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Back stable Schubert calculus

Published online by Cambridge University Press:  30 April 2021

Thomas Lam
Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI48109,
Seung Jin Lee
Department of Mathematical Sciences, Research Institute of Mathematics, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul151-747, Republic of
Mark Shimozono
Department of Mathematics, MC 0123, Virginia Tech, 460 McBryde Hall, 255 Stanger St., Blacksburg, VA24061,
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We study the back stable Schubert calculus of the infinite flag variety. Our main results are:

  • a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;

  • a novel definition of double and triple Stanley symmetric functions;

  • a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;

  • the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;

  • the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;

  • equivariant Pieri rules for the homology of the infinite Grassmannian;

  • homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

Research Article
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1. Introduction

1.1 Flag varieties and Schubert polynomials

The flag variety $\textrm {Fl}_n$ is the smooth projective algebraic variety classifying full flags inside an $n$-dimensional complex vector space ${\mathbb {C}}^n$. The cohomology ring $H^*(\textrm {Fl}_n)$ was determined by Borel [Reference BorelBor53]: it is the quotient of the polynomial ring ${\mathbb {Q}}[x_1,\ldots ,x_n]$ by the ideal generated by symmetric functions in $x_1,\ldots ,x_n$ of positive degree.

The flag variety has a distinguished stratification by Schubert varieties, and the cohomology classes of Schubert varieties form a basis of $H^*(\textrm {Fl}_n)$, called the Schubert basis. Bernstein, Gelfand, and Gelfand [Reference Bernstein, Gel'fand and Gel'fandBGG73] and Demazure [Reference DemazureDem74] found formulae for the Schubert basis in terms of divided difference operators. Lascoux and Schützenberger [Reference Lascoux and SchützenbergerLS82] defined and studied polynomial representatives for the Schubert classes, called the Schubert polynomials ${\mathfrak {S}}_w \in {\mathbb {Q}}[x_1,\ldots ,x_n]$. Lascoux and Schützenberger furthermore defined the double Schubert polynomials ${\mathfrak {S}}_w(x;a)$ that represent Schubert classes in the torus-equivariant cohomology ring $H^*_T(\textrm {Fl}_n)$.

There is a rich combinatorial theory for Schubert polynomials. Among the fundamental results crucial to us is the formula of Billey, Jockusch and Stanley [Reference Billey, Jockusch and StanleyBJS93] for the monomial expansion of ${\mathfrak {S}}_w$.

1.2 Back stable Schubert polynomials

In this work, we consider limits of Schubert polynomials called back stable Schubert polynomials

\[ \overleftarrow{\mathfrak{S}}_w := \lim_{\substack{p\to-\infty \\q\to\infty}} {\mathfrak{S}}_w(x_p,x_{p+1},\ldots,x_q), \]

for $w\in S_{\mathbb {Z}}$, the group of permutations of ${\mathbb {Z}}$ moving finitely many elements. Two of us (T.L. and M.S.) first learnt of this construction from Knutson (personal communication). Buch (personal communication, 2018) was also aware of how to back stabilize (double) Schubert polynomials. Finally, one of us (S.-J. Lee) found them on his own independently.

Define the ring of back symmetric formal power series

\[ \overleftarrow{R} := \Lambda \otimes {\mathbb{Q}}[\ldots,x_{{-}1},x_0,x_1,\ldots] \]

where $\Lambda$ denotes the symmetric functions in $\ldots ,x_{-1},x_0$. In Theorem 3.5, we show that the back stable Schubert polynomials $\overleftarrow {\mathfrak {S}}_w$ form a basis of the ring $\overleftarrow {R}$. As far as we are aware, the ring $\overleftarrow {R}$ has not previously been explicitly studied.

1.3 Coproduct formula

Stanley [Reference StanleySta84] defined the Stanley symmetric functions $F_w \in \Lambda$, for $w \in S_{\mathbb {Z}}$ to study the enumeration of reduced words of permutations. It is well known that the symmetric functions $F_w$ can be obtained as ‘forward limits’ of the Schubert polynomials ${\mathfrak {S}}_w$. We give a new construction of $F_w$ from back stable Schubert polynomials. Namely, we define a natural algebra homomorphism $\eta _0: \overleftarrow {R} \to \Lambda$ and show that Stanley's definition of $F_w$ agrees with $\eta _0({\mathfrak {S}}_w)$. This is closely related to, and explains, a formula of Li [Reference LiLi14]. In contrast, the map sending ${\mathfrak {S}}_w$ to $F_w$ is not multiplicative.

We prove that back stable Schubert polynomials satisfy the ‘coproduct formula’ (Theorem 3.14)

(1.1)\begin{equation} \overleftarrow{\mathfrak{S}}_w = \sum_{w \doteq uv} F_u \otimes {\mathfrak{S}}_v, \end{equation}

where $w\doteq uv$ denotes a length-additive factorization such that $v$ is a permutation not using the reflection $s_0$. The coproduct formula decomposes $\overleftarrow {\mathfrak {S}}_w$ into a ‘symmetric’ part and a ‘finite polynomial’ part. We do not know of an analogue of the coproduct formula for finite Schubert polynomials.

1.4 Double Stanley symmetric functions

Back stable double Schubert polynomials $\overleftarrow {\mathfrak {S}}_w(x;a)$ can also be defined in a similar manner (though the existence of the limit is less clear; see Proposition 4.3), and we show (Theorem 4.7) that they form a basis of the back symmetric double power series ring $\overleftarrow {R}(x;a) :=\Lambda (x||a) \otimes _{{\mathbb {Q}}[a]} {\mathbb {Q}}[x,a]$, where ${\mathbb {Q}}[x,a]:={\mathbb {Q}}[x_i,a_i\mid i\in {\mathbb {Z}}]$ and $\Lambda (x||a)$ is the ring of double symmetric functions. The ring $\Lambda (x||a)$ is the polynomial ${\mathbb {Q}}[a]={\mathbb {Q}}[\dotsc ,a_{-1},a_0,a_1,\dotsc ]$-algebra generated by the double power sums $p_k(x||a):=\sum _{i \leqslant 0} x_i^k- \sum _{i \leqslant 0} a_i^k$. The ring $\Lambda (x||a)$ is a ${\mathbb {Q}}[a]$-Hopf algebra with basis the double Schur functions $s_\lambda (x||a)$, and is studied in detail by Molev [Reference MolevMol09].

Generalizing $\eta _0$, there is an algebra homomorphism $\eta _a: \overleftarrow {R}(x;a) \to \Lambda (x||a)$. We define the double Stanley symmetric functions $F_w(x||a) \in \Lambda (x||a)$ by $F_w(x||a):= \eta _a(\overleftarrow {R}(x;a))$. As far as we are aware, the symmetric functions $F_w(x||a)$ are novel. When $w$ is 321-avoiding, the double Stanley symmetric function is equal to the skew double Schur function which was studied by Molev [Reference MolevMol09]; see Proposition A.2.

One of our main theorems (Theorem 4.22) is a proof that the double Edelman–Greene coefficients $j_\lambda ^w(a) \in {\mathbb {Q}}[a]$ given by the expansion of double Stanley symmetric functions

\[ F_w(x||a) = \sum_{\lambda} j_\lambda^w(a) s_\lambda(x||a) \]

into double Schur functions $s_\lambda (x||a)$ are positive polynomials in certain linear forms $a_i - a_j$. The usual Edelman–Greene coefficients $j_\lambda ^w(0):=j_\lambda ^w(a)|_{a_i \to 0}$ are known to be positive by the influential works of Edelman and Greene [Reference Edelman and GreeneEG87] and Lascoux and Schützenberger [Reference Lascoux and SchützenbergerLS85]. Molev [Reference MolevMol09] has given a combinatorial rule for the expansion coefficients of skew double Schurs into double Schurs (that is, for $j_\lambda ^w(a)$ where $w$ is $321$-avoiding) but it does not exhibit the above positivity.

Back stable double Schubert polynomials satisfy (Theorem 4.16) the same kind of coproduct formula (1.1) as the nondoubled version, with the double Stanley symmetric functions $F_w(x||a)$ replacing $F_w$ and double Schubert polynomials ${\mathfrak {S}}_w(x;a)$ replacing the usual finite Schubert polynomials ${\mathfrak {S}}_w$.

1.5 Bumpless pipedreams

We introduce a combinatorial object called bumpless pipedreams, to study the monomial expansion of back stable double Schubert polynomials. These are pipedreams where pipes are not allowed to bump against each other, or equivalently, the ‘bumping’ or ‘double elbow tile’ is forbidden.

Using bumpless pipedreams, we obtain:

  • an expansion for double Schubert polynomials ${\mathfrak {S}}_w(x;a)$ in terms of products of binomials $\prod (x_i - a_j)$; (Our formula is different from the classical pipe-dream formula of Fomin and Kirillov [Reference Fomin and KirillovFK96] for double Schubert polynomials: unlike theirs, our formula is obviously back stable. Hence we also obtain such an expansion for back stable double Schubert polynomials.)

  • a positive expression for the coefficient of $s_\lambda (x||a)$ in $\overleftarrow {\mathfrak {S}}(x;a)$ (Theorem 5.11);

  • a new combinatorial interpretation of Edelman–Greene (EG) coefficients $j_\lambda ^w(0)$ as the number of certain EG pipedreams (Theorem 5.14).

Our bumpless pipedreams are a streamlined version of the interval positroid pipedreams defined by Knutson [Reference KnutsonKnu14]. Heuristically, our formula for $\overleftarrow {\mathfrak {S}}_w(x;a)$ is obtained by ‘pulling back’ a Schubert variety in $\textrm {Fl}$ to various Grassmannians where it can be identified (after equivariant shifts) with graph Schubert varieties, a special class of positroid varieties. This connects our work with that of Knutson et al. [Reference Knutson, Lam and SpeyerKLS13], who identified the equivariant cohomology classes of positroid varieties with affine double Stanley symmetric functions.

When presenting our findings we were informed by Anna WeigandtFootnote 1 that Lascoux's use [Reference LascouxLas02] of alternating sign matrices (ASMs) in a formula for Grothendieck polynomials is very close to our pipedreams; ours correspond to the subset of reduced ASMs. Our construction has the advantage that the underlying permutation is evident; in the ASM one must go through an algorithm to extract this information. Lascoux's ASMs naturally compute in $K$-theory rather than in cohomology.

1.6 Infinite flag variety

Whereas Schubert polynomials represent Schubert classes in the cohomology of the flag variety, back stable Schubert polynomials represent Schubert classes in the cohomology of an appropriate infinite flag variety.

The infinite Grassmannian ${\mathrm {Gr}}$ is an ind-finite variety over ${\mathbb {C}}$, the points of which are identified with (infinite-dimensional over ${\mathbb {C}}$) admissible subspaces $\Lambda \subset F$, where $F = {\mathbb {C}}((t))$ (see § 6). The infinite Grassmannian can be presented as an infinite union of finite-dimensional Grassmannians. The infinite flag variety $\textrm {Fl}$ is an ind-finite variety over ${\mathbb {C}}$, the points of which are identified with admissible flags

\[ \Lambda_\bullet = \{\cdots \subset \Lambda_{{-}1} \subset \Lambda_0 \subset \Lambda_1 \subset \cdots\}. \]

Under an isomorphism between $\overleftarrow {R}$ and the cohomology of $\textrm {Fl}$, we show in Theorem 6.7 that back stable Schubert polynomials represent Schubert classes of $\textrm {Fl}$. For the infinite Grassmannian it is well known that Schur functions represent Schubert classes. Our $\textrm {Fl}$ differs somewhat from other infinite-dimensional flag varieties we have seen in the literature (see for example [Reference Pressley and SegalPS86]), and thus we give a reasonably independent development in § 6.

The infinite flag variety $\textrm {Fl}$ is the union of finite-dimensional flag varieties, and any product $\xi ^x \xi ^y$ of two Schubert classes $\xi ^x,\xi ^y \in H^*(\textrm {Fl})$ can be computed within some finite-dimensional flag variety. Naively, as some subset of the authors had mistakenly assumed, no interesting and new phenomena would arise in the infinite case. To the contrary, in this article we present our findings of entirely new phenomena that have no classical counterpart.

1.7 Localization and infinite nilHecke algebra

The torus-equivariant cohomology $H^*_T(\textrm {Fl}_n)$ of the flag variety can be studied by localizing to the torus fixed points, giving an injection $H^*_T(\textrm {Fl}_n) \hookrightarrow {\bigoplus} _{v \in S_n} H^*_T(\mathrm {pt}) \simeq {\mathbb {Q}}[a_1,\ldots ,a_n]$. It is known [Reference BilleyBil99, Remark 1] that the localization $\xi ^v|_w$ of a Schubert class indexed by $v \in S_n$ at the torus fixed-point indexed $w \in S_n$ is given by the evaluation ${\mathfrak {S}}_v(wa;a) \in {\mathbb {Q}}[a]$. We prove in Proposition 7.9 an analogous result for the equivariant cohomology ring $H^*_{T_{\mathbb {Z}}}(\textrm {Fl})$: the localization of a Schubert class $\xi ^v$ at a $T_{\mathbb {Z}}$-fixed point $w \in S_{\mathbb {Z}}$ is equal to a specialization $\overleftarrow {\mathfrak {S}}_v(wa;a)$ of the back stable double Schubert polynomial.

Kostant and Kumar [Reference Kostant and KumarKK86] studied the torus-equivariant cohomology of Kac–Moody flag varieties (including the usual flag variety) using the action of the nilHecke ring on these cohomologies. We construct in § 7 an action of the infinite nilHecke ring ${\mathbb {A}}'$ on $H^*_{T_{\mathbb {Z}}}(\textrm {Fl})$, giving an infinite rank variant of the results of Kostant and Kumar.

1.8 Homology

The torus-equivariant cohomology ring $H_{T_{\mathbb {Z}}}^*({\mathrm {Gr}})$ of the infinite Grassmannian is isomorphic to the ring $\Lambda (x||a)$ of double symmetric functions (see Theorem 6.6). The (appropriately completed) equivariant homology $H_*^{T_{\mathbb {Z}}}({\mathrm {Gr}})$ of the infinite Grassmannian is Hopf-dual to the Hopf algebra $\Lambda (x||a)$. Nonequivariantly, this can be explained by the homotopy equivalence ${\mathrm {Gr}} \cong \Omega SU(\infty )$ with a group. Restricting to a one-dimensional torus ${\mathbb {C}}^\times \subset T_{\mathbb {Z}}$, the multiplication of $H_*^{{\mathbb {C}}^\times }({\mathrm {Gr}})$ is induced by the direct sum operation on finite Grassmannians, and was studied in some detail by Knutson and Lederer [Reference Knutson and LedererKL15]. The geometry of the full multiplication on $H_*^{T_{\mathbb {Z}}}({\mathrm {Gr}})$ is still mysterious to us, and we hope to study it in the context of the affine infinite Grassmannian in the future.

Molev [Reference MolevMol09] studied the Hopf algebra $\hat {\Lambda }(y||a)$ Hopf-dual to $\Lambda (x||a)$, and defined the basis ${\hat {s}}_\lambda (y||a)$ of dual Schur functions in $\hat {\Lambda }(y||a)$, dual to the double Schur functions. We identify (Proposition 8.1) the Schubert basis of $H_*^{T_{\mathbb {Z}}}({\mathrm {Gr}})$ with Molev's dual Schur functions ${\hat {s}}_\lambda (y||a)$ [Reference MolevMol09]. We use this to resolve (Theorem 8.12) a question posed in [Reference Knutson and LedererKL15]: to find deformations of Schur functions that have structure constants equal to the Knutson-Lederer direct sum product.

One of our main results (Theorem 8.6) is a recursive formula for the dual Schur functions ${\hat {s}}_\lambda (x||a)$ in terms of novel homology divided difference operators, which are divided difference operators on equivariant variables, but conjugated by the equivariant Cauchy kernel. A similar formula had previously been found independently by Nakagawa and Naruse [Reference Nakagawa and NaruseNN18], who was studying the homology of the infinite Lagrangian Grassmannian. Our construction is also closely related to the presentation of the equivariant homology of the affine Grassmannian given by Bezrukavnikov et al. [Reference Bezrukavnikov, Finkelberg and MirkovićBFM05]. We hope to return to the affine setting in the future.

We compute the ring structure of this equivariant homology ring by giving a positive Pieri rule (Theorem 8.18). Our computation of the Pieri structure constants relies on some earlier work of Lam and Shimozono [Reference Lam and ShimozonoLS12] in the affine case, and on triple Stanley symmetric functions $F_w(x||a||b)$ that we define in § 10. The double Stanley symmetric functions $F_w(x||a)$ are recovered from $F_w(x||a||b)$ by setting $b = a$. The triple Stanley symmetric functions distinguish ‘stable’ phenomena from ‘unstable’ phenomena in the limit from the affine to the infinite setting.

1.9 Affine Schubert calculus

Our study of back stable Schubert calculus is to a large extent motivated by our study of the Schubert calculus of the affine flag variety $\widetilde {\textrm {Fl}}$, and in particular Lee's recent definition of affine Schubert polynomials [Reference LeeLee19]. There is a surjection $H^*(\textrm {Fl}) \to H^*(\widetilde {\textrm {Fl}}_n)$ from the cohomology of the infinite flag variety to that of the affine flag variety of ${\mathrm {SL}}(n)$. A complete understanding of this map yields a presentation for the cohomology of the affine flag variety. Thus this project can be considered as a first step towards understanding the geometry and combinatorics of affine Schubert polynomials and their equivariant analogues.

We shall apply back stable Schubert calculus to affine Schubert calculus in future work. In particular, analogues of our coproduct formulae (Theorems 3.14 and 4.16) hold for equivariant Schubert classes in the affine flag variety of any semisimple group $G$ [Reference Lam, Lee and ShimozonoLLS21].

1.10 Peterson subalgebra

The (finite) torus-equivariant cohomology ring $H^*_T(\widetilde {\textrm {Fl}}_n)$ of the affine flag variety $\widetilde {\textrm {Fl}}_n$ has an action of the level zero affine nilHecke ring ${\tilde {{\mathbb {A}}}}$. Peterson [Reference PetersonPet97, Reference LamLam08] constructed a subalgebra ${\tilde {{\mathbb {P}}}} \subset {\tilde {{\mathbb {A}}}}$ (recalled in Appendix C) and showed that the torus-equivariant homology $H_*^T({\widetilde {{\mathrm {Gr}}}}_n)$ of the affine Grassmannian ${\widetilde {{\mathrm {Gr}}}}_n$ is isomorphic to ${\tilde {{\mathbb {P}}}}$. We refer the reader to [Reference Lam, Lapointe, Morse, Schilling, Shimozono and ZabrockiLLM+14] for an introduction to affine Grassmannian Schubert calculus.

While Kostant and Kumar's definition of the nilHecke algebra applies to any Kac–Moody flag variety, the definition of the Peterson algebra is special to the case of the affine flag variety (of a semisimple group). Thus it came as a surprise that we are able to construct (Theorem 9.8) a subalgebra ${\mathbb {P}}' \subset {\mathbb {A}}'$ of the infinite nilHecke ring that is an analogue of the Peterson subalgebra in the affine case. While the infinite symmetric group $S_{\mathbb {Z}}$ is not an affine Coxeter group, we are able to define elements in ${\mathbb {A}}'$ that behave like translation elements in affine Coxeter groups.

Our infinite Peterson algebra ${\mathbb {P}}'$ is in a precise sense the limit of Peterson algebras for affine type $A$. This allows us to apply known positivity results in affine Schubert calculus to deduce the positivity (Theorem 4.22) of double Edelman–Greene coefficients.

1.11 Other directions

Most of the results of the present work have $K$-theoretic analogues. We plan to address $K$-theory in a separate work (Lam, Lee and Shimozono, Back stable K-theory Schubert calculus, in preparation).

The results in this paper (for e.g. § 9.2) suggests the study of the affine infinite flag variety $\widetilde {\textrm {Fl}}$, an ind-variety whose torus-fixed points are the affine infinite symmetric group $S_{\mathbb {Z}} \ltimes Q_{{\mathbb {Z}}}^\vee$, where $Q_{\mathbb {Z}}^\vee$ is the ${\mathbb {Z}}$-span of root vectors $e_i-e_j$ for $i\ne j$ integers and $e_i$ is the standard basis of a lattice with $i\in {\mathbb {Z}}$. Curiously, Schubert classes of $\widetilde {\textrm {Fl}}$ can have infinite codimension (elements of $S_{\mathbb {Z}} \ltimes Q_{{\mathbb {Z}}}^\vee$ can have infinite length) and should lead to new phenomena in Schubert calculus.

2. Schubert polynomials

We recall known results concerning Lascoux and Schützenberger's (double) Schubert polynomials. None of the results in this section are new, but for completeness we provide short proofs for many of them.

2.1 Notation

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

2.1.1 Permutations

Let $S_{\mathbb {Z}}$ denote the subgroup of permutations of ${\mathbb {Z}}$ generated by $s_i$ for $i\in {\mathbb {Z}}$ where $s_i$ exchanges $i$ and $i+1$. This is the group of permutations of ${\mathbb {Z}}$ that move finitely many elements. Let $S_+$ (respectively $S_-$, respectively $S_n$) be the subgroup of $S_{\mathbb {Z}}$ generated by $s_1, s_2,\ldots$ (respectively $s_{-1},s_{-2},\ldots$, respectively $s_1,s_2,\dotsc ,s_{n-1}$). We have $S_+ = {\bigcup} _{n\geqslant 1} S_n$. We write $S_{\ne 0} = S_-\times S_+$. For $w\in S_{\mathbb {Z}}$ denote by $\ell (w)$ the length of $w$ and $\textrm {Red}(w)$ for the set of reduced words of $w$ [Reference HumphreysHum90, § 1.6]. For $x,y,z \in S_{\mathbb {Z}}$, we write $z \doteq xy$ if $z = xy$ and $\ell (z) = \ell (x) + \ell (y)$. This notation generalizes to longer products $z \doteq x_1 x_2 \cdots x_r$. Let $w_0^{(n)}\in S_n$ be the longest element [Reference HumphreysHum90, § 1.8]. Let $\gamma :S_{\mathbb {Z}}\to S_{\mathbb {Z}}$ be the ‘shifting’ automorphism $\gamma (s_i)=s_{i+1}$ for all $i\in {\mathbb {Z}}$.

Let $\leqslant$ be the (strong) Bruhat order on $S_{\mathbb {Z}}$ [Reference HumphreysHum90, § 5.9]. For a fixed $k\in {\mathbb {Z}}$, say that $w\in S_{\mathbb {Z}}$ is $k$-Grassmannian if $w<ws_i$ (equivalently, $w(i) < w(i+1)$ viewing $w$ as a function ${\mathbb {Z}}\to {\mathbb {Z}}$) for all $i\in {\mathbb {Z}}-\{k\}$. We write $S_{\mathbb {Z}}^0$ for the set of $0$-Grassmannian permutations.

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}

Lemma 2.3 Both the kernel of $A_i$ and the image of $A_i$ are the subalgebra of $s_i$-invariant elements.

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).

Lemma 2.4 ${\mathfrak {S}}_w(x_+)$ is well defined for $w\in S_+$.

Proof. It suffices to show that the definitions of ${\mathfrak {S}}_{w^{(n)}_0}$ and ${\mathfrak {S}}_{w^{(n+1)}_0}$ are consistent. Using $w_0^{(n+1)}\doteq w_0^{(n)} s_n \dotsm s_2 s_1$ we have $A_n \dotsm A_2 A_1(x_1^nx_2^{n-1}\dotsm x_{n}^1) = x_1^{n-1}x_2^{n-2}\dotsm x_{n-1}^1$.

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 ${\mathbb {Q}}$.

Proof. For uniqueness, by induction we may assume that ${\mathfrak {S}}_{ws_i}(x_+)$ is uniquely determined for all $i$ such that $ws_i<w$. Since the applications of all the $A_i$ are specified on ${\mathfrak {S}}_w$, the difference of any two solutions of (2.15), being in the kernel of all $A_i$, is $S_+$-invariant by Lemma 2.3. But ${\mathbb {Q}}[x_+]^{S_+} = {\mathbb {Q}}$, so the homogeneity assumption implies that the two solutions must be equal.

For existence, we note that the Schubert polynomials satisfy (2.13)–(2.15) when $ws_i<w$. When $ws_i>w$, we have ${\mathfrak {S}}_w=A_i {\mathfrak {S}}_{ws_i}$ by (2.15) applied for $ws_i$. The element ${\mathfrak {S}}_w$, being in the image of $A_i$, is $s_i$-invariant and therefore is in $\ker A_i$ by Lemma 2.3. That is, $A_i {\mathfrak {S}}_w=0$, establishing (2.15).

The basis property holds by Proposition 2.6.

Remark 2.8 All the basis theorems for Schubert polynomials and their relatives, such as Theorem 2.7, hold over ${\mathbb {Z}}$.

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}

Proof. This is easily verified using Proposition 2.11.

3. Back stable Schubert polynomials

We define the ring of back symmetric formal power series, and study the basis of back stable Schubert polynomials.

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}

Proof. It is straightforward to verify that $\overleftarrow {R}$ is a ${\mathbb {Q}}$-subalgebra of $R$ containing $\Lambda$ and ${\mathbb {Q}}[x]$. Suppose $f\in R$ is back symmetric. Let $b\in {\mathbb {Z}}$ be such that $s_i(f)=f$ for all $i< b$. Then $f \in \Lambda (x_{\leqslant b}) \otimes {\mathbb {Q}}[x_{b+1},x_{b+2},\dotsc ]$ is a polynomial in the power sums $p_k(x_{\leqslant b})$ and the variables $x_{b+1},x_{b+2},\dotsc$. But $p_k(x_{\leqslant b}) - p_k(x_{\leqslant 0}) \in {\mathbb {Q}}[x]$. It follows that $f\in \Lambda \otimes {\mathbb {Q}}[x]$.

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}

Proof. The well-definedness of the series and its monomial expansion follows by taking the limit of (2.10). Let $w\in S_{\mathbb {Z}}$. For $i\ll 0$ we have $ws_i>w$. By (2.15) and Lemma 2.3, $\overleftarrow {\mathfrak {S}}_w$ is $s_i$-symmetric. Thus $\overleftarrow {\mathfrak {S}}_w$ is back symmetric.

Properties (3.6), (3.7) and (3.8) hold for $\overleftarrow {\mathfrak {S}}_w$ by the corresponding parts of Theorem 2.7 for usual Schubert polynomials.

Proposition 3.3 For $w \in S_{\mathbb {Z}}$, we have $\gamma (\overleftarrow {\mathfrak {S}}_w) = \overleftarrow {\mathfrak {S}}_{\gamma (w)}$.

Proposition 3.4 For $\lambda \in {\mathbb {Y}}$, we have $\overleftarrow {\mathfrak {S}}_{w_\lambda } = s_\lambda \in \Lambda ( x_-)$, the Schur function.

Proof. Let $0<k<n$ be large enough such that $\lambda$ is contained in the $k \times (n-k)$ rectangular partition. For such partitions the map $\lambda \mapsto \gamma ^k(w_\lambda )$ defines a bijection to the $k$-Grassmannian elements of $S_n$. It is well known that $\mathfrak {S}_{\gamma ^k(w_\lambda )} = s_\lambda (x_1,\dotsc ,x_k)$ [Reference FultonFul97, Chapter 10, Proposition 8]. Applying $\gamma ^{-k}$ we have $\mathfrak {S}^{[1-k,n-k]}_{w_\lambda } = s_\lambda (x_{1-k}\dotsc ,x_{-1},x_0)$. The result follows by letting $k,n\to \infty$.

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}

Theorem 3.5 The back stable Schubert polynomials form a ${\mathbb {Q}}$-basis of $\overleftarrow {R}$.

Proof. By (3.9) the back stable Schubert polynomials are linearly independent. For spanning, using Proposition 3.3 and applying $\gamma ^n$ for $n$ sufficiently large, it suffices to show that any element of $\Lambda ( x_-) \otimes {\mathbb {Q}}[x_+]$ is a ${\mathbb {Q}}$-linear combination of finitely many back stable Schubert polynomials. This holds due to the unitriangularity (3.9) of back stable Schubert polynomials with monomials and the following facts: (i) the reverse-lex leading monomial $x^\beta$ in any nonzero element of $\Lambda ( x_-)\otimes {\mathbb {Q}}[x_+]$ satisfies $\dotsm \leqslant \beta _{-2} \leqslant \beta _{-1} \leqslant \beta _0$; (ii) if $w\in S_{\mathbb {Z}}$ is such that $c(w)=\beta$ for such a $\beta$, then $\dotsm <w(-2)<w(-1)<w(0)$; (iii) for such $w$, $\overleftarrow {\mathfrak {S}}_w$ is symmetric in $x_-$ so that $\overleftarrow {\mathfrak {S}}_w\in \Lambda ( x_-)\otimes {\mathbb {Q}}[x_+]$; (iv) there are finitely many $\gamma$ below $\beta$ in reverse-lex order such that $x^\gamma$ and $x^\beta$ have the same degree, and satisfying $\dotsm \leqslant \gamma _{-2}\leqslant \gamma _{-1}\leqslant \gamma _0$.

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.

Remark 3.6 The map $\eta _0$ ‘knows’ the difference between $x_i\in {\mathbb {Q}}[x]$ and the $x_i$ that appear in $\Lambda =\Lambda ( x_-)$.

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.

Lemma 3.7 For $f \in \overleftarrow {R}$, we have $\eta _0(\gamma (f))=\eta _0(f)$.

Proof. This holds since $\eta _0$ is a $\mathbb {Q}$-algebra homomorphism and the claim is easily verified for the algebra generators of $\overleftarrow {R}$.

Corollary 3.8 For $w\in S_{\mathbb {Z}}$, we have $F_{\gamma (w)} = F_w$.

Proof. Using Lemma 3.7 and Proposition 3.3, we have $F_{\gamma (w)} = \eta _0(\overleftarrow {\mathfrak {S}}_{\gamma (w)}) = \eta _0(\gamma (\overleftarrow {\mathfrak {S}}_w)) = \eta _0(\overleftarrow {\mathfrak {S}}_w) = F_w$.

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}

Proof. By Corollary 3.8 we may assume that $w\in S_+$. Since $ws_i>w$ for $i<0$, $\overleftarrow {\mathfrak {S}}_w$ is $s_i$-symmetric for $i<0$, that is, $\overleftarrow {\mathfrak {S}}_w \in \Lambda \otimes {\mathbb {Q}}[x_+]$. Therefore $F_w$ is obtained from $\overleftarrow {\mathfrak {S}}_w$ by setting $x_i=0$ for $i\geqslant 1$. Making this substitution in (3.5) yields (3.11).

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 3.10 For $w\in S_{\mathbb {Z}}$, we have $F_{w^{-1}} = \omega (F_w) = F_{\omega (w)}$.

Proof. Reversal of a reduced word gives a bijection $\textrm {Red}(w) \to \textrm {Red}(w^{-1})$ that sends a Coxeter–Knuth class of shape $\lambda$ (see § 5.8) to a Coxeter–Knuth class of shape $\lambda '$. The first equality follows.

Negating each entry of a reduced word gives a bijection $\textrm {Red}(w) \to \textrm {Red}(\omega (w))$ which sends a Coxeter–Knuth class of shape $\lambda$ to a Coxeter–Knuth class of shape $\lambda '$. The second equality follows.

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).

Example 3.12 For $u=s_{-3}s_{-2}s_{-1}$, we have $\omega (u)=s_3s_2s_1$, $\mathfrak {S}_{s_3s_2s_1}(x_+) = x_1^3$, and $\mathfrak {S}_u( x_-)=-x_0^3$. For $i > 0$ we have $\mathfrak {S}_{s_{-i}} = \omega \mathfrak {S}_{s_i} =\omega (x_1+\dotsm +x_i) = -(x_0+x_{-1}+\dotsm +x_{1-i})$.

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}

Proposition 3.13 For $w \in S_{\ne 0}$, we have $\omega (\mathfrak {S}_w(x)) = \mathfrak {S}_{\omega (w)}(x)$.

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.

Proposition 3.15 For all $w\in S_{\mathbb {Z}}$, we have $\omega (\overleftarrow {\mathfrak {S}}_w) = \overleftarrow {\mathfrak {S}}_{\omega (w)}$.

Proof. This follows immediately from Theorem 3.14 and Propositions 3.10 and 3.13.

Remark 3.16 The elements $\{s_\lambda \otimes {\mathfrak {S}}_v \mid \lambda \in {\mathbb {Y}} \text { and } v \in S_{\neq 0}\}$ form a ${\mathbb {Q}}$-basis of $\overleftarrow {R}$. It follows from Theorem 3.14 that the coefficient of $s_\lambda \otimes {\mathfrak {S}}_v$ in $\overleftarrow {\mathfrak {S}}_w$ is equal to $j_\lambda ^{wv^{-1}}$ if $\ell (wv^{-1}) = \ell (w) - \ell (v)$, and 0 otherwise.

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}

Proposition 3.18

  1. (i) For $u,v,w\in S_+$, we have $c^w_{uv} = \overleftarrow {c}^w_{uv}$.

  2. (ii) Every back stable Schubert structure constant is a usual Schubert structure constant.

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).

Example 3.19 We have $\overleftarrow {\mathfrak {S}}_{s_1}^2 = \overleftarrow {\mathfrak {S}}_{s_2s_1} + \overleftarrow {\mathfrak {S}}_{s_0s_1}$ and ${\mathfrak {S}}_{s_1}^2 = {\mathfrak {S}}_{s_2s_1}$. Shifting forward by one we obtain $\overleftarrow {\mathfrak {S}}_{s_2}^2 = \overleftarrow {\mathfrak {S}}_{s_3s_2}+\overleftarrow {\mathfrak {S}}_{s_1s_2}$ and ${\mathfrak {S}}_{s_2}^2 = {\mathfrak {S}}_{s_3s_2}+{\mathfrak {S}}_{s_1s_2}$.

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. Back stable double Schubert polynomials

We define the back symmetric double power series ring, and study the basis of double back stable Schubert polynomials.

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.

Lemma 4.1 We have $p_k(x||a)|_w = \sum _{i\in I_{w,+}} a_i^k - \sum _{i\in I_{w,-}} a_i^k$.

Example 4.2 Using $w=w_\lambda$ of Example 2.2 we have $I_{w,+}=\{1,3\}$ and $I_{w,-}=\{-1,0\}$. Therefore $p_k(x||a)|_w = a_1^k+a_3^k-a_{-1}^k-a_0^k$.

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 4.4 For $w \in S_{\mathbb {Z}}$, we have $\gamma (\overleftarrow {\mathfrak {S}}_w(x;a)) = \overleftarrow {\mathfrak {S}}_{\gamma (w)}(x;a)$.

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