mathbb {Q}}$-algebra map sending $p_k\mapsto 0$ for all $k\geqslant 1$. The antipode is the ${\mathbb {Q}}$-algebra automorphism sending $p_k\mapsto -p_k$ for all $k\geqslant 1$. For a symmetric function $f(x)$ we write $f(/x)$ for its image under the antipode.

The superization map

(3.1)\begin{equation} \Lambda\to \Lambda\otimes\Lambda, \quad f \mapsto f(x/a) \end{equation}

is the ${\mathbb {Q}}$-algebra homomorphism defined by applying the coproduct $\Delta$ followed by applying the antipode in the second factor. Equivalently, it is the ${\mathbb {Q}}$-algebra homomorphism sending $p_k\mapsto p_k( x_-)-p_k(a_-)$. In particular, $f(x/a)$ is symmetric in $x_-$ and symmetric in $a_-$. We use the notation $f(x/a)$ instead of $f( x_-/a_-)$ for the sake of simplicity.

3.2 Back symmetric formal power series

Let $R$ be the ${\mathbb {Q}}$-algebra of formal power series $f$ in the variables $x_i$ for $i\in {\mathbb {Z}}$ such that $f$ has bounded total degree (there is an $M$ such that all monomials in $f$ have total degree at most $M$) and the support of $f$ is bounded above (there is an $N$ such that the variables $x_i$ do not appear in $f$ for $i>N$). The group $S_{\mathbb {Z}}$ acts on $R$ by permuting variables. Say that $f\in R$ is back symmetric if there is a $b\in {\mathbb {Z}}$ such that $s_i(f)=f$ for all $i< b$. Let $\overleftarrow {R}$ be the subset of back symmetric elements of $R$.

Proposition 3.1 We have the equality

(3.2)\begin{equation} \overleftarrow{R} = \Lambda \otimes {\mathbb{Q}}[x]. \end{equation}

We emphasize that $\overleftarrow {R}$ is a polynomial ${\mathbb {Q}}$-algebra with algebraically independent generators $p_k$ for $k\geqslant 1$ and $x_i$ for $i\in {\mathbb {Z}}$. The restriction of the action of $S_{\mathbb {Z}}$ from $R$ to $\overleftarrow {R}$ is given on algebra generators by

\begin{align*} w(x_i) &= x_{w(i)}, \\ s_i(p_k) &= \begin{cases} p_k & \text{if } i\ne0 , \\ p_k - x_0^k + x_1^k & \text{if } i=0. \end{cases} \end{align*}

For $s_0(p_k)$ we use the computation

\[ s_0 \sum_{i\leqslant 0} (x_i^k-a_i^k) = \sum_{i\leqslant -1} (x_i^k-a_i^k) + s_0(x_0^k-a_0^k) = \sum_{i\leqslant -1} (x_i^k-a_i^k) + x_1^k-a_0^k = p_k -x_0^k + x_1^k. \]

The divided difference operators $A_i$ for $i \in {\mathbb {Z}}$ act on $\overleftarrow {R}$ using the same formula as (2.3).

3.3 Back stable limit

Let $\gamma : \overleftarrow {R} \to \overleftarrow {R}$ be the ${\mathbb {Q}}$-algebra automorphism shifting all $x$ variables, that is,

(3.3)\begin{gather} \gamma(x_i) = x_{i+1}, \quad \gamma^{{-}1}(x_i) = x_{i-1}, \end{gather}

(3.4)\begin{gather}\gamma(p_k) = p_k + x_1^k, \quad \gamma^{{-}1}(p_k) = p_k - x_0^k. \end{gather}

Given $w\in S_{{\mathbb {Z}}}$, let $[p,q]\subset {\mathbb {Z}}$ be an interval that contains all nonfixed points of $w$. Let ${\mathfrak {S}}_{w}^{[p,q]}$ be the usual Schubert polynomial but computed using the variables $x_p,x_{p+1},\dotsc ,x_q$ instead of starting with $x_1$. This is the same as shifting $w$ to start at $1$ instead of $p$, constructing the Schubert polynomial, and then shifting variables to start at $x_p$ instead of $x_1$. That is,

\[ {\mathfrak{S}}_w^{[p,q]}(x_p,\dotsc,x_q) = \gamma^{p-1}({\mathfrak{S}}_{\gamma^{1-p}(w)}(x_+)). \]

We say that the limit of a sequence $f_1,f_2,\ldots$ of formal power series is equal to a formal power series $f$ if, for each monomial $M$, the coefficient of $M$ in $f_1,f_2,\ldots$ eventually stabilizes and equals the coefficient in $f$.

Theorem 3.2 For $w\in S_{\mathbb {Z}}$, there is a well-defined power formal series $\overleftarrow {\mathfrak {S}}_w\in \overleftarrow {R}$ given by

\[ \overleftarrow{\mathfrak{S}}_w := \lim_{\substack{p \to -\infty \\ q\to \infty}} {\mathfrak{S}}_{w}^{[p,q]} \]

called the back stable Schubert polynomial. It has the monomial expansion

(3.5)\begin{equation} \overleftarrow{\mathfrak{S}}_w = \sum_{a_1a_2 \cdots a_\ell \in {\rm Red}(w)} \sum_{\substack{b_1 \leqslant b_2 \leqslant \cdots \leqslant b_\ell\\ a_i<a_{i+1} \implies b_i < b_{i+1} \\ b_i \leqslant a_i}} x_{b_1} x_{b_2} \cdots x_{b_\ell} \end{equation}

in which $b_i\in {\mathbb {Z}}$. Moreover, the back stable Schubert polynomials are the unique family $\{\overleftarrow {\mathfrak {S}}_w \in \overleftarrow {R} \mid w \in S_{\mathbb {Z}}\}$ of elements satisfying the following conditions:

(3.6)\begin{gather} \overleftarrow{\mathfrak{S}}_\mathrm{id} = 1, \end{gather}

(3.7)\begin{gather}\overleftarrow{\mathfrak{S}}_w \text{ is homogeneous of degree } \ell(w) , \end{gather}

(3.8)\begin{gather}A_i \overleftarrow{\mathfrak{S}}_w = \begin{cases} \overleftarrow{\mathfrak{S}}_{w s_i} & \text{if } ws_i < w, \\ 0 & \text{otherwise.} \end{cases} \end{gather}

By Propositions 3.3 and 2.6 we have

(3.9)\begin{equation} \overleftarrow{\mathfrak{S}}_w = x^{c(w)} + \text{reverse-lex lower terms}. \end{equation}

3.4 Stanley symmetric functions

Stanley [Reference StanleySta84] defined Stanley symmetric functions $F_w(x_+)$ to enumerate reduced decompositions of permutations. These symmetric functions are also called stable Schubert polynomials, and are usually defined by $F_w(x_+):= \lim _{n\to \infty } {\mathfrak {S}}_{\gamma ^n(w)}(x_+)$. Our definition $F_w$ of Stanley symmetric function agrees (by Theorem 3.9) with the standard definition up to using $x_-$ instead of $x_+$.

There is a ${\mathbb {Q}}$-algebra map $\eta _0: {\mathbb {Q}}[x] \to {\mathbb {Q}}$ given by evaluation at zero: $x_i \mapsto 0$ for all $i\in {\mathbb {Z}}$. This induces a ${\mathbb {Q}}$-algebra map $1 \otimes \eta _0: \overleftarrow {R} \to \Lambda \otimes _{\mathbb {Q}} {\mathbb {Q}} \cong \Lambda$, which we simply denote by $\eta _0$ as well.

For $w \in S_{\mathbb {Z}}$, we define the Stanley symmetric function by

(3.10)\begin{equation} F_w := \eta_0(\overleftarrow{\mathfrak{S}}_w)\in\Lambda. \end{equation}

Recall the shifting automorphism $\gamma :S_{\mathbb {Z}}\to S_{\mathbb {Z}}$ from § 2.1.

Theorem 3.9 (cf. [Reference StanleySta84])

For $w\in S_{\mathbb {Z}}$, we have

(3.11)\begin{equation} F_w = \sum_{a_1a_2 \cdots a_\ell \in {\rm Red}(w)} \sum_{\substack{b_1 \leqslant b_2 \leqslant \cdots \leqslant b_\ell \leqslant 0 \\ a_i<a_{i+1} \implies b_i < b_{i+1} }} x_{b_1} x_{b_2} \cdots x_{b_\ell} . \end{equation}

The Edelman–Greene coefficients $j_\lambda ^w \in {\mathbb {Z}}$ are defined by

(3.12)\begin{equation} F_w = \sum_\lambda j_\lambda^w s_\lambda. \end{equation}

These coefficients are known to be nonnegative and have a number of combinatorial interpretations: leaves of the transition tree [Reference Lascoux and SchützenbergerLS85], promotion tableaux [Reference HaimanHai92], and peelable tableaux [Reference Reiner and ShimozonoRS98]. In particular, by [Reference Edelman and GreeneEG87] $j^w_\lambda$ is equal to the number of reduced word tableaux for $w$: that is, row strict and column strict tableaux of shape $\lambda$ whose row-reading words are reduced words for $w$.

Let $\omega$ be the involutive ${\mathbb {Q}}$-algebra automorphism of $\Lambda$ defined by $\omega (p_r) = (-1)^{r-1} p_r$ for $r\geqslant 1$. We have $\omega (s_\lambda )=s_{\lambda '}$ for $\lambda \in {\mathbb {Y}}$. The action of $\omega$ on a homogeneous element of degree $d$ is equal to that of the antipode times $(-1)^d$. Let $\omega$ also denote the automorphism of $S_{{\mathbb {Z}}}$ given by $s_i\mapsto s_{-i}$ for all $i\in {\mathbb {Z}}$.

Proposition 3.11 For $w\in S_{\mathbb {Z}}$, we have

(3.13)\begin{align} \Delta(F_w) &= \sum_{w\doteq uv} F_u \otimes F_v, \end{align}

(3.14)\begin{align}F_w({/}x) &= ({-}1)^{\ell(w)} F_{w^{{-}1}}(x), \end{align}

(3.15)\begin{align}F_w(x/a) &= \sum_{w\doteq uv} ({-}1)^{\ell(u)} F_{u^{{-}1}}(a) F_v(x), \end{align}

(3.16)\begin{align}F_w(a/a) &= \delta_{w,\mathrm{id}}, \end{align}

(3.17)\begin{align}F_w(x) &= \sum_{w\doteq uvz} ({-}1)^{\ell(u)} F_{u^{{-}1}}(a) F_v(x) F_z(a). \end{align}

Proof. Equation (3.13) follows by plugging in two set of variables into (3.11). Equation (3.14) follows from Proposition 3.10. Equation (3.15) is obtained by combining (3.13) and (3.14). Equation (3.16) follows from the Hopf algebra axiom which asserts that superization followed by multiplication is the counit. For (3.17) we have

\begin{align*} \sum_{w \doteq uvz} ({-}1)^{\ell(u)} F_{u^{{-}1}}(a) F_v(x) F_z(a) &= \sum_{w \doteq uvz} ({-}1)^{\ell(u)} F_{u^{{-}1}}(a) F_v(a) F_z(x) \\ &= \sum_{w \doteq yz} F_y(a/a) F_z(x) = F_w(x) \end{align*}

using cocommutativity, (3.15), and (3.16).

3.5 Negative Schubert polynomials

The following Schubert polynomials are indexed by permutations in $S_-$, contain variables indexed by nonpositive integers, and may contain signs. Recall $S_-$ and $S_{\ne 0}$ from § 2.1 and the automorphism $\omega$ of $S_{{\mathbb {Z}}}$. It restricts to an isomorphism $S_-\to S_+$. Let $\omega :{\mathbb {Q}}[x]\to {\mathbb {Q}}[x]$ be the ${\mathbb {Q}}$-algebra automorphism defined by $\omega (x_i)=-x_{1-i}$ for $i\in {\mathbb {Z}}$. For $u\in S_-$, define ${\mathfrak {S}}_u( x_-)\in {\mathbb {Q}}[ x_-]$ by

(3.18)\begin{equation} {\mathfrak{S}}_u( x_-) := \omega({\mathfrak{S}}_{\omega(u)}(x_+)). \end{equation}

That is, in $w$ replace the negatively indexed reflections with positively indexed ones, take the usual Schubert polynomial in positively indexed variables, and then use $\omega$ to substitute nonpositively indexed $x$ variables for the positively indexed ones (with signs).

By Theorem 2.5, we have

\[ {\mathfrak{S}}_u( x_-) = \sum_{a_1a_2 \cdots a_\ell \in R(u)} \sum_{\substack{ 0 \geqslant b_1 \geqslant b_2 \geqslant \cdots \geqslant b_\ell \\ a_i > a_{i+1} \implies b_i > b_{i+1} \\ b_i \geqslant a_i+1}} x_{b_1} x_{b_2} \cdots x_{b_\ell}\quad\text{for } u\in S_-. \]

Note the $+1$ in $b_i \geqslant a_i+1$.

For $w\in S_{\ne 0}$, define ${\mathfrak {S}}_w(x)\in {\mathbb {Q}}[x]$ by

(3.19)\begin{equation} {\mathfrak{S}}_w(x) := {\mathfrak{S}}_u( x_-){\mathfrak{S}}_v(x_+)\quad\text{where } w=uv \hbox{ with } u\in S_- \hbox{ and } v\in S_+. \end{equation}

3.6 Coproduct formula

There is a coaction $\Delta : \overleftarrow {R} \to \Lambda \otimes \overleftarrow {R}$ of $\Lambda$ on $\overleftarrow {R}$, defined by the comultiplication on the first factor of the tensor product $\Lambda \otimes _{\mathbb {Q}} {\mathbb {Q}}[x]$.

Theorem 3.14 Let $w \in S_{\mathbb {Z}}$. We have the coproduct formulae

(3.20)\begin{align} \Delta(\overleftarrow{\mathfrak{S}}_w) &= \sum_{w \doteq xy} F_x \otimes \overleftarrow{\mathfrak{S}}_y , \end{align}

(3.21)\begin{align}\overleftarrow{\mathfrak{S}}_w &= \sum_{\substack{w \doteq xy \\ y \in S_{{\neq} 0}}} F_x \; {\mathfrak{S}}_y . \end{align}

Proof. Equation (3.20) can be deduced from (3.21) and Proposition 3.11:

\[ \Delta(\overleftarrow{\mathfrak{S}}_w) = \sum_{\substack{w\doteq xy \\ y\in S_{\ne0}}} \Delta(F_x) {\mathfrak{S}}_y = \sum_{\substack{w\doteq uvy \\ y\in S_{\ne0}}} F_u \otimes F_v {\mathfrak{S}}_y = \sum_{w\doteq uz} F_u \otimes \overleftarrow{\mathfrak{S}}_z. \]

We prove (3.21) by a cancellation argument. We say that a pair of integer sequences $({\mathbf {a}},{\mathbf {b}})$ of the same length is a compatible pair, if ${\mathbf {b}}$ is weakly increasing and $a_i < a_{i+1} \implies b_i < b_{i+1}$.

Let $(x,y,{\mathbf {a}},{\mathbf {b}})$ index a monomial $x_{\mathbf {b}} = x_{b_1} \cdots x_{b_\ell }$ on the right-hand side, corresponding to the term $F_x \; {\mathfrak {S}}_y$ and reduced word ${\mathbf {a}} = a_1 a_2 \cdots a_\ell$. By convention, to obtain ${\mathbf {a}}$, we always factorize $y \in S_{\ne 0}$ as $y = y'y''$ with $y' \in S_-$ and $y'' \in S_+$. We will provide a partial sign-reversing involution $\iota$ on the quadruples $(x,y,{\mathbf {a}},{\mathbf {b}})$; the left-over monomials will give the left-hand side.

Suppose $\ell (x) = r$, $\ell (y') = s$, $\ell (y'') = t$, and set $\ell = r+s+t$. Call an index $i \in [1,\ell ]$ bad if $b_i > a_i$, and good if $b_i \leqslant a_i$. It follows from the definitions that all indices $i \in [r+s+1,\ell ]$ are good, while all indices $i \in [r+1,r+s]$ are bad. Furthermore, if $i \in [1,r]$ is bad, then $a_i < 0$.

Let $k$ be the largest bad index in $[1,r]$, which we assume exists. We claim that $s_{a_k}$ commutes with $s_{a_{k+1}} \cdots s_{a_r}$. To see this, observe that if $a_{k'} \in \{a_k-1,a_k,a_k+1\}$ where $k < k' \leqslant r$ then we must have $b_{k'} > a_{k'}$, contradicting our choice of $k$. If $s = 0$, we set

(3.22)\begin{equation} \iota(x,y,{\mathbf{a}},{\mathbf{b}}) = (a_1 \cdots \hat a_k \cdots a_{r} |a_k a_{r+1} \cdots a_t, b_1 \cdots \hat b_k \cdots b_r| b_k b_{r+1} \cdots b_t) = (\tilde x,\tilde y,\tilde{{\mathbf{a}}}, \tilde{{\mathbf{b}}}) \end{equation}

where the vertical bar separates $x$ from $y$. Thus $\tilde {y}' = s_{a_k}$. If $s > 0$, we compare $b_k$ with $b_{r+1}$. If $b_k > b_{r+1}$ or ($b_k = b_{r+1}$ and $a_k < a_{r+1}$), then we again make the definition (3.22) where now $\tilde {y}' = s_{a_k} y'$. We call this CASE A.

Suppose still that $s > 0$. If ($b_k < b_{r+1}$) or ($b_k = b_{r+1}$ and $a_k \geqslant a_{r+1}$) or ($k$ does not exist) then there is a unique index $j \in [k,r]$ so that

\[ \iota(x,y,{\mathbf{a}},{\mathbf{b}}) = (a_1 \cdots a_j a_{r+1} a_{j+1} \cdots a_{r} |a_{r+2} \cdots a_t, b_1 \cdots b_j b_{r+1} b_{j+1} \cdots b_r | b_{r+2} \cdots b_t) = (\tilde x,\tilde y,\tilde {\mathbf{a}}, \tilde {\mathbf{b}}) \]

has the property that $(a_1 \cdots a_j a_{r+1} a_{j+1} \cdots a_{r} , b_1 \cdots b_j b_{r+1} b_{j+1} \cdots b_r)$ is a compatible sequence. In this case, $s_{a_{r+1}}$ commutes with $s_{a_{j+1}} \cdots s_{a_{r}}$. We call this CASE B.

Finally, if $s = 0$ and $k$ does not exist, then $\iota$ is not defined.

It remains to observe that CASE A and CASE B are sent to each other via $\iota$, which keeps $x_{\mathbf {b}}$ constant and changes $\ell (y')$ by 1.

Let $\omega$ be the involutive ${\mathbb {Q}}$-algebra automorphism of $\overleftarrow {R}$ given by combining the maps $\omega$ on $\Lambda$ from § 3.4 and on ${\mathbb {Q}}[x]$ from § 3.5.

Remark 3.17 Let $\nu _\lambda : \overleftarrow {R} \to {\mathbb {Q}}[x]$ denote the linear map given by ‘taking the coefficient of $s_\lambda$’. Then

(3.23)\begin{equation} \nu_\lambda(\overleftarrow{\mathfrak{S}}_w) = \sum_{\substack{v \in S_{{\neq} 0} \\ \ell(wv^{{-}1}) = \ell(w) - \ell(v)}} j_\lambda^{wv^{{-}1}} {\mathfrak{S}}_v. \end{equation}

We will give an explicit description of the polynomial $\nu _\lambda (\overleftarrow {\mathfrak {S}}_w)$ in Theorem 5.11.

3.7 Back stable Schubert structure constants

For $u,v,w\in S_+$, define the usual Schubert structure constants $c^w_{uv}$ by

(3.24)\begin{equation} \mathfrak{S}_u \mathfrak{S}_v = \sum_{w\in S_+} c^w_{uv} \mathfrak{S}_w. \end{equation}

For $u,v,w\in S_{\mathbb {Z}}$, define the back stable Schubert structure constants $\overleftarrow {c}^w_{uv}\in {\mathbb {Q}}$ by

(3.25)\begin{equation} \overleftarrow{\mathfrak{S}}_u \overleftarrow{\mathfrak{S}}_v = \sum_{w\in S_{\mathbb{Z}}} \overleftarrow{c}^w_{uv} \overleftarrow{\mathfrak{S}}_w. \end{equation}

By Proposition 3.3, we have

(3.26)\begin{equation} \overleftarrow{c}^{\gamma^n(w)}_{\gamma^n(u),\gamma^n(v)} = \overleftarrow{c}^w_{uv}\quad\text{for all } u,v,w\in S_{\mathbb{Z}} \hbox{ and } n\in{\mathbb{Z}}. \end{equation}

Proof. Consider the ${\mathbb {Q}}$-algebra homomorphism $\pi _+:\overleftarrow {R}\to {\mathbb {Q}}[x_+]$ sending $p_r\mapsto 0$ for $r\geqslant 1$, $x_i\mapsto 0$ for $i\leqslant 0$ and $x_i\mapsto x_i$ for $i>0$. Applying $\pi _+$ to Theorem 3.14 for $y\in S_{\mathbb {Z}}$ we have

(3.27)\begin{equation} \pi_+(\overleftarrow{\mathfrak{S}}_y) = \begin{cases} {\mathfrak{S}}_y & \text{if } y\in S_+ , \\ 0 & \text{otherwise,} \end{cases} \end{equation}

because $\pi _+$ kills all symmetric functions with no constant term and all negative Schubert polynomials of positive degree. Now let $u,v\in S_+$. Applying $\pi _+$ to (3.25) and using (3.27), (i) follows.

For (ii), let $u,v\in S_{\mathbb {Z}}$. By (3.26), we may assume that $u,v\in S_+$ and that the finitely many $w$ appearing in (3.25) are also in $S_+$. The proof is completed by applying part (i).

We derive a relation involving back stable Schubert structure constants and Edelman–Greene coefficients.

Proposition 3.20 Let $u\in S_m$, $v\in S_n$ and $\lambda \in {\mathbb {Y}}$. Let $u\times v:=u \gamma ^m(v)\in S_{m+n}\subset S_+$. Then

(3.28)\begin{equation} j^{u \times v}_\lambda = \sum_{w\in S_{\mathbb{Z}}} \overleftarrow{c}^w_{uv} j^w_\lambda. \end{equation}

Proof. Since $u\times v\in S_m \times S_n \subset S_{m+n}$ it follows that ${\mathfrak {S}}_{u\times v} = {\mathfrak {S}}_u {\mathfrak {S}}_{\gamma ^m(v)}$. We deduce that $\overleftarrow {\mathfrak {S}}_{u \times v} = \overleftarrow {\mathfrak {S}}_u \gamma ^m(\overleftarrow {\mathfrak {S}}_v)$. Using the algebra map $\eta _0$ several times we obtain

\[ F_{u\times v} = F_u F_v = \eta_0(\overleftarrow{\mathfrak{S}}_u \overleftarrow{\mathfrak{S}}_v) = \eta_0\bigg(\sum_{w\in S_{\mathbb{Z}}} \overleftarrow{c}^w_{uv} \overleftarrow{\mathfrak{S}}_w\bigg) = \sum_{w\in S_{\mathbb{Z}}} \overleftarrow{c}^w_{uv} F_w. \]

Taking the coefficient of $s_\lambda$ we obtain (3.28).

4.1 Double symmetric functions

Let $p_k(x||a) := p_k(x/a) =\sum _{i \leqslant 0} x_i^k - a_i^k$, a formal power series in variables $x_i$ and $a_i$; it is the image of $p_k$ under superization. Let $\Lambda (x||a)$ be the ${\mathbb {Q}}[a]$-algebra generated by the elements $p_1(x||a), p_2(x||a),\ldots$, which are algebraically independent over ${\mathbb {Q}}[a]$. We call $\Lambda (x||a)$ the ring of double symmetric functions (see [Reference MolevMol09] for more details). For $\lambda = (\lambda _1,\lambda _2,\ldots ,\lambda _{\ell }) \in {\mathbb {Y}}$, we denote $p_\lambda (x||a):= p_{\lambda _1}(x||a) \cdots p_{\lambda _\ell }(x||a)$.

The algebra $\Lambda (x||a)$ is a Hopf algebra over ${\mathbb {Q}}[a]$ with primitive generators $p_k(x||a)$ for $k\geqslant 1$. The counit is the ${\mathbb {Q}}[a]$-algebra homomorphism $\epsilon :\Lambda (x||a)\to {\mathbb {Q}}[a]$ given by $p_k(x||a)\mapsto 0$ for $k\geqslant 1$. The antipode is the ${\mathbb {Q}}[a]$-algebra homomorphism defined by $p_k(x||a)\mapsto -p_k(x||a)$ for $k\geqslant 1$.

4.2 Back symmetric double power series

Define the back symmetric double power series ring $\overleftarrow {R}(x;a):= \Lambda (x||a) \otimes _{{\mathbb {Q}}[a]} {\mathbb {Q}}[x,a]$, where ${\mathbb {Q}}[x,a]:={\mathbb {Q}}[x_i,a_i\mid i\in {\mathbb {Z}}]$. The ring $\overleftarrow {R}(x;a)$ has two actions of $S_{\mathbb {Z}}$: one that acts on all the $x$ variables and one that acts on all the $a$ variables, including those in $\Lambda (x||a)$. More precisely for $i\in {\mathbb {Z}}$ let $s_i^x$ (respectively $s_i^a)$ act on $\overleftarrow {R}(x;a)$ by exchanging $x_i$ and $x_{i+1}$ (respectively $a_i$ and $a_{i+1}$) while leaving the other polynomial generators of ${\mathbb {Q}}[x,a]$ alone and

(4.1)\begin{gather} s_i^x(p_k(x||a)) = \begin{cases} p_k(x||a) & \text{if $i\ne0$}, \\ p_k(x||a) - x_0^k + x_1^k & \text{if $i=0$}, \end{cases}, \end{gather}

(4.2)\begin{gather} s_i^a(p_k(x||a)) = \begin{cases} p_k(x||a) & \text{if $i\ne0$}, \\ p_k(x||a) + a_0^k - a_1^k & \text{if $i=0$}.\end{cases}. \end{gather}

For $w\in S_{\mathbb {Z}}$, we write $w^x$ (respectively $w^a$) for this action of $w$ on the $x$-variables (respectively $a$-variables).

4.3 Localization of back symmetric formal power series

Let $\epsilon : \overleftarrow {R}(x;a)\to {\mathbb {Q}}[a]$ be the ${\mathbb {Q}}[a]$-algebra homomorphism which extends the counit $\epsilon$ of $\Lambda (x||a)$ via

(4.3)\begin{align} \epsilon(p_k(x||a)) &= 0\quad\text{for all $k\geqslant1$}, \end{align}

(4.4)\begin{align} \epsilon(x_i) &= a_i\quad\text{for all $i\in\mathbb{Z}$.} \end{align}

In other words $\epsilon$ ‘sets all $x_i$ to $a_i$’ including those in $p_k(x||a)$. Define

(4.5)\begin{equation} f|_w = \epsilon(w^x(f)))=f(wa;a)\quad\text{for } f(x,a)\in \overleftarrow{R}(x;a) \hbox{ and } w\in S_{\mathbb{Z}}. \end{equation}

For any $w\in S_{\mathbb {Z}}$, let

(4.6)\begin{gather} I_{w,+} := {\mathbb{Z}}_{{>}0} \cap w({\mathbb{Z}}_{\leqslant0}), \end{gather}

(4.7)\begin{gather}I_{w,-} := {\mathbb{Z}}_{\leqslant0} \cap w({\mathbb{Z}}_{{>}0}). \end{gather}

The map $w\mapsto (I_{w,+},I_{w,-})$ is a bijection from $S_{\mathbb {Z}}^0$ to pairs of finite sets $(I_+,I_-)$ such that $I_+\subset {\mathbb {Z}}_{>0}$, $I_-\subset {\mathbb {Z}}_{\leqslant 0}$, and $|I_+|=|I_-|$. Then the following holds.

4.4 Back stable double Schubert polynomials

Let $\gamma$ be the ${\mathbb {Q}}$-algebra automorphism of $\overleftarrow {R}(x;a)$ which shifts all variables forward by $1$ in $\overleftarrow {R}(x;a)$. That is, $\gamma (x_i)=x_{i+1}$, $\gamma (a_i)=a_{i+1}$, and $\gamma (p_k(x||a)) = p_k(x||a) + x_1^k - a_1^k$. As before, let $[p,q]$ be an interval of integers containing all integers moved by $w\in S_{\mathbb {Z}}$. Define

(4.8)\begin{equation} {\mathfrak{S}}_w^{[p,q]}(x;a) := \gamma^{p-1}({\mathfrak{S}}_{\gamma^{1-p}(w)}(x_+;a_+)). \end{equation}

For $w\in S_{\mathbb {Z}}$, define the back stable double Schubert polynomial $\overleftarrow {\mathfrak {S}}_w(x;a)$ by

(4.9)\begin{equation} \overleftarrow{\mathfrak{S}}_w(x;a) := \lim_{\substack{p\to-\infty\\ q\to\infty}} {\mathfrak{S}}_w^{[p,q]}(x;a). \end{equation}

There is a double version of the monomial expansion (Theorem 2.5) of Schubert polynomials; see for example [Reference Fomin and KirillovFK96]. However, the well-definedness of $\overleftarrow {\mathfrak {S}}_w(x;a)$ is not apparent from that expansion. In Theorem 5.13 we give a new combinatorial formula for ${\mathfrak {S}}_w(x_+;a_+)$ using bumpless pipedreams as a sum of products of binomials $x_i-a_j$. Theorem 5.13 is compatible with the back stable limit and yields a monomial formula (Theorem 5.2) for the back stable double Schubert polynomials.

Proposition 4.3 For $w \in S_{\mathbb {Z}}$, ${\mathfrak {S}}_w(x;a)$ is a well-defined series such that

(4.10)\begin{equation} \overleftarrow{\mathfrak{S}}_w(x;a) = \sum_{w\doteq uv} ({-}1)^{\ell(u)} \overleftarrow{\mathfrak{S}}_{u^{{-}1}}(a) \overleftarrow{\mathfrak{S}}_v(x). \end{equation}

Proof. Since length-additive factorizations are well behaved under shifting it follows that

\begin{align*} {\mathfrak{S}}_w^{[p,q]}(x;a) &= \gamma^{p-1}({\mathfrak{S}}_{\gamma^{1-p}(w)}(x_+;a_+)) \\ &= \gamma^{p-1}\bigg( \sum_{w \doteq uv} ({-}1)^{\ell(u)} {\mathfrak{S}}_{\gamma^{1-p}(u^{{-}1})}(a_+) {\mathfrak{S}}_{\gamma^{1-p}(v)}(x_+) \bigg)\\ &= \sum_{w\doteq uv} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}}^{[p,q]}(a) {\mathfrak{S}}_v^{[p,q]}(x) \end{align*}

using Proposition 2.11. Taking the limit as $p\to -\infty$ and $q\to \infty$ we obtain (4.10).

Corollary 4.5 For $w \in S_{\mathbb {Z}}$, we have

(4.11)\begin{equation} \overleftarrow{\mathfrak{S}}_w(x;a) = \sum_{\substack{w\doteq uvz \\ u,z\in S_{\ne0}}} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}}(a) F_v(x/a) {\mathfrak{S}}_z(x). \end{equation}

In particular, $\overleftarrow {\mathfrak {S}}_w(x;a)\in \overleftarrow {R}(x;a)$.

Proof. Using (3.21) and Propositions 4.3 and 3.11 we have

\begin{align*} \overleftarrow{\mathfrak{S}}_w(x;a) &= \sum_{w\doteq uv} ({-}1)^{\ell(u)} \overleftarrow{\mathfrak{S}}_{u^{{-}1}}(a) \overleftarrow{\mathfrak{S}}_v(x) \\ &= \sum_{w\doteq uv} \sum_{\substack{u^{{-}1}\doteq u_1v_1 \\ v_1\in S_{\ne0}}} \sum_{\substack{v\doteq u_2v_2 \\ v_2 \in S_{\ne0}}} ({-}1)^{\ell(u)} F_{u_1}(a) {\mathfrak{S}}_{v_1}(a) F_{u_2}(x) {\mathfrak{S}}_{v_2}(x) \\ &=\sum_{\substack{w\doteq v_1^{{-}1} u_1^{{-}1} u_2 v_2 \\ v_1,v_2\in S_{\ne0}}} ({-}1)^{\ell(u_1)+\ell(v_1)} {\mathfrak{S}}_{v_1}(a) F_{u_1}(a) F_{u_2}(x) {\mathfrak{S}}_{v_2}(x) \\ &= \sum_{\substack{w\doteq v_1^{{-}1} u v_2 \\ v_1,v_2\in S_{\ne0}}} ({-}1)^{\ell(v_1)}{\mathfrak{S}}_{v_1}(a) F_u(x/a) {\mathfrak{S}}_{v_2}(x). \end{align*}

Example 4.6 We have

\begin{align*} \overleftarrow{\mathfrak{S}}_{s_i}(x;a) &={-} {\mathfrak{S}}_{s_i}(a) + F_{s_i}(x/a) ={-}{\mathfrak{S}}_{s_i}(a) + s_1(x/a) = s_1[x_{\leqslant0} - a_{\leqslant i}], \\ \overleftarrow{\mathfrak{S}}_{s_1s_0}(x;a) &={-} {\mathfrak{S}}_{s_1}(a) F_{s_0}(x/a) + F_{s_1s_0}(x/a) ={-}a_1 s_1(x/a) + s_2(x/a), \\ \overleftarrow{\mathfrak{S}}_{s_{{-}1}s_0} &={-}{\mathfrak{S}}_{s_{{-}1}}(a) F_{s_0}(x/a) + F_{s_{{-}1}s_0}(x/a) = a_0 s_1(x/a) + s_{11}(x/a), \\ \overleftarrow{\mathfrak{S}}_{s_0s_{{-}1}} &= F_{s_0s_{{-}1}}(x/a) + F_{s_0} {\mathfrak{S}}_{s_{{-}1}}(x) = s_2(x/a) + s_1(x/a) ({-}x_0). \end{align*}

Theorem 4.7 The back stable double Schubert polynomials $\{\overleftarrow {\mathfrak {S}}_w(x;a) \in \overleftarrow {R}(x;a) \mid w \in S_{\mathbb {Z}}\}$ form the unique family of power series satisfying the following conditions:

(4.12)\begin{align} \overleftarrow{\mathfrak{S}}_\mathrm{id} &= 1, \end{align}

(4.13)\begin{align}\overleftarrow{\mathfrak{S}}_w(a;a) &= 0 \quad {\rm if}\ w\neq {\rm id},\end{align}

(4.14)\begin{align}A_i \overleftarrow{\mathfrak{S}}_w(x;a) &= \begin{cases} \overleftarrow{\mathfrak{S}}_{w s_i}(x;a) & \text{if $ws_i < w$}, \\ 0 & \text{otherwise.} \end{cases} \end{align}

The elements $\{\overleftarrow {\mathfrak {S}}_w(x;a) \mid w \in S_{\mathbb {Z}}\}$ form a basis of $\overleftarrow {R}(x;a)$ over ${\mathbb {Q}}[a]$.

Proof. Uniqueness follows as in the proof of Theorem 3.2. Since the double Schubert polynomials are related by divided differences, the corresponding fact (4.14) also holds. For (4.13), applying the map $\epsilon$ of § 4.3 to (4.11) and using (3.16), we have

\[ \overleftarrow{\mathfrak{S}}_w(a;a) = \sum_{\substack{w\doteq uz \\ u,z\in S_{\ne0}}} ({-}1)^{\ell(u)} {\mathfrak{S}}_u(a) {\mathfrak{S}}_z(a). \]

This is $0$ automatically if $w\not \in S_{\ne 0}$. If $w\in S_{\ne 0}\setminus \{\mathrm {id}\}$ then the vanishing follows from the straightforward generalization of Lemma 2.10 to $\mathfrak {S}_w$ for $w\in S_{\ne 0}$.

The basis property follows from the fact that setting $a_i=0$ for all $i\in {\mathbb {Z}}$ gives $\overleftarrow {\mathfrak {S}}_w(x,0)=\overleftarrow {\mathfrak {S}}_w(x)$ and the latter are a basis of $\Lambda \otimes {\mathbb {Q}}[x]$.

The back stable double Schubert polynomials localize the same way that ordinary double Schubert polynomials do in the following sense.

Proof. By Corollary 4.4 we may assume that $[p,q]=[1,n]$ for some $n$ so that $v,w\in S_n$. We are specializing $x_i\mapsto a_{w(i)}$ for all $i$, and in particular $x_i\mapsto a_i$ for all $i\leqslant 0$. Under this substitution all the super Stanley functions in (4.11) vanish except those indexed by the identity. Using Proposition 2.11, we have

\[ \overleftarrow{\mathfrak{S}}_v(wa;a) = \sum_{\substack{v \doteq uz \\ u,z\in S_{\ne0}}} ({-}1)^{\ell(u)} {\mathfrak{S}}_{u^{{-}1}}(a) \mathfrak{S}_z(wa) = \sum_{\substack{v \doteq uz \\ u,z\in S_n}} ({-}1)^{\ell(u)} \mathfrak{S}_{u^{{-}1}}(a_+) {\mathfrak{S}}_z(wa_+) = {\mathfrak{S}}_v(wa_+;a_+). \]

Let $s_i^a$ and $A_i^a$ be the reflection and divided difference operators acting on the $a$-variables in both ${\mathbb {Q}}[a]$ and in $p_r(x||a)$.

Proposition 4.9 For all $i\in {\mathbb {Z}}$ and $w \in S_{\mathbb {Z}}$, we have

\[ A_i^a \overleftarrow{\mathfrak{S}}_w(x;a) = \begin{cases} -\overleftarrow{\mathfrak{S}}_{s_iw}(x;a) & \text{if $s_iw<w$}, \\ 0 & \text{otherwise.} \end{cases} \]

4.5 Double Schur functions

We realize the double Schur functions (see [Reference MolevMol09]) as the Grassmannian back stable double Schubert polynomials. As such our double Schur functions are symmetric in $x_-$. In Appendix A a precise dictionary is given which connects our conventions with the literature, which uses symmetric functions in $x_+$.

Let $\gamma _a$ be the shift of all of the $a$-variables, that is, the ${\mathbb {Q}}$-algebra automorphism of $\Lambda (x||a)$ given by

(4.15)\begin{gather} \hskip-.8pc\gamma_a(a_i) = a_{i+1}, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \gamma_a^{{-}1}(a_i) = a_{i- 1}, \end{gather}

(4.16)\begin{gather}\hskip6pt\gamma_a(p_k(x||a)) = p_k(x||a) - a_1^k, \quad\quad\quad\quad \hskip-4pt\gamma_a^{{-}1}(p_k(x||a)) = p_k(x||a) + a_0^k. \end{gather}

By definition $\gamma _a$ leaves the $x$ variables alone. For $\lambda \in {\mathbb {Y}}$ define the double Schur function $s_\lambda (x||a)\in \Lambda (x||a)$ by

(4.17)\begin{equation} h_r(x||a) := \gamma_a^{r-1}(h_r(x/a)) \quad s_\lambda(x||a) := \det \gamma_a^{1-j}(h_{\lambda_i-i+j}(x||a)). \end{equation}

Example 4.10 The double Schur functions for $\lambda=(r)$ and $\lambda=(1,1)$ are given by

\begin{align*} h_r(x||a) &= h_r(x_{\leqslant0}/a_{\leqslant r-1})\quad \text{for } r\geqslant1 , \\ s_{11}(x||a) &= \det \begin{pmatrix} h_1(x||a) & \gamma_a^{{-}1}(h_2(x||a)) \\ h_0(x||a) & \gamma_a^{{-}1}(h_1(x||a)) \end{pmatrix}\\ &= h_1(x_{\leqslant0}/a_{\leqslant0}) h_1(x_{\leqslant0}/a_{\leqslant-1}) - h_2(x_{\leqslant0}/a_{\leqslant 0}) \\ &= h_1(x/a) (h_1(x/a)+a_0) - h_2(x/a) \\ &= s_{11}(x/a) + a_0 s_1(x/a) = \overleftarrow{\mathfrak{S}}_{s_{{-}1}s_0}(x;a) \end{align*}

by Example 4.6 for $w_{(1,1)}=s_{-1}s_0$.

4.6 Double Stanley symmetric functions

We introduce the double Stanley symmetric functions $F_w(x||a)$ for $w\in S_{\mathbb {Z}}$. If $w$ is a $321$-avoiding permutation, we recover Molev's skew double Schur function; see Appendix A.4.

Let $\eta _a$ be the ${\mathbb {Q}}[a]$-algebra homomorphism ${\mathbb {Q}}[x,a] \to {\mathbb {Q}}[a]$ given by $x_i \mapsto a_i$. This induces a ${\mathbb {Q}}[a]$-algebra map $1 \otimes \eta _a: \overleftarrow {R}(x;a) \to \Lambda (x||a) \otimes _{{\mathbb {Q}}[a]} {\mathbb {Q}}[a] \cong \Lambda (x||a)$, which we simply denote by $\eta _a$ as well.

For $w \in S_{\mathbb {Z}}$, define the double Stanley symmetric function $F_w(x||a) \in \Lambda (x||a)$ by

(4.18)\begin{equation} F_w(x||a) := \eta_a(\overleftarrow{\mathfrak{S}}_w(x;a)). \end{equation}

Proposition 4.13 For $w \in S_{\mathbb {Z}}$, we have

(4.19)\begin{equation} F_w(x||a) = \sum_{\substack{w \doteq uvz\\ u,z\in S_{\ne0}}} ({-}1)^u {\mathfrak{S}}_{u^{{-}1}}(a) F_v(x/a) {\mathfrak{S}}_z(a). \end{equation}

4.7 Negative double Schubert polynomials

Let $\omega$ be the involutive ${\mathbb {Q}}$-algebra automorphism of ${\mathbb {Q}}[x;a]$ given by $\omega (x_i)=-x_{1-i}$ and $\omega (a_i)=-a_{1-i}$ for $i\in {\mathbb {Z}}$. For $w\in S_-$, define the negative double Schubert polynomial ${\mathfrak {S}}_w( x_-;a_-)\in {\mathbb {Q}}[ x_-,a_-]$ by

(4.20)\begin{equation} {\mathfrak{S}}_w( x_-;a_-) := \omega({\mathfrak{S}}_{\omega(w)}(x_+;a_+)) \quad\text{for } w\in S_-. \end{equation}

Define ${\mathfrak {S}}_w(x;a)\in {\mathbb {Q}}[x;a]$ for $w\in S_{\ne 0}$ by

(4.21)\begin{equation} {\mathfrak{S}}_w(x;a) := {\mathfrak{S}}_u(x_+;a_+) {\mathfrak{S}}_v( x_-;a_-)\quad\text{where $w=uv$ with $u\in S_+$ and $v\in S_-$}. \end{equation}

Proposition 4.15 For $w\in S_{\ne 0}$, we have

(4.22)\begin{align} {\mathfrak{S}}_w(x;a) &= \sum_{w\doteq uv} ({-}1)^u {\mathfrak{S}}_{u^{{-}1}}(a) {\mathfrak{S}}_v(x) , \end{align}

(4.23)\begin{align}{\mathfrak{S}}_w(x) &= \sum_{w\doteq uv} {\mathfrak{S}}_u(a) {\mathfrak{S}}_v(x;a). \end{align}

4.8 Coproduct formula

There is a coaction $\Delta : \overleftarrow {R}(x;a) \to \Lambda (x||a) \otimes _{{\mathbb {Q}}[a]} \overleftarrow {R}(x||a)$ of $\Lambda (x||a)$ on $\overleftarrow {R}(x;a)$, defined by the comultiplication on the first factor of the tensor product $\Lambda (x||a) \otimes _{{\mathbb {Q}}[a]} {\mathbb {Q}}[x,a]$.

Theorem 4.16 Let $w \in S_{\mathbb {Z}}$. We have the coproduct formulae

(4.24)\begin{align} \Delta(\overleftarrow{\mathfrak{S}}_w(x;a)) &= \sum_{\substack{w \doteq uv }} F_u(x||a) \otimes \overleftarrow{\mathfrak{S}}_v(x;a), \end{align}

(4.25)\begin{align}\overleftarrow{\mathfrak{S}}_w(x;a) &= \sum_{\substack{w \doteq uv \\ v \in S_{{\neq} 0}}} F_u(x||a) \; {\mathfrak{S}}_v(x;a). \end{align}

Proof. We first deduce (4.24) from (4.25). Using Corollary 4.5, Proposition 4.13 and Lemma 2.10 we have

\begin{align*} &\sum_{w \doteq uv } F_u(x||a) \otimes \overleftarrow{\mathfrak{S}}_v(x;a) \\ &\quad= \sum_{\substack{w \doteq u_1v_1z_1u_2v_2z_2 \\ u_i,z_j \in S_{\ne 0}}} ({-}1)^{\ell(u_1)+\ell(u_2)} {\mathfrak{S}}_{u_1^{{-}1}}(a) F_{v_1}(x/a) {\mathfrak{S}}_{z_1}(a) \otimes {\mathfrak{S}}_{u_2^{{-}1}}(a) F_{v_2}(x/a) {\mathfrak{S}}_{z_2}(x) \\ &\quad= \sum_{\substack{w \doteq u_1v_1v_2z_2 \\ u_1,z_2 \in S_{\ne 0}}} ({-}1)^{\ell(u_1)} {\mathfrak{S}}_{u_1^{{-}1}}(a) F_{v_1}(x/a) \otimes F_{v_2}(x/a) {\mathfrak{S}}_{z_2}(x) \\ &\quad= \Delta\bigg(\sum_{\substack{w \doteq u_1vz_2 \\ u_1,z_2 \in S_{\ne 0}}} ({-}1)^{\ell(u_1)} {\mathfrak{S}}_{u_1^{{-}1}}(a) F_v(x/a) {\mathfrak{S}}_{z_2}(x)\bigg) \\ &\quad= \Delta(\overleftarrow{\mathfrak{S}}_w(x;a)). \end{align*}

For (4.25), using Propositions 4.13 and 4.15 we have

\begin{align*} \sum_{\substack{w \doteq uv \\ v \in S_{{\neq} 0}}} F_u(x||a) \; {\mathfrak{S}}_v(x;a) &= \sum_{\substack{w \doteq u_1v_1z_1 v \\ u_1,z_1,v \in S_{\ne 0}}} ({-}1)^{\ell(u_1)} {\mathfrak{S}}_{u_1{^{{-}1}}}(a) F_{v_1}(x/a) {\mathfrak{S}}_{z_1}(a) {\mathfrak{S}}_v(x;a) \\ &= \sum_{\substack{w \doteq u_1v_1z \\ u_1,z \in S_{\ne 0}}} ({-}1)^{\ell(u_1)} {\mathfrak{S}}_{u_1{^{{-}1}}}(a) F_{v_1}(x/a) {\mathfrak{S}}_z(x) \\ &= \overleftarrow{\mathfrak{S}}_w(x;a). \end{align*}

Corollary 4.17 Let $w \in S_{\mathbb {Z}}$. Then

\[ \Delta(F_w(x||a)) = \sum_{w \doteq uv} F_u(x||a) \otimes F_v(x||a). \]

Recall the definition of $w_{\lambda /\mu }$ from (2.2).

Corollary 4.18 For $\lambda \in {\mathbb {Y}}$, we have

(4.26)\begin{equation} \Delta(s_\lambda(x||a)) = \sum_{\mu \subset \lambda} F_{w_{\lambda/\mu}}(x||a) \otimes s_\mu(x||a). \end{equation}

4.9 Dynkin reversal

Extend the ${\mathbb {Q}}$-algebra automorphism $\omega$ of ${\mathbb {Q}}[x;a]$ to $\overleftarrow {R}(x;a)$ by $\omega (p_k(x||a)) = (-1)^{k-1}p_k(x||a)$.

Proposition 4.19 We have

(4.27)\begin{align} \omega(f)|_{\omega(v)} &= \omega(f|_v) \quad \text{for } f\in \overleftarrow{R}(x;a) \hbox{ and } v\in S_{\mathbb{Z}}, \end{align}

(4.28)\begin{align}\omega(\gamma_a(f)) &= \gamma_a^{{-}1}(\omega(f))\quad \text{for } f\in \overleftarrow{R}(x;a). \end{align}

Moreover,

(4.29)\begin{align} \omega(F_v(x/a)) &= F_{\omega(v)}(x/a)\ \quad\text{for } v \in S_{\mathbb{Z}} , \end{align}

(4.30)\begin{align} \omega(\overleftarrow{\mathfrak{S}}_v(x;a))&= \overleftarrow{\mathfrak{S}}_{\omega(v)}(x;a)\quad\text{for } v \in S_{\mathbb{Z}}, \end{align}

(4.31)\begin{align} \omega(F_v(x||a)) &= F_{\omega(v)}(x||a)\ \quad\text{for } v \in S_{\mathbb{Z}}, \end{align}

(4.32)\begin{align} \omega(s_\lambda(x||a)) &= s_{\lambda'}(x||a)\ \qquad\text{for } \lambda\in{\mathbb{Y}}. \end{align}

4.10 Double Edelman–Greene coefficients

Define the double Edelman–Greene coefficients $j_\lambda ^w(a) \in {\mathbb {Q}}[a]$ by the equality

(4.33)\begin{equation} F_w(x||a) = \sum_\lambda j_\lambda^w(a) s_\lambda(x||a). \end{equation}

Theorem 4.22 Let $x \in S_{\mathbb {Z}}$ and $v \in S_{\mathbb {Z}}^0$. Then $j^x_v(a) \in {\mathbb {Q}}[a]$ is a positive integer polynomial in the linear forms $a_i - a_j$ where $i \prec j$ under the total ordering of ${\mathbb {Z}}$ given by

\[ 1 \prec 2 \prec 3 \prec \cdots\prec{-}2 \prec{-}1 \prec 0. \]

Theorem 4.22 will be proven in § 9.9.

Define the coproduct structure constants $\hat {c}^\lambda _{\mu \nu }(a)\in {\mathbb {Q}}[a]$ for $\lambda ,\mu ,\nu \in {\mathbb {Y}}$ by

(4.34)\begin{equation} \Delta(s_\lambda(x||a)) = \sum_{\mu,\nu\in{\mathbb{Y}}} \hat{c}^\lambda_{\mu\nu}(a) s_\mu(x||a) \otimes s_\nu(x||a). \end{equation}

Let $d(\lambda )$ be the Durfee square of $\lambda \in {\mathbb {Y}}$, the maximum index $d$ such that $\lambda _d \geqslant d$. The following change of basis coefficients between the double and super Schur bases were previously computed in [Reference Olshanski, Regev and VershikORV03] expressed as a determinantal formula and in [Reference MolevMol09] by a tableau formula. We give them as (signed) Schubert polynomials.

Proposition 4.25 For $\lambda \in {\mathbb {Y}}$, we have

(4.35)\begin{align} s_\lambda(x||a) &= \sum_{\substack{\mu\subset\lambda\\ d(\mu)=d(\lambda)}} ({-}1)^{|\lambda/\mu|} {\mathfrak{S}}_{w_{\lambda/\mu}^{{-}1}} (a) s_\mu(x/a) , \end{align}

(4.36)\begin{align}s_\lambda(x/a) &= \sum_{\substack{\mu\subset\lambda\\ d(\mu)=d(\lambda)}} {\mathfrak{S}}_{w_{\lambda/\mu}} (a) s_\mu(x||a). \end{align}

5.1 $S_{\mathbb {Z}}$-bumpless pipedreams

Let $w \in S_{\mathbb {Z}}$. A $w$-bumpless pipedream is a bumpless pipedream covering the whole plane, satisfying the following conditions.

1. (i) There is a bijective labeling of pipes by integers.

2. (ii) The pipe labeled $i$ eventually heads south in column $i$ and heads east in row $w^{-1}(i)$.

3. (iii) Two pipes cannot cross more than once.

4. (iv) For all $N \gg 0$ and all $N \ll 0$, the pipe labeled $N$ travels north from $(\infty ,N)$ to the square $(N,N)$ where it turns east and travels towards $(N,\infty )$.

Because of condition (ii), every pipe has to make at least one turn. We call pipe $i$ standard if it makes exactly one turn and this turn is at the diagonal square $(i,i)$. By (iv), all but finitely many pipes are standard. We often omit standard pipes from our drawings of pipedreams. The weight ${\rm wt}(P) := \prod (x_i - a_j)$ of a pipedream $P$ is the product of $x_i - a_j$ over all empty tiles $(i,j)$.

Example 5.1 Let $w = s_3s_0s_1$. In one line notation, $w(-2,-1,0,1,2,3,4) = (-2,-1,1,2,0,4,3)$ and the rest are fixed points. Figure 1 shows a $w$-bumpless pipedream, where we have only drawn the region $\{(i,j) \mid i,j \in [-2,4]\}$. In the left picture, the empty tiles have been indicated, as have the row and column numbers. The label of a pipe is the column number to which its south end is attached. In the right picture, we have indicated the labels of the pipes instead of the row numbers. The one-line notation of $w$ can then be read off the east border.

Figure 1. A bumpless pipedream with weight ${\rm wt} = (x_{-1}-a_{-1})(x_1-a_0)(x_1-a_2)$.

The proof of Theorem 5.2 is delayed to after Theorem 5.13.

5.2 Drooping and the Rothe pipedream

A $w$-bumpless pipedream is uniquely determined by the location of the two kinds of elbow tiles. Each pipe has to turn at least once. There is a unique $w$-bumpless pipedream such that for all $i$, pipe $i$ turns right from south to east in the square $(w^{-1}(i),i)$. We call this the Rothe pipedream $D(w)$ of $w$. The empty tiles of the Rothe pipedream form what is commonly known as the Rothe diagram of $w$.

Let $P$ be a $w$-bumpless pipedream. A droop is a local move that swaps an SE elbow $e$ with an empty tile $t$, when the SE elbow lies strictly to the northwest of the empty tile. Let $R$ be the rectangle with northwest corner $e$ and southeast corner $t$ and let $p$ be the pipe passing through $e$. After the droop, the pipe $p$ travels along the southmost row and eastmost column of $R$; a NW elbow occupies the square that used to be empty while the square that contained an SE elbow becomes empty. The droop is allowed only if all the following hold.

1. (i) The westmost column and northmost row of $R$ contains $p$.

2. (ii) The rectangle $R$ contains only one elbow which is at $e$.

3. (iii) After the droop we obtain a bumpless pipedream $P'$.

Pipes $p' \neq p$ do not move in a droop. We denote a droop by $P \searrow P'$. Figure 2 shows a Rothe bumpless pipedream followed by a sequence of two droops.

Proposition 5.3 Every $w$-bumpless pipedream can be obtained from the Rothe pipedream $D(w)$ of $w$ by a sequence of droops.

Figure 2. A Rothe bumpless pipedream $P$, and a sequence of two droops.

5.3 Halfplane crossless pipedreams

Let $P$ be an $S_{\mathbb {Z}}$-bumpless pipedream. By (iv) of the definition, only finitely many crossings appear. If we cut off, using a horizontal line, the bottom part of $P$ containing all crossing tiles, we will obtain a picture that we call a halfplane crossless pipedream. It turns out that the double Schur function is a generating function of such pipedreams.

For $\lambda \in {\mathbb {Y}}$, a $\lambda$-halfplane pipedream is a bumpless pipedream in the upper halfplane $H={\mathbb {Z}}_{\leqslant 0}\times {\mathbb {Z}}$ such that the crossing tile is not used, and:

1. (i) there are (unlabeled) pipes entering from the southern boundary in the columns indexed by $I \subset {\mathbb {Z}}$;

2. (ii) setting $(I_+,I_-) = (I \cap {\mathbb {Z}}_{>0}, {\mathbb {Z}}_{>0} \setminus I)$, we have $I_{\pm } = I_{w_\lambda ,\pm }$ (see (4.6), (4.7), and (2.1));

3. (iii) the $i$th eastmost pipe entering from the south heads off to the east in row $1-i$. (Equivalently, for every row $i \in {\mathbb {Z}}_{\leqslant 0}$, there is some pipe heading towards $(i,\infty )$.)

As before, the weight of a $\lambda$-halfplane pipedream is ${\rm wt}(P) = \prod (x_i - a_j)$, where the product is over all empty tiles $(i,j)$ in the halfplane $H$.

For example, taking $\lambda = (2,1,1)$ we have $w_\lambda = s_{-2}s_{-1}s_1 s_0$ and $(I_+,I_-) = (\{2\},\{-2\})$. Figure 3 shows a $\lambda$-halfplane pipedream.

Lemma 5.5 The number of empty tiles in a $\lambda$-halfplane pipedream is equal to $|\lambda |$.

Figure 3. A $(2,1,1)$-halfplane pipedream with weight ${\rm wt} = (x_{-3}-a_{-3})(x_{-2}-a_{-3})(x_{-1}-a_0)(x_0-a_{-2})$.

The proof of Theorem 5.6 is delayed to after Theorem 5.13.

A ${\mathbb {Z}}_{\leqslant 0}$-semistandard Young tableau (SSYT) of shape $\lambda$ is a filling of the Young diagram $\lambda$ (in English notation) with the integers $0,-1,-2,\ldots$ such that rows are weakly increasing and columns are strictly increasing. The weight ${\rm wt}(T)$ of a ${\mathbb {Z}}_{\leqslant 0}$-SSYT is the product ${\rm wt}(T) = \prod _{(i,j) \in T} (x_{T(i,j)} - a_{T(i,j)+c(i,j)})$ where $c(i,j) = j-i$ is the content of the square $(i,j)$ in row $i$ and column $j$.

5.4 Rectangular $S_n$-bumpless pipedreams

Let $w \in S_n$. A $w$-rectangular bumpless pipedream is a bumpless pipedream in the $n \times 2n$ rectangular region

\[ R_n := \{(i,j) \mid i \in [1,n] \text{ and } j \in [1-n,n]\}. \]

The pipes are labeled $1-n,2-n,\ldots ,0,1,\ldots ,n$, entering the south boundary from left to right. The positively labeled pipes exit the east boundary: pipe $i$ exits in row $i$. The nonpositively labeled pipes exit the north boundary. Two pipes intersect at most once, and a nonpositively labeled pipe cannot intersect any other pipe. As before, the weight of a rectangular $S_n$-bumpless pipedream $P$ is given by ${\rm wt}(P) = \prod (x_i - a_j)$, with the product over all empty tiles $(i,j)$.

We also associate a partition $\lambda (P)$ to an $S_n$-rectangular bumpless pipedream: it is obtained by reading the north boundary edges from right to left, to then obtain the boundary of a partition inside a $n \times n$ box, where empty edges correspond to steps to the left, and edges with a pipe exiting correspond to downward steps. See Figure 4, where empty edges are marked $e$ and edges with a pipe exiting are marked $x$.

Lemma 5.9 Let $w \in S_n$ and $P$ be a $w$-bumpless pipedream. We have $\ell (w) = |\lambda (P)| + {\rm deg}({\rm wt}(P))$.

Figure 4. The partition of a rectangular $S_n$-bumpless pipedream.

Example 5.10 Let $w = 2143$. In Figure 5 is a complete list of all $w$-bumpless pipedreams. The nonpositively labeled pipes are those which enter from the bottom to the left of the dotted line.

Figure 5. Rectangular $S_n$-bumpless pipedreams for $w=2143$.

Theorem 5.11 is proved in § 12.4.

5.5 Square $S_n$-bumpless pipedreams

Let $w \in S_n$. A $w$-square bumpless pipedream is a bumpless pipedream in the $n \times n$ square region

\[ S_n := \{(i,j) \mid i \in [1,n] \text{ and } j \in [1,n]\}. \]

The pipes are labeled $1,\ldots ,n$, entering the south boundary from left to right. The pipes exit the east boundary: pipe $i$ exits in row $i$. Two pipes intersect at most once. As before, the weight of a square $S_n$-bumpless pipedream $P$ is given by ${\rm wt}(P) = \prod (x_i - a_j)$, with the product over all empty tiles $(i,j)$. In Example 5.10, if we erase the left half and all nonpositively labeled pipes, we obtain a square $2143$-bumpless pipedream.

Proof. By Theorem 4.16 and Lemma 4.20, when $\overleftarrow {\mathfrak {S}}_w(x_+;a_+)$ is expanded in terms of $\{s_\lambda (x||a) \mid \lambda \in {\mathbb {Y}}\}$, the coefficient of $s_\emptyset (x||a)$ is equal to ${\mathfrak {S}}_w(x_+;a_+)$. By Theorem 5.11, we thus have

\[ {\mathfrak{S}}_w(x_+;a_+) = \sum_P {\rm wt}(P) \]

summed over $w$-rectangular bumpless pipedreams $P$ satisfying $\lambda (P) = \emptyset$. The condition $\lambda (P) = \emptyset$ is equivalent to all nonpositively labeled pipes in $P$ being completely vertical. In particular, the nonpositively labeled pipes stay within the left $n \times n$ square of $P$. Such pipedreams are in weight-preserving bijection with $w$-square bumpless pipedreams.

5.6 EG pipedreams

Let $w \in S_n$. Let $P$ be a $w$-square bumpless pipedream. We call $P$ a $w$-EG pipedream if all the empty tiles are in the northeast corner, where they form a partition shape $\lambda = \lambda (P)$, called the shape of $P$. See Figure 6.

Theorem 5.14 The Edelman–Greene coefficient $j_\lambda ^w = j_\lambda ^w(0)$ of (3.12) is equal to the number of $w$-EG pipedreams $P$ satisfying $\lambda (P) = \lambda$.

Figure 6. A 213-EG pipedream with partition $(1)$.

An empty tile $T$ in a bumpless pipedream $D$ is called a floating tile if there exists a pipe that is northwest of $T$. A bumpless pipedream $D$ is called near EG if it has a single floating tile.

5.7 Column moves

We define column moves that modify a bumpless pipedream in two adjacent columns; see Figure 7. Only one of the pipes (the active pipe) is drawn in these pictures. For the move to be allowed, the southeastmost tile must be an empty tile (before the move), and it must be the only empty tile. Thus the move takes the empty tile from the southeastmost position to the northwestmost position. There are usually other pipes in the move, and the ‘kinks are shifted left’ if necessary; see the move on the right of Figure 7.