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Involution pipe dreams

Published online by Cambridge University Press:  14 May 2021

Zachary Hamaker
Department of Mathematics, University of Florida, Gainesville, FL, USA e-mail:
Eric Marberg*
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
Brendan Pawlowski
Department of Mathematics, University of Southern California, Los Angeles, CA, USA e-mail:
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Involution Schubert polynomials represent cohomology classes of K-orbit closures in the complete flag variety, where K is the orthogonal or symplectic group. We show they also represent $\mathsf {T}$ -equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$ , and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey–Jockusch–Stanley formula for Schubert polynomials. In Knutson and Miller’s approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.

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1 Introduction

One can identify the equivariant cohomology rings for the spaces of symmetric and skew-symmetric complex matrices with multivariate polynomial rings. Under this identification, we show that the classes of certain natural subvarieties of (skew-)symmetric matrices are given by the involution Schubert polynomials introduced by Wyser and Yong in [Reference Wyser and Yong45]. These classes of varieties generalize various others studied in the settings of degeneracy loci and combinatorial commutative algebra, for instance the (skew-)symmetric determinantal varieties studied by Harris and Tu [Reference Harris and Tu16].

Involution Schubert polynomials have a combinatorial formula for their monomial expansion [Reference Hamaker, Marberg and Pawlowski13]. As a consequence of our geometric results, they must also expand as sums of products of binomials $x_i + x_j$ . We give a combinatorial description of these expansions, which is a new analogue of the classic Billey–Jockusch–Stanley expansion for ordinary Schubert polynomials [Reference Billey, Jockusch and Stanley3]. This description is far more compact than the monomial expansion. Our formulas involve novel objects that we call involution pipe dreams. Involution pipe dreams appear to be the fundamental objects necessary to replicate Knutson and Miller’s program [Reference Knutson and Miller23] to understand our varieties from a commutative algebra perspective.

1.1 Three flavors of matrix Schubert varieties

Fix a positive integer n. Let $\mathsf {GL}_n$ denote the general linear group of complex $n\times n$ invertible matrices, and write $\mathsf {B}$ and $\mathsf {B}^+$ for the Borel subgroups of lower- and upper-triangular matrices in $\mathsf {GL}_n$ . Our work aims to extend what is known about the geometry of the $\mathsf {B}$ -orbits on matrix space to symmetric and skew-symmetric matrix spaces.

We begin with some classical background. Consider the type A flag variety $\textsf {Fl}_{n} = \mathsf {B} \backslash \mathsf {GL}_n$ . The subgroup $\mathsf {B}^+$ acts on $\textsf {Fl}_{n} $ with finitely many orbits, which are naturally indexed by permutations w in the symmetric group $S_n$ of permutations of $\{1,2,\dots ,n\}$ . These orbits afford a CW decomposition of $\textsf {Fl}_{n}$ , so the cohomology classes of their closures $X_w$ , the Schubert varieties, form a basis for the integral singular cohomology ring $H^*(\textsf {Fl}_{n})$ . Borel’s isomorphism explicitly identifies $H^*(\textsf {Fl}_{n})$ with a quotient of the polynomial ring $\mathbb {Z}[x_1,\dots ,x_n]$ , and the Schubert polynomials $\mathfrak {S}_w \in \mathbb {Z}[x_1,\dots ,x_n]$ are (nonunique) representatives for the Schubert classes $[X_w] \in H^*(\textsf {Fl}_{n})$ .

The maximal torus $\mathsf {T}$ of diagonal matrices in $\mathsf {GL}_n$ also acts on $\textsf {Fl}_{n}$ , so we can instead consider the equivariant cohomology ring $H^*_{\mathsf {T}}(\textsf {Fl}_{n})$ . Via an extension of Borel’s isomorphism, this ring is isomorphic to a quotient of $\mathbb {Z}[x_1, \ldots , x_n, y_1, \ldots , y_n]$ . Lascoux and Schützenberger [Reference Lascoux and Schützenberger27] introduced the double Schubert polynomials $\mathfrak {S}_w(x,y)$ to represent the equivariant classes $[X_w]_{\mathsf {T}}\in H^*_{\mathsf {T}}(\textsf {Fl}_{n})$ . These representatives are distinguished in the following sense.

Let $\textsf {Mat}_n$ be the set of $n\times n$ complex matrices and write $\iota :\mathsf {GL}_n \hookrightarrow \textsf {Mat}_n$ for the obvious inclusion. The product group $ \mathsf {T} \times \mathsf {T}$ acts on $A\in \textsf {Mat}_n$ by $(t_1, t_2) \cdot A = t_1 A t_2^{-1}$ . The matrix Schubert variety of a permutation $w \in S_n$ is $MX_w = \overline {\iota (X_w)}$ . Since $M_n$ is $\mathsf {T} \times \mathsf {T}$ -equivariantly contractible, $H^*_{\mathsf {T} \times \mathsf {T}}(\textsf {Mat}_n) \cong H^*_{\mathsf {T} \times \mathsf {T}}(\text {point}) \cong \mathbb {Z}[x_1,\dots ,x_n,y_1,\dots ,y_n]$ . The launching point for Knutson and Miller’s program is the following theorem:

Theorem 1.1 [Reference Knutson and Miller23]

For all $w \in S_n$ , we have $\mathfrak {S}_w(x,y) = [MX_w] \in H^*_{\mathsf {T} \times \mathsf {T}}(\textsf {Mat}_n)$ .

As mentioned in the historical notes at the end of [Reference Miller and Sturmfels36, Chapter 15], Theorem 1.1 is equivalent to Fulton’s characterization of each $\mathfrak {S}_w(x,y)$ as the class of a certain degeneracy locus for vector bundle morphisms [Reference Fulton11].

Our results are related to the geometry of certain spherical varieties studied by Richardson and Springer in [Reference Richardson and Springer40]. Specifically, define the orthogonal group $\mathsf {O}_n$ as the subgroup of $\mathsf {GL}_n$ preserving a fixed nondegenerate symmetric bilinear form on $\mathbb {C}^n$ , and when n is even define the symplectic group $\mathsf {Sp}_n$ as the subgroup of $\mathsf {GL}_n$ preserving a fixed nondegenerate skew-symmetric bilinear form.

We consider the actions of $\mathsf {O}_n$ and $\mathsf {Sp}_n$ (when n is even) on $\mathsf {Fl}_{n}$ . The associated orbit closures $\hat X_y$ and $\hat X^{\mathsf {{FPF}}}_z$ are indexed by arbitrary involutions y and fixed-point-free involutions z in $S_n$ . Let $\kappa (y)$ denote the number of two-cycles in an involution $y=y^{-1} \in S_n$ . Wyser and Yong [Reference Wyser and Yong45] constructed certain polynomials $\hat {\mathfrak {S}}_y, \hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z \in \mathbb {Z}[x_1,\dots ,x_n]$ and showed that the classes $[\hat X_y]$ and $[\hat X^{\mathsf {{FPF}}}_z]$ are represented in $H^*(\mathsf {Fl}_{n})$ by $2^{\kappa (y)}\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ . We refer to $\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ as involution Schubert polynomials; for their precise definitions, see Section 2.1.

Write $\mathsf {SMat}_n$ and $\mathsf {SSMat}_n$ for the sets of symmetric and skew-symmetric $n\times n$ complex matrices. Let $t \in \mathsf {T}$ act on these spaces by $t\cdot A = tAt$ . One can identify the $\mathsf {T}$ -equivariant cohomology rings of both spaces with $\mathbb {Z}[x_1, \ldots , x_n]$ ; see the discussion in Section 2.2. For each involution $y \in S_n$ , let $M\hat X_y = MX_y \cap \mathsf {SMat}_n$ . Similarly, for each fixed-point-free involution $z \in S_n$ , let $M\hat X^{\mathsf {{FPF}}}_z = MX_z \cap \mathsf {SSMat}_n$ . Our first main result is a (skew-)symmetric analogue of Theorem 1.1:

Theorem 1.2 For all involutions y and fixed-point-free involution z in $S_n$ , we have $ 2^{\kappa (y)}\hat {\mathfrak {S}}_y = [M\hat X_y] \in H_{\mathsf {T}}^*(\mathsf {SMat}_n) $ and $ \hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z = [M\hat X^{\mathsf {{FPF}}}_z] \in H_{\mathsf {T}}^*(\mathsf {SSMat}_n). $ Thus, involution Schubert polynomials are also equivariant cohomology representatives for symmetric and skew-symmetric matrix varieties.

Our proof of this theorem appears in Section 2.3. An extension of Theorem 1.2 to complex K-theory appears in [Reference Marberg and Pawlowski31]. Theorem 1.2 was first announced in a conference proceedings before the appearance of the preprint version of [Reference Marberg and Pawlowski31], which precedes the preprint version of this article. The proof of Theorem 1.2 is a special case of results of [Reference Marberg and Pawlowski31].

Remark Another family of varieties in $\mathsf {SMat}_n$ indexed by permutations in $S_n$ has been studied by Fink et al. [Reference Fink, Rajchgot and Sullivant6]. However, their varieties are cut out by northeast rank conditions, while $M\hat X_y$ and $M\hat X^{\mathsf {{FPF}}}_z$ are cut out by northwest rank conditions (see (2.3) and (2.4) in Section 2.3). The varieties in [Reference Fink, Rajchgot and Sullivant6] are closely related to type C Schubert calculus and generally do not coincide with our $M\hat X_y$ varieties.

1.2 Three flavors of pipe dreams

If Z is a closed subvariety of $\mathsf {SMat}_n$ or $\mathsf {SSMat}_n$ , then its $\mathsf {T}$ -equivariant cohomology class is a positive integer combination of products of binomials $x_i + x_j$ (see Corollary 2.10). Our second main result gives a combinatorial description of such an expansion for $\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ .

Let $[n] = \{1, 2, \ldots , n\}$ and . Consider a subset . One associates to D a wiring diagram by replacing the cells by tiles of two types, given either by a crossing of two paths (drawn as a tile) if $(i,j) \in D$ or by two paths bending away from each other (drawn as a tile) if $(i,j) \notin D$ . Connecting the endpoints of adjacent tiles yields a union of n continuously differentiable paths, which we refer to as “pipes.” For example:


Definition 1.3 A subset is a reduced pipe dream if no two pipes in the associated wiring diagram cross more than once.

This condition holds in the example (1.1). Pipe dreams as described here were introduced by Bergeron and Billey [Reference Bergeron and Billey1], inspired by related diagrams of Fomin and Kirillov [Reference Fomin and Kirillov9]. Bergeron and Billey originally referred to pipe dreams as reduced-word compatible sequence graphs or rc-graphs for short.

A reduced pipe dream D determines a permutation $w \in S_n$ in the following way. Label the left endpoints of the pipes in D’s wiring diagram by $1, 2, \ldots , n$ from top to bottom, and the top endpoints by $1, 2, \ldots , n$ from left to right. Then the associated permutation $w \in S_n$ is the element such that the pipe with left endpoint i has top endpoint $w(i)$ . For instance, the permutation of $D=\{(1,3),(2,1)\}$ is $w = 1423 \in S_4$ . Let $\mathcal {PD}(w)$ denote the set of all reduced pipe dreams associated to $w \in S_n$ .

Pipe dreams are of interest for their role in formulas for $\mathfrak {S}_w$ and $\mathfrak {S}_w(x,y)$ . Lascoux and Schützenberger’s original definition of these Schubert polynomials in [Reference Lascoux and Schützenberger28] is recursive in terms of divided difference operators. However, by results of Fomin and Stanley [Reference Fomin and Stanley10, Section 4] we also have

(1.2) $$ \begin{align} \mathfrak{S}_w = \sum_{D \in \mathcal{PD}(w)} \prod_{(i,j) \in D} x_i \quad\text{and}\quad \mathfrak{S}_w(x,y) = \sum_{D \in \mathcal{PD}(w)} \prod_{(i,j) \in D} (x_i - y_j). \end{align} $$

The first identity is the Billey–Jockusch–Stanley formula for Schubert polynomials [Reference Billey, Jockusch and Stanley3].

There are analogues of this formula for the involution Schubert polynomials $\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ , which involve the following new classes of pipe dreams. A reduced pipe dream is symmetric if $(i,j) \in D$ implies $(j,i) \in D$ , and almost-symmetric if both of the following properties hold:

  • If $(i, j) \in D$ where $i< j$ then $(j,i) \in D$ .

  • If $(j, i) \in D$ where $i< j$ but $(i, j) \notin D$ , then the pipes crossing at $(j, i)$ in the wiring diagram of D are also the pipes that avoid each other at $(i, j)$ .

Equivalently, D is almost-symmetric if it is as symmetric as possible while respecting the condition that no two pipes cross twice, and any violation of symmetry forced by this condition takes the form of a crossing $(j,i)$ below the diagonal rather than at the transposed position $(i,j)$ .

Let $\mathcal {I}_n=\{ w \in S_n : w=w^{-1}\}$ and write $\mathcal {I}^{\mathsf {{FPF}}}_n$ for the subset of fixed-point-free elements of $\mathcal {I}_n$ . Note that n must be even for $\mathcal {I}^{\mathsf {{FPF}}}_n$ to be nonempty. Also let

Definition 1.4 The set of involution pipe dreams for $y \in \mathcal {I}_n$ is

The set of fpf-involution pipe dreams for $z \in \mathcal {I}^{\mathsf {{FPF}}}_{n}$ is

By convention, (fpf-)involution pipe dreams are always instances of reduced pipe dreams. It would be more precise to call our objects “reduced involution pipe dreams,” but since we will never consider any pipe dreams that are unreduced, we opt for more concise terminology.

We now state our second main result, which will reappear as Theorems 4.25 and 4.36.

Theorem 1.5 If $y \in \mathcal {I}_n$ and $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ then

$$ \begin{align*} \hat{\mathfrak{S}}_y = \sum_{D \in \mathcal{ID}(y)} \prod_{(i,j) \in D} 2^{-\delta_{ij}}(x_i + x_j) \quad\text{and}\quad \hat{\mathfrak{S}}^{\mathsf{{FPF}}}_z = \sum_{D \in \mathcal{FD}(z)} \prod_{(i,j) \in D} (x_i + x_j), \end{align*} $$

where $\delta _{ij}$ denotes the usual Kronecker delta function.

Example 1.6 The involution $y = 1432 = (2,4) \in \mathcal {I}_4$ has five reduced pipe dreams:

Only the last two of these are almost-symmetric, so $|\mathcal {ID}(y)| = 2$ and Theorem 1.5 reduces to the formula $\hat {\mathfrak {S}}_{y} = (x_2+x_1)(x_3+x_1) + (x_2+x_1)(x_2 + x_2)/2 = (x_2 + x_1)(x_3 + x_1 +x_2)$ . The monomial expansion has six terms, as opposed to two. In general, the expansion in Theorem 1.5 uses roughly a factor of $2^{\deg \hat {\mathfrak {S}}_y}$ fewer terms.

Remark There is an alternate path toward establishing the fact that the class of a matrix Schubert variety is represented by the weighted sum of reduced pipe dreams. The defining ideal of $MX_w$ has a simple set of generators due to Fulton [Reference Fulton11]. Knutson and Miller showed that Fulton’s generators form a Gröbner basis with respect to any anti-diagonal term order [Reference Knutson and Miller23]. The Gröbner degeneration of this ideal decomposes into a union of coordinate subspaces indexed by reduced pipe dreams. Our hope is that a similar program can be implemented in the (skew-)symmetric setting, which would give a geometric proof of Theorem 1.5. We discuss this in greater detail in Section 6.2.

In addition to Theorem 1.5, we also prove a number of results about the properties of involution pipe dreams. An outline of the rest of this article is as follows.

Section 2 contains some preliminaries on involution Schubert polynomials along with a proof of Theorem 1.2. In Section 3, we give several equivalent characterizations of $\mathcal {ID}(y)$ and $\mathcal {FD}(z)$ in terms of reduced words for permutations. Section 4 contains our proof of Theorem 1.5, which uses ideas from recent work of Knutson [Reference Knutson22] along with certain transition equations for $\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ given in [Reference Hamaker, Marberg and Pawlowski14]. In Section 5, we show that both families of involution pipe dreams are obtained from distinguished “bottom” elements by repeatedly applying certain simple transformations. These transformations are extensions of the ladder moves for pipe dreams described by Bergeron and Billey in [Reference Bergeron and Billey1]. In Section 6, finally, we describe several related open problems and conjectures.

2 Schubert polynomials and matrix varieties

Everywhere in this paper, n denotes a fixed positive integer. For convenience, we realize the symmetric group $S_n$ as the group of permutations of $\mathbb {Z}_{>0} = \{1,2,3,\dots \}$ fixing all $i>n$ , so that there is an automatic inclusion $S_n \subset S_{n+1}$ . In this section, we present some relevant background on involution Schubert polynomials and equivariant cohomology, and then prove Theorem 1.2.

2.1 Involution Schubert polynomials

To start, we provide a succinct definition of $\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ in terms of the ordinary Schubert polynomials $\mathfrak {S}_w$ given by (1.2). Let $s_i = (i,i+1) \in S_n$ for each $i \in [n-1]$ . A reduced word for $w \in S_n$ is a minimal-length sequence $a_1a_2 \cdots a_l$ such that $w=s_{a_1}s_{a_2} \cdots s_{a_l}$ . Let $\mathcal {R}(w)$ denote the set of reduced words for w. The length $\ell (w)$ of $w \in S_n$ is the length of any word in $\mathcal {R}(w)$ . One has $\ell (ws_i)=\ell (w)+1>\ell (w)$ if and only if $w(i)< w(i+1)$ .

Proposition 2.1 [Reference Humphreys20, Theorem 7.1]

There is a unique associative operation $\circ : S_n \times S _n \to S_n$ , called the Demazure product, with $s_i \circ s_i = s_i$ for all $i \in [n-1]$ and $v\circ w = vw$ for all $v,w \in S_n$ with $\ell (vw) = \ell (v)+\ell (w)$ .

An involution word for $y \in \mathcal {I}_n = \{ w \in S_n : w=w^{-1}\}$ is a minimal-length word $a_1a_2 \cdots a_l$ with

(2.1) $$ \begin{align} y = s_{a_l} \circ \cdots \circ s_{a_2}\circ s_{a_1} \circ 1 \circ s_{a_1} \circ s_{a_2} \circ \cdots \circ s_{a_l}. \end{align} $$

Note that we could replace $s_{a_1}\circ 1 \circ s_{a_1}$ in this expression by $s_{a_1} = s_{a_1}\circ s_{a_1}= s_{a_1}\circ 1 \circ s_{a_1}$ . An atom for $y \in \mathcal {I}_n$ is a minimal-length permutation $w \in S_n$ with $ y = w^{-1} \circ w. $ Let $\hat {\mathcal {R}}(y)$ be the set of involution words for $y \in \mathcal {I}_n$ and let $\mathcal {A}(y)$ be the set of atoms for y. The associativity of the Demazure product implies that $ \hat {\mathcal {R}}(y) = \bigsqcup _{w \in \mathcal {A}(y)} \mathcal {R}(w) $ .

Example 2.2 If $y = 1432 $ then $\hat {\mathcal {R}}(y) = \{ 23, 32\}$ and $\mathcal {A}(y) = \{1342, 1423\}$ .

One can show that $ \mathcal {I}_n = \{ w^{-1} \circ w : w \in S_n \} $ , so $\hat {\mathcal {R}}(y)$ and $\mathcal {A}(y)$ are nonempty for all $y \in \mathcal {I}_n$ . Involution words are a special case of a more general construction of Richardson and Springer [Reference Richardson and Springer40], and have been studied by various authors [Reference Can, Joyce and Wyser5Reference Hansson and Hultman15Reference Hu and Zhang17Reference Hultman18]. Our notation follows [Reference Hamaker, Marberg and Pawlowski12Reference Hamaker, Marberg and Pawlowski13].

Definition 2.3 The involution Schubert polynomial of $y \in \mathcal {I}_n$ is $ \hat {\mathfrak {S}}_y = \sum _{w \in \mathcal {A}(y)} \mathfrak {S}_w. $

Wyser and Yong [Reference Wyser and Yong45] originally defined these polynomials recursively using divided difference operators; work of Brion [Reference Brion4] implies that our definition agrees with theirs. For a detailed explanation of the equivalence among these definitions, see [Reference Hamaker, Marberg and Pawlowski13].

Example 2.4 If $z = 1432 \in \mathcal {I}_4$ then $\mathcal {A}(z) = \{1342, 1423\}$ and

$$ \begin{align*} \hat{\mathfrak{S}}_{z} = \mathfrak{S}_{1342} + \mathfrak{S}_{1423} = (x_2 x_3 + x_1 x_3 + x_1 x_2) + (x_2^2 + x_1 x_2 + x_1^2). \end{align*} $$

Assume n is even, so $\mathcal {I}^{\mathsf {{FPF}}}_n = \{ z \in \mathcal {I}_n: i \neq z(i)\text { for all}\ i \in [n]\}$ is nonempty, and let

$$ \begin{align*}1^{\mathsf{{FPF}}}_n = 2143\dots n\ n{-}1 = s_1s_3\cdots s_{n-1} \in \mathcal{I}^{\mathsf{{FPF}}}_n.\end{align*} $$

An fpf-involution word for $z \in \mathcal {I}^{\mathsf {{FPF}}}_{n}$ is a minimal-length word $a_1a_2 \cdots a_l$ with

$$ \begin{align*} z = s_{a_l} \cdots s_{a_2} s_{a_1} 1^{\mathsf{{FPF}}}_n s_{a_1} s_{a_2} \cdots s_{a_l} .\end{align*} $$

This formulation avoids the Demazure product, but there is an equivalent definition that more closely parallels (2.1). Namely, by [Reference Hamaker, Marberg and Pawlowski12, Corollary 2.6], an fpf-involution word for $z \in \mathcal {I}^{\mathsf {{FPF}}}_{n}$ is also a minimal-length word $a_1a_2 \cdots a_l$ with

$$ \begin{align*} z = s_{a_l} \circ \cdots \circ s_{a_2} \circ s_{a_1} \circ 1^{\mathsf{{FPF}}}_n \circ s_{a_1} \circ s_{a_2} \circ \cdots \circ s_{a_l} .\end{align*} $$

An fpf-atom for $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ is a minimal length permutation $w \in S_n$ with $z = w^{-1} 1^{\mathsf {{FPF}}}_n w.$ Let $\mathcal {A}^{\mathsf {{FPF}}}(z)$ be the set of fpf-atoms for z, and let $ \hat {\mathcal {R}}^{\mathsf {{FPF}}}(z) $ be the set of fpf-involution words for z. The basic properties of reduced words imply that

$$ \begin{align*} \hat{\mathcal{R}}^{\mathsf{{FPF}}}(z) = \bigsqcup_{w \in \mathcal{A}^{\mathsf{{FPF}}}(z)} \mathcal{R}(w). \end{align*} $$

Example 2.5 If $z = 4321 $ then $\hat {\mathcal {R}}^{\mathsf {{FPF}}}(z) = \{ 23, 21\}$ and $\mathcal {A}^{\mathsf {{FPF}}}(z) = \{1342, 3124\}$ .

Note that $a_1a_2\cdots a_l $ belongs to $\hat {\mathcal {R}}^{\mathsf {{FPF}}}(z)$ if and only if $135\cdots (n-1)a_1a_2\cdots a_l $ belongs to $\hat {\mathcal {R}}(z)$ . If $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ then $\hat {\mathcal {R}}^{\mathsf {{FPF}}}(z) = \hat {\mathcal {R}}^{\mathsf {{FPF}}}(zs_{n+1}) $ and $ \mathcal {A}^{\mathsf {{FPF}}}(z) = \mathcal {A}^{\mathsf {{FPF}}}(zs_{n+1}).$

Fpf-involution words are special cases of reduced words for quasiparabolic sets [Reference Rains and Vazirani39]. Since $\mathcal {I}^{\mathsf {{FPF}}}_n$ is a single $S_n$ -conjugacy class, each $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ has at least one fpf-involution word and fpf-atom.

Definition 2.6 The fpf-involution Schubert polynomial of $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ is

$$ \begin{align*} \hat{\mathfrak{S}}^{\mathsf{{FPF}}}_z = \sum_{w \in \mathcal{A}^{\mathsf{{FPF}}}(z)} \mathfrak{S}_w. \end{align*} $$

These polynomials were also introduced in [Reference Wyser and Yong45]. If $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ then $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_{z} =\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_{zs_{n+1}}$ .

Example 2.7 If $z = 532614 \in \mathcal {I}^{\mathsf {{FPF}}}_6$ then $\mathcal {A}^{\mathsf {{FPF}}}(z) = \{13452, 31254\}$ and $ \hat {\mathfrak {S}}^{\mathsf {{FPF}}}_{z} = \mathfrak {S}_{13452} + \mathfrak {S}_{31254} = (x_2 x_3 x_4 + x_1 x_3 x_4 + x_1 x_2 x_4 + x_1 x_2 x_3) + (x_1^2 x_4 + x_1^2 x_3 + x_1^2 x_2 + x_1^3). $

2.2 Torus-equivariant cohomology

Suppose V is a finite-dimensional rational representation of a torus $\mathsf {T} \simeq (\mathbb {C}^\times )^n$ . A character $\lambda \in \operatorname {Hom}(\mathsf {T}, \mathbb {C}^\times )$ is a weight of V if the weight space $V_\lambda = \{v \in V : tv = \lambda (t)v\ \text {for all}\ t \in \mathsf {T}\}$ is nonzero. Any nonzero $v \in V_{\lambda }$ is a weight vector, and V has a basis of weight vectors. Let $\operatorname {\mathrm {wt}}(V)$ denote the set of weights of V. After fixing an isomorphism $\mathsf {T} \simeq (\mathbb {C}^\times )^n$ , we identify the character $(t_1, \ldots , t_n) \mapsto t_1^{a_1} \cdots t_n^{a_n}$ with the linear polynomial $a_1 x_1 + \cdots + a_n x_n \in \mathbb {Z}[x_1, \ldots , x_n]$ .

The equivariant cohomology ring $H_{\mathsf {T}}(V)$ is isomorphic to $\mathbb {Z}[x_1, \ldots , x_n]$ , an identification we make without comment from now on. Each $\mathsf {T}$ -invariant subscheme $X \subseteq V$ has an associated class $[X] \in H_{\mathsf {T}}(V)$ , which we describe following [Reference Miller and Sturmfels36, Chapter 8].

First, if X is a linear subspace then we define $[X] = \prod _{\lambda \in \operatorname {\mathrm {wt}}(X)} \lambda $ , where we identify each character $\lambda $ with a linear polynomial as above. More generally, fix a basis of weight vectors of V, and let $z_1, \ldots , z_n \in V^*$ be the dual basis; this determines an isomorphism $\mathbb {C}[V] = \operatorname {Sym}(V^*) \simeq \mathbb {C}[z_1, \ldots , z_n]$ .

Choose a term order on monomials in $z_1, \ldots , z_n$ , and let $\operatorname {\mathrm {\mathsf {init}}}(I)$ denote the ideal generated by the leading terms of all members of a given set $I \subseteq \mathbb {C}[V]$ . Given that $\operatorname {\mathrm {\mathsf {init}}}(I)$ is a monomial ideal, one can show that each of its associated primes $\mathfrak {p}$ is also a monomial ideal, and hence of the form $\langle z_{i_1}, \ldots , z_{i_r} \rangle $ . The corresponding subscheme $Z(\mathfrak {p})$ is a $\mathsf {T}$ -invariant linear subspace of V. Now define

(2.2) $$ \begin{align} [X] = \sum_{\mathfrak{p}} \operatorname{mult}_{\mathfrak{p}}(\operatorname{\mathrm{\mathsf{init}}} I(X)) [Z(\mathfrak{p})], \end{align} $$

where $I(X)$ is the ideal of X and $\mathfrak {p}$ runs over the associated primes of $\operatorname {\mathrm {\mathsf {init}}} I(X)$ .

2.3 Classes of involution matrix Schubert varieties

The matrix Schubert varieties in Theorem 1.1 can be described in terms of rank conditions, namely:

$$ \begin{align*} MX_w = \{A \in \mathsf{Mat}_n: \operatorname{\mathrm{rank}} A_{[i][j]} \leq \operatorname{\mathrm{rank}} w_{[i][j]}\ \text{for}\ i,j \in [n]\}, \end{align*} $$

where $\mathsf {Mat}_n$ is the variety of $n\times n$ matrices, $A_{[i][j]}$ denotes the upper-left $i \times j$ corner of $A \in \mathsf {Mat}_n$ , and we identify $w \in S_n$ with the $n\times n$ permutation matrix having $1$ ’s in positions $(i,w(i))$ .

The varieties $M\hat X_y$ and $M\hat X^{\mathsf {{FPF}}}_z$ from Theorem 1.2 can be reformulated in a similar way. Specifically, we define the involution matrix Schubert variety of $y \in \mathcal {I}_n$ by

(2.3) $$ \begin{align} M\hat X_y &= MX_y \cap \mathsf{SMat}_n \nonumber\\&= \{A \in \mathsf{SMat}_n : \operatorname{\mathrm{rank}} A_{[i][j]} \leq \operatorname{\mathrm{rank}} y_{[i][j]}\ \text{for}\ i,j \in [n]\}, \end{align} $$

where $\mathsf {SMat}_n$ is the subvariety of symmetric matrices in $\mathsf {Mat}_n$ . When n is even, we define the fpf-involution matrix Schubert variety of $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ by

(2.4) $$ \begin{align} M\hat X^{\mathsf{{FPF}}}_z &= MX_z \cap \mathsf{SSMat}_n \nonumber\\&= \{A \in \mathsf{SSMat}_n : \operatorname{\mathrm{rank}} A_{[i][j]} \leq \operatorname{\mathrm{rank}} z_{[i][j]}\ \text{for}\ i,j \in [n]\}, \end{align} $$

where $\mathsf {SSMat}_n$ is the subvariety of skew-symmetric matrices in $\mathsf {Mat}_n$ .

Example 2.8 Suppose $y = 132 =\left [\begin {smallmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end {smallmatrix}\right ]\in \mathcal {I}_3$ . Setting $R_{ij} = \operatorname {\mathrm {rank}} y_{[i][j]}$ , we have $R = \left [\begin {smallmatrix} 1 & 1 & 1 \\ 1 & 1 & 2\\ 1 & 2 & 3 \end {smallmatrix}\right ]$ . The conditions $\operatorname {\mathrm {rank}} A_{[i][j]} \leq R_{ij}$ for $i,j \in [3]$ defining $M\hat X_y$ are all implied by the single condition $\operatorname {\mathrm {rank}} A_{[2][2]} \leq R_{22} = 1$ . Thus,

$$ \begin{align*}M\hat X_y = \left\{\left[\begin{smallmatrix} z_{11} & z_{21} & z_{31} \\ z_{21} & z_{22} & z_{32} \\ z_{31} & z_{32} & z_{33} \end{smallmatrix} \right] : z_{11}z_{22} - z_{21}^2 = 0\right\}.\end{align*} $$

Let $\mathsf {T} \subseteq \mathsf {GL}_n$ be the usual torus of invertible diagonal matrices. Recall that $\kappa (y) = |\{ i : y(i) < i\}|$ for $y \in \mathcal {I}_n$ , and that $ \mathsf {T}$ acts on matrices in $ \mathsf {Mat}_n$ by $t\cdot A = tA$ and on symmetric matrices in $ \mathsf {SMat}_n$ by $t \cdot A = tAt$ . We can now prove Theorem 1.2, which states that if $y \in \mathcal {I}_n$ and $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ then $2^{\kappa (y)}\hat {\mathfrak {S}}_y = [M\hat X_y] \in H_{\mathsf {T}}^*(\mathsf {SMat}_n)$ while $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z = [M\hat X^{\mathsf {{FPF}}}_z] \in H_{\mathsf {T}}^*(\mathsf {SSMat}_n)$ .

Remark It is possible, though a little cumbersome, to derive Theorem 1.2 from [Reference Marberg and Pawlowski31, Theorem 2.17 and Lemma 3.1], which provide a similar statement in complex K-theory. We originally announced Theorem 1.2 in an extended abstract for this paper which preceded the appearance of [Reference Marberg and Pawlowski31]. However, as the argument below is similar to the proofs of the results in [Reference Marberg and Pawlowski31], we will be somewhat curt here in our presentation of the details.

For $w = w_1 \dots w_n \in S_n$ , let $w \times 1^k = w_1 \dots w_n\ n{+}1 \dots n{+}k \in S_{n+k}$ . Similarly, for n even define $w \times (21)^k = w \times 1^{2k} \cdot (1^{\mathsf {{FPF}}}_n \cdot 1^{\mathsf {{FPF}}}_{n+2k})$ . Our proof of Theorem 1.2 relies on the following characterizations of $\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ :

Theorem 2.9 [Reference Wyser and Yong45, Theorem 2]

If $y \in \mathcal {I}_n$ and $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ , then $2^{\kappa (y)}\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ are the unique representatives for $[\hat X_y]$ and $[\hat X^{\mathsf {{FPF}}}_z]$ with $2^{\kappa (y)}\hat {\mathfrak {S}}_y = 2^{\kappa (y)}\hat {\mathfrak {S}}_{y\times 1^k}$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z = \hat {\mathfrak {S}}^{\mathsf {{FPF}}}_{z \times (21)^k}$ for all $k \geq 1$ .

Proof of Theorem 1.2 If X and Y are complex varieties with $\mathsf {T}$ -actions, and $f : X \to Y$ is a $\mathsf {T}$ -equivariant morphism, then there is a pullback homomorphism $f^* : H_{\mathsf {T}}^*(Y) \to H_{\mathsf {T}}^*(X)$ . If f is a flat morphism (e.g., an inclusion of an open subset, a projection of a fiber bundle, or a composition of flat morphisms), then $f^*([Z]) = [f^{-1}(Z)]$ for any subscheme $Z \subseteq Y$ .

Because $\mathsf {T}$ acts freely on $\mathsf {GL}_n$ and since $\mathsf {T}\backslash \mathsf {GL}_n \twoheadrightarrow \mathsf {B} \backslash \mathsf {GL}_n \simeq \mathsf {Fl}_{n}$ is a homotopy equivalence (see, e.g., [Reference McGovern35, Section 8.1]), one has $H_{\mathsf {T}}^*(\mathsf {GL}_n) \simeq H^*(\mathsf {T} \backslash \mathsf {GL}_n) \simeq H^*(\mathsf {Fl}_{n})$ . If $Z \subseteq \mathsf {GL}_n$ is a $\mathsf {B}$ -invariant subvariety, then $[Z] \in H_{\mathsf {T}}^*(\mathsf {GL}_n)$ corresponds to the class of $\mathsf {B}\backslash Z = \{\mathsf {B} g : g \in Z\}$ in $H^*(\mathsf {Fl}_{n})$ . Fix $y \in \mathcal {I}_n$ and define $\sigma : \mathsf {GL}_n \to \mathsf {SMat}_n$ by $\sigma (g) = gg^T$ . Let $\iota : \mathsf {GL}_n \hookrightarrow M_n$ be the obvious inclusion and consider the diagram


Realize $\mathsf {O}_n$ as the group $\{g \in \mathsf {GL}_n : gg^T = 1\}$ . The map $\sigma $ is flat because it is the composition $\mathsf {GL}_n \twoheadrightarrow \mathsf {GL}_n/\mathsf {O}_n \hookrightarrow \mathsf {SMat}_n$ , where the second map sends $g\mathsf {O}_n \mapsto gg^T$ and may be identified with the open inclusion $\mathsf {GL}_n \cap \mathsf {SMat}_n \hookrightarrow \mathsf {SMat}_n$ . For fixed $i \in [n]$ , one checks using the prescription of Section 2.2 that $2x_i$ represents both the class of $Z = \{A \in \mathsf {SMat}_n : A_{ii} = 0\}$ in $H_{\mathsf {T}}^*(\mathsf {SMat}_n)$ and the class of $Z' = \{A \in M_n : (AA^T)_{ii} = 0\}$ in $H_{\mathsf {T}}^*(M_n)$ . Since $\sigma ^*[Z] = [\sigma ^{-1}(Z)] = [\iota ^{-1}(Z')] = \iota ^*[Z']$ , this calculation implies that (2.5) commutes.

Now set $ \hat X_y = \mathsf {B} \backslash \sigma ^{-1}(M\hat X_y) = \{\mathsf {B} g \in \mathsf {Fl}_{n} : \operatorname {\mathrm {rank}} (gg^T)_{[i][j]} \leq \operatorname {\mathrm {rank}} y_{[i][j]} \text { for}\ i,j \in [n]\},$ so that the path through the upper-left corner of (2.5) sends the polynomial $[M\hat X_y]$ to $[\hat X_y]$ . The variety $\hat X_y$ is the closure of an $\mathsf {O}_n$ -orbit on $\mathsf {Fl}_{n}$ [Reference Wyser44, Section 2.1.2]. The path through the lower-right corner of (2.5) is simply the classical Borel map $\mathbb {Z}[x_1, \ldots , x_n] \to H^*(\mathsf {Fl}_{n})$ . We claim $[M\hat X_{y \times 1^m}]$ is constant for fixed y and varying m. Since $[M\hat X_{y}]$ is a representative for $[\hat X_y]$ , the result then follows by Theorem 2.9.

For $y \neq 1 \in S_n$ , define $\operatorname {maxdes}(y) = \max \{i \in \mathbb {Z}_{\geq 0} : y(i)> y(i+1)\}$ . Replacing $[n]$ in the definition (2.3) by $[\operatorname {maxdes}(y)]$ yields exactly the same variety $M\hat X_y$ . Since $\operatorname {maxdes}(y \times 1^m)$ is independent of m, as is $\operatorname {\mathrm {rank}}(y \times 1^m)_{[i][j]}$ for $i, j \in [\operatorname {maxdes}(y)]$ , it follows that the ideals of $M\hat X_{y \times 1^m}$ for fixed y and varying m have a common generating set. It is clear from §2.2 that this means that the polynomial $[M\hat X_{y \times 1^m}]$ is independent of m.

The proof for the skew-symmetric case is the same, replacing $\mathsf {O}_n$ by $\mathsf {Sp}_n$ and the map $\sigma : g \mapsto gg^T$ by $g \mapsto g\Omega g^T$ , where $\Omega \in \mathsf {GL}_n$ is the nondegenerate skew-symmetric form preserved by $\mathsf {Sp}_n$ .

Corollary 2.10 The polynomial $2^{\kappa (y)}\hat {\mathfrak {S}}_y$ (respectively, $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ ) is a positive integer linear combination of products of terms $x_i+x_j$ for $1 \leq i \leq j \leq n$ (respectively, $1 \leq i < j \leq n$ ).

Proof The weights of $\mathsf {T}$ acting on $\mathsf {SMat}_n$ are $x_i+x_j$ for $1 \leq i \leq j \leq n$ , while the weights of $\mathsf {SSMat}_n$ are the same with the added restriction $i < j$ . The expression (2.2) makes clear that the classes $[M\hat X_y]$ and $[M\hat X^{\mathsf {{FPF}}}_z]$ are positive integer linear combinations of products of these weights.▪

Remark Let $\mathsf {S}$ be a maximal torus in $\mathsf {O}_n$ . Let $\mathsf {T} \times \mathsf {S}$ act on $\mathsf {GL}_n$ by $(t,s) \cdot g = tgs^{-1}$ and on $\mathsf {SMat}_n$ by $(t,s) \cdot A = tAt$ . The map $\sigma : \mathsf {GL}_n \to \mathsf {SMat}_n$ , $g \mapsto gg^T$ considered above is then $\mathsf {T} \times \mathsf {S}$ -equivariant. Since the second factor of $\mathsf {T} \times \mathsf {S}$ acts trivially on $\mathsf {SMat}_n$ , the polynomial $2^{\kappa (y)}\hat {\mathfrak {S}}_y$ still represents the class $[M\hat X_y] \in H_{\mathsf {T} \times \mathsf {S}}(\mathsf {SMat}_n)$ . It follows as in the proof of Theorem 1.2 that $2^{\kappa (y)}\hat {\mathfrak {S}}_y$ also represents the class $[\hat X_y]_{\mathsf {S}} \in H_{\mathsf {S}}(\mathsf {Fl}_{n})$ . The latter fact was proven by Wyser and Yong [Reference Wyser and Yong45], but our approach gives an explanation for the surprising existence of a representative for $[\hat X_y]_{\mathsf {S}}$ not involving the $\mathsf {S}$ -weights. Similar remarks apply in the skew-symmetric case.

3 Characterizing pipe dreams

The rest of this article is focused on the combinatorial properties of involution pipe dreams and their role in the formulas in Theorem 1.5 that manifest Corollary 2.10. In the introduction, we defined (fpf-)involution pipe dreams via simple symmetry conditions. In this section, we give an equivalent characterization in terms of “compatible sequences” related to involution words.

3.1 Reading words

For $p \in \mathbb {Z}$ , the $p{th}$ antidiagonal in $\mathbb {Z}_{>0} \times \mathbb {Z}_{>0}$ is the set

$$ \begin{align*}\{(i,j) \in \mathbb{Z}_{>0}\times \mathbb{Z}_{>0} : i+j-1 = p\}.\end{align*} $$

The $p{th}$ diagonal in $\mathbb {Z}_{>0} \times \mathbb {Z}_{>0}$ is the set

$$ \begin{align*}\{(i,j) \in \mathbb{Z}_{>0}\times \mathbb{Z}_{>0}: j-i = p\}.\end{align*} $$

Labeling the elements of $\{1,2,3\}\times \{1,2,3\}$ by their antidiagonal and diagonal gives

$$ \begin{align*} \begin{array}{ccc} 1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5\\ \end{array} \qquad \text{and} \qquad \begin{array}{rrr} 0 & -1 & -2\\ 1 & 0 & -1\\ 2 & 1 & 0 \end{array} \end{align*} $$

respectively. Let $\mathsf {adiag}: \mathbb {Z}_{>0}\times \mathbb {Z}_{>0} \to \mathbb {Z}_{>0}$ be the map sending $(i,j) \mapsto i+j-1$ .

Definition 3.1 The standard reading word of $D\subseteq [n]\times [n]$ is the sequence

$$ \begin{align*} \mathsf{word}(D) = \mathsf{adiag}(\alpha_1) \mathsf{adiag}(\alpha_2) \cdots \mathsf{adiag}(\alpha_{|D|}),\end{align*} $$

where $\alpha _1,\alpha _2,\dots ,\alpha _{|D|}$ are the positions of D read row-by-row from right to left, starting with the top row.

If one also records the row indices of the positions $\alpha _i$ as a second word, then the resulting words uniquely determine D and are the same data as a compatible sequence for $\mathsf {word}(D)$ (see [Reference Billey, Jockusch and Stanley3, (1)]).

Example 3.2 The subset $D= \{ (1,3),(1,2),(2,3),(2,2),(3,2)\}$ has $\mathsf {word}(D) = 32434$ .

We introduce a more general class of reading words. Suppose $\omega : [n] \times [n] \to [n^2]$ is a bijection. For a subset $D \subseteq [n] \times [n]$ with $\omega (D) = \{ i_1 < i_2< \dots < i_{m}\}$ , let

$$ \begin{align*}\mathsf{word}(D,\omega) =\mathsf{adiag}(\omega^{-1}(i_1))\mathsf{adiag}(\omega^{-1}(i_2))\cdots \mathsf{adiag}(\omega^{-1}(i_m)).\end{align*} $$

The standard reading word of $D\subseteq [n]\times [n]$ corresponds to $\omega : (i,j) \mapsto ni - j + 1$ .

Example 3.3 If $n=2$ and $\omega $ is such that $ \left [ \begin {array}{@{}rr@{}} \omega (1,1) & \omega (1,2) \\ \omega (2,1) & \omega (2,2) \end {array}\right ] = \left [ \begin {array}{@{}cc@{}} 3 & 1 \\ 4 & 2 \end {array} \right ] $ then we would have $\mathsf {word}([n]\times [n],\omega ) = 2312$ , while if $D = \{(1,1),(2,2)\}$ then $\mathsf {word} (D,\omega ) = 31$ .

For us, a linear extension of a finite poset $(P,\preceq )$ with size $m=|P|$ is a bijection $\omega : P \to [m]$ such that $\omega (s) < \omega (t)$ whenever $s \prec t$ in P.

Definition 3.4 A reading order on $[n]\times [n]$ is a linear extension of the partial order $\leq _{\mathsf {NE}}$ on $[n] \times [n]$ that has $(i,j) \leq _{\mathsf {NE}} (i',j')$ if and only if both $i \leq i'$ and $j \geq j'$ . If $\omega $ is a reading order, then we refer to $\mathsf {word}(D,\omega )$ as a reading word of $D\subseteq [n]\times [n]$ .

The Coxeter commutation class of a finite sequence of integers is its equivalence class under the relation that lets adjacent letters commute if their positive difference is at least two. For example, $\{1324, 3124,1342,3142, 3412\}$ is a single Coxeter commutation class. Fix a set $D \subseteq [n] \times [n]$ .

Lemma 3.5 All reading words of D are in the same Coxeter commutation class.

This result can be derived using Viennot’s theory of heaps of pieces; see [Reference Viennot43, Lemma 3.3].

Proof Let $s_p \in S_{n^2}$ be the simple transposition interchanging p and $p+1$ , and choose a reading order $\omega $ on $[n]\times [n]$ . The sequence $\mathsf {word}(D, s_p \omega )$ is equal to $\mathsf {word}(D, \omega )$ when $\{p,p+1\}\not \subset \omega (D)$ , and otherwise is obtained by interchanging two adjacent letters in $\mathsf {word}(D,\omega )$ . In the latter case, if $\omega ^{-1}(p) = (i,j)$ and $\omega ^{-1}(p+1) = (i',j')$ are not in adjacent antidiagonals, then $\mathsf {word}(D,\omega )$ and $\mathsf {word}(D, s_p \omega )$ are in the same Coxeter commutation class.

Now suppose $\upsilon $ is a second reading order on $[n]\times [n]$ . We claim that one can pass from $\omega $ to $\upsilon $ by composing $\omega $ with a sequence of simple transpositions obeying the condition just described. To check this, we induct on the number of inversions in the permutation $\upsilon \omega ^{-1} \in S_{n^2}$ . If $\upsilon \omega ^{-1}$ is not the identity, then there exists p with $\upsilon (\omega ^{-1}(p))> \upsilon (\omega ^{-1}(p+1))$ . Since $\upsilon $ and $\omega $ are both linear extensions of $\leq _{\mathsf {NE}}$ , we can have neither $\omega ^{-1}(p) \leq _{\mathsf {NE}} \omega ^{-1}(p+1)$ nor $\omega ^{-1}(p+1) \leq _{\mathsf {NE}} \omega ^{-1}(p)$ , so the cells $\omega ^{-1}(p)$ and $\omega ^{-1}(p+1)$ are not in adjacent antidiagonals. Therefore $\mathsf {word}(D,\omega )$ and $\mathsf {word}(D, s_p \omega )$ are in the same Coxeter commutation class, which by induction also includes $\mathsf {word}(D,\upsilon )$ .▪

Each diagonal is an antichain for $\leq _{\mathsf {NE}}$ , so if $\omega $ first lists the elements on diagonal $-(n-1)$ in any order, then lists the elements on diagonal $-(n-2)$ , and so on, then $\omega $ is a reading order.

Definition 3.6 The unimodal-diagonal reading order on $[n]\times [n]$ is the reading order that lists the elements of the pth diagonal from bottom to top if $p < 0$ , and from top to bottom if $p \geq 0$ . The unimodal-diagonal reading word of $D\subseteq [n]\times [n]$ , denoted $\mathsf {udiag}(D)$ , is the associated reading word.

The unimodal-diagonal reading order on $\{1,2,3,4\} \times \{1,2,3,4\}$ has values

$$ \begin{align*} \begin{array}{ccccc} 7 & 6 & 3 & 1 \\ 11 & 8 & 5 & 2 \\ 14 & 12& 9 & 4 \\ 16& 15& 13& 10 \\ \end{array} \end{align*} $$

and if $D = \{1,2,3,4\} \times \{1,2,3,4\}$ then $\mathsf {udiag}(D) = 4536421357246354$ .

3.2 Pipe dreams

Recall the definitions of the sets of reduced words $\mathcal {R}(w)$ , involution words $\hat {\mathcal {R}}(y)$ , and fpf-involution words $\hat {\mathcal {R}}^{\mathsf {{FPF}}}(z)$ for $w \in S_n$ , $y \in \mathcal {I}_n$ , and $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ from Section 2.1. For the standard reading word, the following theorem is well-known from [Reference Bergeron and Billey1]. The main new results of this section are versions of this theorem for involution pipe dreams and fpf-involution pipe dreams.

Theorem 3.7 A subset $D \subseteq [n]\times [n]$ is a reduced pipe dream for $w \in S_n$ if and only if some (equivalently, every) reading word of D is a reduced word for w.

Proof Fix $D \subseteq [n]\times [n]$ and $w \in S_n$ . The set $\mathcal {R}(w)$ is a union of Coxeter commutation classes, so $\mathsf {word}(D)\in \mathcal {R}(w)$ if and only every reading word of D belongs to $\mathcal {R}(w)$ by Lemma 3.5. Saying that D is a reduced pipe dream for w if and only if $\mathsf {word}(D) \in \mathcal {R}(w)$ is Bergeron and Billey’s original definition of an rc-graph in [Reference Bergeron and Billey1, Section 3], and it is clear from the basic properties of permutation wiring diagrams that this is equivalent to the definition of a reduced pipe dream in the introduction.▪

Corollary 3.8 [Reference Bergeron and Billey1, Lemma 3.2]

If D is a reduced pipe dream for $w \in S_n$ then $D^T$ is a reduced pipe dream for $w^{-1}$ .

Recall that the set $\mathcal {ID}(z)$ of involution pipe dreams for $z \in \mathcal {I}_n$ consists of all intersections where D is a reduced pipe dream for z that is almost-symmetric and .

Theorem 3.9 Suppose $z \in \mathcal {I}_n$ and $D \subseteq [n]\times [n]$ . The following are equivalent:

  1. (a) Some reading word of D is an involution word for z.

  2. (b) Every reading word of D is an involution word for z.

  3. (c) The set D is a reduced pipe dream for some atom of z.

Moreover, if then $D\in \mathcal {ID}(z)$ if and only if these equivalent conditions hold.

Remark Although this theorem implies that $\mathcal {ID}(z) \subseteq \bigsqcup _{w \in \mathcal {A}(z)} \mathcal {PD}(w)$ , it is possible for an atom $w \in \mathcal {A}(z)$ to have no reduced pipe dreams contained in , in which case $\mathcal {ID}(z)$ and $\mathcal {PD}(w)$ are disjoint. See Example 3.10 for an illustration of this.

Proof Recall that $\hat {\mathcal {R}}(z)$ is the disjoint union of the sets $\mathcal {R}(w)$ , running over all atoms $w \in \mathcal {A}(z)$ . The equivalences (a) $\Leftrightarrow $ (b) $\Leftrightarrow $ (c) are clear from Lemma 3.5 and Theorem 3.7. Assume . To prove the final assertion, it suffices to show that $D\in \mathcal {ID}(z)$ if and only if the unimodal-diagonal reading word of D from Definition 3.6 is an involution word of z.

Suppose $|D| = m$ and $\mathsf {udiag}(D) = a_1a_2\cdots a_m$ . We construct a sequence $w_0,w_1,w_2,\dots ,w_m$ of involutions as follows: start by setting $w_0 = 1$ , and for each $i \in [m]$ define $ w_i = s_{a_i} w_{i-1} s_{a_i}$ if we have $w_{i-1} s_{a_i} \neq s_{a_i} w_{i-1}$ , or else set $w_i = w_{i-1} s_{a_i} = s_{a_i} w_{i-1}$ . For example, if $m=5$ and $a_1a_2a_3a_4a_5 =13235$ then this sequence has

$$ \begin{align*} w_1 &=s_1, \\ w_2&= s_1s_3, \\ w_3 &= s_2s_1s_3s_2, \\ w_4 &= s_3 s_2s_1s_3s_2 s_3, \\ w_5 &=s_3 s_2s_1s_3s_2 s_3 s_5. \end{align*} $$

Let $b_l \cdots b_2 b_1$ be the subword of $a_m \cdots a_2 a_1$ which contains $a_i$ if and only if $w_i = s_{a_i} w_{i-1} s_{a_i}$ . In our example with $m=5$ and $a_1a_2a_3a_4a_5= 13235$ , we have $l=2$ and $ b_2 b_1 = a_4a_3= 32$ . Let $(p_1,q_1)$ , $(p_2,q_2)$ , …, $(p_m,q_m)$ be the cells in D listed in the unimodal-diagonal reading order and define $E = D\sqcup \{ (q_i,p_i) : w_i = s_{a_i} w_{i-1} s_{a_i}\}.$ If $\mathsf {udiag}(D) = 13235$ then we could have

$$ \begin{align*}D = \left\{\begin{smallmatrix} + & \cdot & \cdot & \cdot \\ + & + & \cdot & \cdot \\ + & \cdot & \cdot & \cdot\\ \cdot & + & \cdot & \cdot \end{smallmatrix}\right\} \qquad\text{then}\qquad E = \left\{\begin{smallmatrix} + & + & + & \cdot \\ + & + & \cdot & \cdot \\ + & \cdot & \cdot & \cdot \\ \cdot & + & \cdot & \cdot \end{smallmatrix}\right\}. \end{align*} $$

By construction $\mathsf {udiag}(E) = b_l \cdots b_2 b_1 a_1a_2\cdots a_m$ is a reduced word for z. It follows that E is almost-symmetric since each $b_i$ has a corresponding $a_j$ and the associated cells are transposes of each other.

The exchange principle (see, e.g., [Reference Hultman18, Lemma 3.4]) implies that if $w \in \mathcal {I}_n$ , $i \in [n-1]$ , and $w(i)< w(i+1)$ , then either $s_i ws_i =w\neq ws_i =s_i w = s_i \circ w \circ s_i$ or $ s_iw s_i = s_i \circ w \circ s_i \neq w$ . From this, it is straightforward to show that $\mathsf {udiag}(D) \in \hat {\mathcal {R}}(z) $ if and only if $\mathsf {udiag}(E) \in \mathcal {R}(z)$ ; this also follows from the results in [Reference Hamaker, Marberg and Pawlowski12, Section 2]. Given the previous paragraph, we conclude that $\mathsf {udiag}(D)\in \mathcal {R}(z)$ if and only if is an involution pipe dream for z.▪

Example 3.10 Let $z = 1432 = \in \mathcal {I}_4$ . Since $ z = s_3 \circ s_2 \circ 1 \circ s_2 \circ s_3 = s_2 \circ s_3 \circ 1 \circ s_3 \circ s_2,$ we have $ 23 \in \hat {\mathcal {R}}(z)$ and $ 32\in \hat {\mathcal {R}}(z)$ . These are the standard reading words of the involution pipe dreams $\{(2,1),(3,1)\}$ and $\{(2,1),(2,2)\}$ , which may be drawn as

The only involution pipe dream for $y = 321 \in \mathcal {I}_3$ is $\{(1,1),(2,1)\}$ which has standard reading word $12$ . Although $\hat {\mathcal {R}}(y) =\{12,21\}$ , there is no involution pipe dream with standard reading word $21$ .

We turn to the fixed-point-free case.

Lemma 3.11 Assume n is even. Suppose $z \in \mathcal {I}_n$ is an involution with a symmetric reduced pipe dream $D=D^T$ . Then $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ if and only if $\{(i,i) : i \in [n/2]\}\subseteq D$ .

Proof In fact, a stronger statement holds: for symmetric D and $i \in [n/2]$ , the pipes in cell $(i,i)$ of the wiring diagram of D are labeled by fixed points of z if and only if $(i,i) \notin D$ . Let a and b be the labels for the pipes entering $(i,i)$ from the left and below, respectively. Since D is symmetric, if $(i,i) \in D$ then $z(a) = b$ (hence $z(b) = a$ ), and if $(i,i) \notin D$ then $z(a) = a$ and $z(b) = b$ .▪

Recall that the set $\mathcal {FD}(z)$ of fpf-involution pipe dreams for $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ consists of all intersections where D is a reduced pipe dream for z that is symmetric and .

Theorem 3.12 Suppose n is even, $z \in \mathcal {I}^{\mathsf {{FPF}}}_n$ , and $D \subseteq [n]\times [n]$ . The following are equivalent:

  1. (a) Some reading word of D is an fpf-involution word for z.

  2. (b) Every reading word of D is an fpf-involution word for z.

  3. (c) The set D is a reduced pipe dream for some fpf-atom of z.

Moreover, if then $D\in \mathcal {FD}(z)$ if and only if these equivalent conditions hold.

Proof Recall that $\hat {\mathcal {R}}^{\mathsf {{FPF}}}(z)$ is the disjoint union of the sets $\mathcal {R}(w)$ , running over all fpf-atoms $w \in \mathcal {A}^{\mathsf {{FPF}}}(z)$ . Properties (a), (b), and (c) are again equivalent by Lemma 3.5 and Theorem 3.7. Assume . To prove the final assertion, it suffices to check that D is an fpf-involution pipe dream for z if and only if $\mathsf {udiag}(D) \in \hat {\mathcal {R}}^{\mathsf {{FPF}}}(z)$ .

To this end, first suppose where $E=E^T \in \mathcal {PD}(z)$ . Then E is also almost-symmetric, so Theorem 3.9 implies that . This combined with Lemma 3.11 implies that , so $\mathsf {udiag}(D) \in \hat {\mathcal {R}}^{\mathsf {{FPF}}}(z)$ .

Conversely, suppose every reading word of D is an fpf-involution word for z, so that $\mathsf {udiag}(D) \in \hat {\mathcal {R}}^{\mathsf {{FPF}}}(z)$ . The set $D' = D \sqcup \{(i,i) : i \in [n-1]\}$ then has $\mathsf {udiag}(D') \in \hat {\mathcal {R}}(z)$ , so there exists an almost-symmetric $D'' \in \mathcal {PD}(z)$ with by Theorem 3.9. By construction , and since $|D''| = \ell (z) = 2|D| + n/2$ it follows that $D''$ is actually symmetric. Therefore, $D\in \mathcal {FD}(z)$ .▪

Example 3.13 Let $z =216543 \in \mathcal {I}^{\mathsf {{FPF}}}_6$ . Then $\ell (z) = 7$ and

$$ \begin{align*} z = s_3 \cdot s_4 \cdot (s _1 \cdot s_3 \cdot s_5) \cdot s_4\cdot s_3 =s_5 \cdot s_4 \cdot (s_1 \cdot s_3 \cdot s_5) \cdot s_4\cdot s_5, \end{align*} $$

so $3413543$ and $5413545$ are reduced words for z. These words are the unimodal-diagonal reading words of the symmetric reduced pipe dreams

so $\{(3,1),(3,2)\}$ and $\{(4,1),(5,1)\}$ are fpf-involution pipe dreams for z, and their standard reading words $43$ and $45$ are fpf-involution words for z.

4 Pipe dreams and Schubert polynomials

In this section, we derive the pipe dream formulas for involution Schubert polynomials given in Theorem 1.5. Our arguments are inspired by a new proof due to Knutson [Reference Knutson22] of the classical pipe dream formula (1.2). Knutson’s approach is inductive. The key step in his argument is to show that the right side of (1.2) satisfies certain recurrences that also apply to double Schubert polynomials [Reference Kohnert and Veigneau25, Section 4].

Similar recurrences for $\hat {\mathfrak {S}}_y$ and $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ appear in [Reference Hamaker, Marberg and Pawlowski14]. Adapting Knutson’s strategy to our setting requires us to show that the right hand expressions in Theorem 1.5 satisfy the same family of identities. This is accomplished in Theorems 4.23 and 4.34. Proving these results involves a detailed analysis of the maximal (shifted) Ferrers diagram contained in a reduced pipe dream, which we refer to as the (shifted) dominant component. We gradually develop the technical properties of these components over the course of this section.

4.1 Dominant components of permutations

The results in this subsection are all straightforward consequences of known results, with the possible exception of Lemma 4.2; see in particular [Reference Knutson22, Section 3]. However, we are unaware of an explicit description of Definition 4.1 in the literature. Since this definition is central to our construction, we give a self-contained treatment of its properties.

A lower set in a poset $(P,<)$ is a subset $L \subset P$ such that if $x \in P$ , $y \in L$ , and $x < y$ , then $x \in L$ . Let $\leq _{\mathsf {NW}}$ be the partial order on $\mathbb {Z}_{>0} \times \mathbb {Z}_{>0}$ with $(i,j) \leq _{\mathsf {NW}} (i',j')$ if $i \leq i'$ and $j \leq j'$ , i.e., if $(i,j)$ is northwest of $(i',j')$ in matrix coordinates.

Definition 4.1 The dominant component $\operatorname {dom}(D)$ of a set $D \subseteq \mathbb {Z}_{>0}\times \mathbb {Z}_{>0}$ is the maximal lower set in $(\mathbb {Z}_{>0}\times \mathbb {Z}_{>0},\leq _{\mathsf {NW}})$ contained in D.

Equivalently, the set $\operatorname {dom}(D)$ consists of all $(i,j) \in D$ such that if $(i',j') \in \mathbb {Z}_{>0}\times \mathbb {Z}_{>0}$ and $(i',j') \leq _{\mathsf {NW}} (i,j)$ then $(i',j') \in D$ . If D is finite, then its dominant component $\operatorname {dom}(D)$ is the Ferrers diagram $ \mathsf {D}_\lambda = \{(i,j) : 1 \leq i \leq \ell (\lambda ),\ 1 \leq j \leq \lambda _i\} $ of some partition $\lambda $ . An outer corner of D is a pair $(i,j) \in (\mathbb {Z}_{>0} \times \mathbb {Z}_{>0}) \setminus D$ such that $\operatorname {dom}(D) \sqcup \{(i,j)\}$ is again a Ferrers diagram of some partition. For example, $(1,2)$ and $(2,1)$ are the outer corners of $D = \{(1,1),(1,3)\}$ , since $\operatorname {dom}(D) = \{(1,1)\}$ .

For distinct $i,j \in [n]$ , let $t_{ij} \in S_n$ be the transposition interchanging i and j.

Lemma 4.2 Suppose $w \in S_n$ and $(i,j)$ is an outer corner of some $D \in \mathcal {PD}(w)$ . Then $w(i) = j$ and $D\sqcup \{(i,j)\}$ is a reduced pipe dream (for a longer permutation).

Proof By hypothesis, D contains every cell above $(i,j)$ in the jth column and every cell to the left of $(i,j)$ in the ith row. This means that in the wiring diagram associated to D, the pipe leaving the top of position $(i,j)$ must continue straight up and terminate in column j on the top side of D, and after leaving the left of position $(i,j)$ , the same pipe must continue straight left and terminate in row i on the left side of D. Thus $w(i) = j$ as claimed. Suppose the other pipe at position $(i,j)$ starts at p on the left and ends at $q=w(p)$ on the top. As this pipe leaves $(i,j)$ rightwards and downwards, we have $p> i$ and $q>j$ , and the pipe only intersects $[i] \times [j]$ at $(i,j)$ , where it avoids the other pipe. Therefore, we have $D\sqcup \{(i,j)\} \in \mathcal {PD}(w')$ for $w' :=wt_{ip} = t_{jq}w\in S_n$ , and it holds that $\ell (w) < \ell (w')$ as $i< p$ and $w(i)< w(p)$ .▪

Example 4.3 Suppose $w=426135 \in S_6$ . If $(i,j) = (2,2)$ and

then in the notation of the proof, we have $p = 3$ , $q=6$ , and $w' = 462135$ .

The Rothe diagram of $w \in S_n$ is

$$ \begin{align*} D(w) = \{(i,j) \in [n] \times [n] : w(i)> j\text{ and } w^{-1}(j) > i\}. \end{align*} $$

It is often useful to observe that the set $D(w)$ is the complement in $[n]\times [n]$ of the union of the hooks $\{ (x,w(i)) : i < x \leq n\} \sqcup \{ (i,w(i))\} \sqcup \{ (i,y) : w(i) < y \leq n\}$ for $i \in [n]$ . It is not hard to show that one always has $|D(w)| = \ell (w)$ .

Definition 4.4 The dominant component of a permutation $w \in S_n$ is $\operatorname {dom}(w) = \operatorname {dom}(D(w)).$ We say that permutation $w \in S_n$ is dominant if $\operatorname {dom}(w) \in \mathcal {PD}(w)$ .

It is more common to define w to be dominant if $D(w)$ is the Ferrers diagram of a partition, or equivalently if w is $132$ -avoiding. The following lemma shows that our definition is equivalent.

Lemma 4.5 A permutation $w \in S_n$ is dominant if and only if it holds that $\mathcal {PD}(w) = \{ \operatorname {dom}(w)\}$ , in which case $\operatorname {dom}(w) = D(w)$ .

Proof If $w \in S_n$ is dominant then $\mathcal {PD}(w) = \{ \operatorname {dom}(w)\}= \{ D(w) \}$ since all reduced pipe dreams for w have size $\ell (w) = |D(w)|$ and contain $\operatorname {dom}(w) \subseteq D(w)$ .▪

Corollary 4.6 Let $w \in S_n$ . Then $\operatorname {dom}(w^{-1}) = \operatorname {dom}(w)^T$ . If $w $ is dominant, then $\operatorname {dom}(w) = \operatorname {dom}(w)^T$ if and only if $w=w^{-1}$ .

Proof The first claim holds since $D(w^{-1}) = D(w)^T$ . If w is dominant and $\operatorname {dom}(w) = \operatorname {dom}(w)^T$ , then $\operatorname {dom}(w) = D(w)$ by Lemma 4.5 so $D(w) = D(w)^T = D(w^{-1})$ and therefore $w=w^{-1}$ .▪

We write $\mu \subseteq \lambda $ for partitions $\mu $ and $\lambda $ to indicate that $\mathsf {D}_\mu \subseteq \mathsf {D}_\lambda $ .

Proposition 4.7 If $\lambda $ is a partition with $\lambda \subseteq (n-1,\dots ,3, 2,1)$ then there exists a unique dominant permutation $w \in S_n$ with $\operatorname {dom}(w) = \mathsf {D}_\lambda $ .

Proof This holds by induction as adding an outer corner to the reduced pipe dream of a dominant permutation yields a reduced pipe dream of a dominant permutation.▪

Write $\leq $ for the Bruhat order on $S_n$ . Since $v \leq w$ if and only if some (equivalently, every) reduced word for w has a subword that is a reduced word for v [Reference Humphreys20, Section 5.10], Theorem 3.7 implies:

Lemma 4.8 If $v,w \in S_n$ then $v \leq w$ if and only if some (equivalently, every) reduced pipe dream for w has a subset that is a reduced pipe dream for v.

Corollary 4.9 Let $v,w \in S_n$ with v dominant. Then $v\leq w$ if and only if $\operatorname {dom}(v) \subseteq D$ for some (equivalently, every) $D \in \mathcal {PD}(w)$ .

Proof This holds since a dominant permutation has only one reduced pipe dream.▪

For each $i \in [n]$ let $c_i(w) = |\{ j : (i,j) \in D(w)\}|.$ The code of $w \in S_n$ is the integer sequence $ c(w) = (c_1(w), \ldots , c_n(w))$ . The bottom pipe dream of $w \in S_n$ is the set

(4.1) $$ \begin{align}D_{\text{bot}}(w)=\{(i,j) \in [n]\times [n] : j \leq c_i(w)\} \end{align} $$

obtained by left-justifying $D(w)$ . It is not obvious that $D_{\text {bot}}(w) \in \mathcal {PD}(w)$ , but this holds by results in [Reference Bergeron and Billey1]; see also Theorem 5.2 below.

Example 4.10 If $w = 35142 \in S_5$ , then $D(w)$ is the set of $+$ ’s below:

$$ \begin{align*} \begin{array}{ccccc} + & + & 1 & \cdot & \cdot\\ + & + & \cdot & + & 1\\ 1 & \cdot & \cdot & \cdot & \cdot\\ \cdot & + & \cdot & 1 & \cdot\\ \cdot & 1 & \cdot & \cdot & \cdot \end{array} \end{align*} $$

so we have $c(w) = (2,3,0,1,0)$ and

$$ \begin{align*} D_{\text{bot}}(w) = \begin{array}{ccccc} + & + & \cdot & \,\cdot\\ + & + & + & \,\cdot\\ \cdot & \cdot & \cdot & \,\cdot \\ + & \cdot & \cdot & \,\cdot \\ \cdot & \cdot & \cdot & \,\cdot \end{array} \end{align*} $$

Proposition 4.11 If $w \in S_n$ and $D \in \mathcal {PD}(w)$ then $\operatorname {dom}(D) = \operatorname {dom}(w)$ .

Proof For each $D \in \mathcal {PD}(w)$ there exists a dominant permutation $v \in S_n$ with $\operatorname {dom}(v) = \operatorname {dom}(D)$ and $v \leq w$ , in which case $\operatorname {dom}(D) \subseteq \operatorname {dom}(E)$ for all $E \in \mathcal {PD}(w)$ by Corollary 4.9. This can only hold if $\operatorname {dom}(D) = \operatorname {dom}(E)$ for all $E \in \mathcal {PD}(w)$ .

To finish the proof, it suffices to show that $\operatorname {dom}(w) = \operatorname {dom}(D_{\text {bot}}(w))$ . It is clear by definition that $\operatorname {dom}(w) \subseteq \operatorname {dom}(D_{\text {bot}}(w))$ . Conversely, each outer corner of $\operatorname {dom}(w)$ has the form $(i,w(i))$ for some $i \in [n]$ but no such cell is in $\operatorname {dom}(D_{\text {bot}}(w))$ , so we cannot have $\operatorname {dom}(w) \subsetneq \operatorname {dom}(D_{\text {bot}}(w))$ .▪

Below, we define an outer corner of $w \in S_n$ to be an outer corner of $\operatorname {dom}(w)$ .

4.2 Involution pipe dream formulas

Recall that $\mathcal {I}_n = \{ w \in S_n : w=w^{-1}\}$ and .

Definition 4.12 The shifted dominant component of $z \in \mathcal {I}_n$ is the set

Fix $z \in \mathcal {I}_n$ . By Proposition 4.11, for all $D \in \mathcal {PD}(z)$ . The shifted Ferrers diagram of a strict partition $\lambda = (\lambda _1>\lambda _2 > \dots > \lambda _k >0)$ is the set

$$ \begin{align*} \mathsf{SD}_\lambda = \{ (i, i+j-1) : 1 \leq i \leq k,\ 1\leq j \leq \lambda_i\},\end{align*} $$

which is formed from $\mathsf {D}_\lambda $ by moving the boxes in row i to the right by $i-1$ columns. Since $\operatorname {dom}(z)$ is a Ferrers diagram, the set $\operatorname {shdom}(z)$ is the transpose of the shifted Ferrers diagram of some strict partition. A pair $(j,i) \in \mathbb {Z}_{>0}\times \mathbb {Z}_{>0}$ with $i\leq j$ is an outer corner of z if and only if the transpose of $\operatorname {shdom}(z) \cup \{(j,i)\}$ is a shifted Ferrers diagram, in which case $z(j) = i$ .

Lemma 4.13 If $z \in \mathcal {I}_n$ then $\operatorname {dom}(z) = \operatorname {shdom}(z) \cup \operatorname {shdom}(z)^T$ .

Proof This holds since $z=z^{-1}$ implies that $\operatorname {dom}(z) = \operatorname {dom}(z)^T$ .▪

Corollary 4.14 If $z \in \mathcal {I}_n$ then $\operatorname {shdom}(z)$ is the union of all lower sets of that are contained in some (equivalently, every) $D \in \mathcal {ID}(z)$ .

Proof This is clear from Proposition 4.11 and Lemma 4.13.▪

The natural definition of “involution” dominance turns out to be equivalent to the usual notion:

Proposition 4.15 Let $z \in \mathcal {I}_n$ . The following are equivalent:

  1. (a) The permutation z is dominant.

  2. (b) It holds that $\mathcal {PD}(z) = \{ \operatorname {dom}(z)\}$ .

  3. (c) It holds that $\mathcal {ID}(z) = \{\operatorname {shdom}(z)\}$ .

Proof We have (a) $\Leftrightarrow $ (b) by Lemma 4.5 and (b) $\Leftrightarrow $ (c) by Lemma 4.13.▪

Proposition 4.16 If $\lambda $ is a strict partition with $\lambda \subseteq (n-1,n-3,n-5,\dots )$ then there exists a unique dominant involution $z \in \mathcal {I}_n$ with $\operatorname {shdom}(z)^T = \mathsf {SD}_\lambda $ .

Proof We have $\mathsf {D}_\mu = \mathsf {SD}_\lambda \cup ( \mathsf {SD}_\lambda )^T$ for some $\mu $ . Let $z \in S_n$ be dominant with $\operatorname {dom}(z) = \mathsf {D}_\mu $ . Then $z \in \mathcal {I}_n$ by Corollary 4.6 and . Uniqueness holds by Lemma 4.13.▪

If $y,z \in \mathcal {I}_n$ , then $y \leq z$ in Bruhat order if and only if some (equivalently, every) involution word for z contains a subword that is an involution word for y (see either [Reference Richardson and Springer40, Corollary 8.10] with [Reference Richardson and Springer41], or [Reference Hultman19, Theorem 2.8]). The following is an immediate corollary of this property and Theorem 3.9.

Lemma 4.17 Let $y,z \in \mathcal {I}_n$ . Then $y \leq z$ if and only if some (equivalently, every) involution pipe dream for z has a subset that is an involution pipe dream for y.

Corollary 4.18 Let $y,z \in \mathcal {I}_n$ with y dominant. Then $y\leq z$ if and only if $\operatorname {shdom}(y) \subseteq D$ for some (equivalently, every) $D \in \mathcal {ID}(z)$ .

Proof This is clear since if $y \in \mathcal {I}_n$ is dominant then $|\mathcal {ID}(y)|=1$ .▪

We will need the following technical property of the Demazure product from [Reference Knutson and Miller24].

Lemma 4.19 [Reference Knutson and Miller24, Lemma 3.4(1)]

If $b_1b_2\cdots b_q$ is a subword of $a_1a_2\cdots a_p$ where each $a_i \in [n-1]$ , then $ s_{b_1} \circ s_{b_2}\circ \dots \circ s_{b_q} \leq s_{a_1} \circ s_{a_2}\circ \dots \circ s_{a_p} \in S_n. $

Corollary 4.20 If $v',w',v,w \in S_n$ , and $v \leq w$ , and $v'\leq w'$ then $v' \circ v \leq w' \circ w$ .

Proof This is clear from Lemma 4.19 given the subword property of $\leq $ .▪

Corollary 4.21 If $v,w \in S_n$ and $v \leq w$ then $v^{-1}\circ v \leq w^{-1} \circ w$ .

Proof Apply Corollary 4.20 with $v' = v^{-1}$ and $w'=w^{-1}$ .▪

Recall $t_{ij} \in S_n$ is a transposition for distinct $i,j \in [n]$ . It is well-known and not hard to check that if $w \in S_n$ then $\ell (w t_{ij}) = \ell (w) +1$ if and only if $w(i)< w(j)$ and no $i< e< j$ has $w(i) < w(e) < w(j)$ .

Given $y \in \mathcal {I}_n$ and $1\leq i < j \leq n$ , let $\mathcal {A}_{ij}(y)= \{ w t_{ij} : w \in \mathcal {A}(y)\text {, }\ell (wt_{ij})=\ell (w)+1\}.$ Each covering relation in $(S_n,\leq )$ arises as the image of right multiplication by some transposition $t_{ij}$ . The following theorem characterizes certain operators $\tau _{ij}$ which play an analogous role for $(\mathcal {I}_n,\leq )$ .

Theorem 4.22 (See [Reference Hamaker, Marberg and Pawlowski14, Section 3])

For each pair of integers $1\leq i < j \leq n$ , there are unique maps $\tau _{ij} : \mathcal {I}_n \to \mathcal {I}_n$ with the following properties:

  1. (a) If $y \in \mathcal {I}_n$ and $\mathcal {A}_{ij}(y)\cap \mathcal {A}(z)\neq \varnothing $ for some $z \in \mathcal {I}_n$ then $\tau _{ij}(y)= z$ .

  2. (b) If $y \in \mathcal {I}_n$ and $\mathcal {A}_{ij}(y)\cap \mathcal {A}(z)=\varnothing $ for all $z \in \mathcal {I}_n$ then $\tau _{ij}(y)= y$ .

Moreover, if $y \in \mathcal {I}_n$ and $y \neq \tau _{ij}(y)=z$ , then $y(i) \neq z(i)$ and $y(j)\neq z(j)$ .

This result has an extension for affine symmetric groups; see [Reference Marberg30Reference Marberg and Zhang34].

Remark The operators $\tau _{ij}$ , which first appeared in [Reference Incitti21], can be given a more explicit definition; see [Reference Hamaker, Marberg and Pawlowski14, Table 1]. However, our present applications only require the properties in the theorem.

For $y \in \mathcal {I}_n$ , let $\hat \ell (y)$ denote the common value of