Involution pipe dreams

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 $w(i)< w(i+1)$ .

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) $ .

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 Wyser5, Reference Hansson and Hultman15, Reference Hu and Zhang17, Reference Hultman18]. Our notation follows [Reference Hamaker, Marberg and Pawlowski12, Reference Hamaker, Marberg and Pawlowski13].

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

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

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 $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_{z} =\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_{zs_{n+1}}$ .

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 $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z = [M\hat X^{\mathsf {{FPF}}}_z] \in H_{\mathsf {T}}^*(\mathsf {SSMat}_n)$ .

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 $\hat {\mathfrak {S}}^{\mathsf {{FPF}}}_z$ :

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

(2.5)

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 $\mathsf {Sp}_n$ .

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)]).

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 $\omega : (i,j) \mapsto ni - j + 1$ .

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.

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 $D \subseteq [n] \times [n]$ .

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

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.

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.

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 .

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.

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 .

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.

4.1 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.

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.

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 $|D(w)| = \ell (w)$ .

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.

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

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:

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

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

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 .

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 $z(j) = i$ .

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

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.

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