Involution pipe dreams

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 $T$-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 Gr\"obner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.


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 [45].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 [16].
Involution Schubert polynomials have a combinatorial formula for their monomial expansion [13].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 [3].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 [23] to understand our varieties from a commutative algebra perspective.

Three flavors of matrix Schubert varieties
Fix a positive integer n.Let GL n denote the general linear group of complex n × n invertible matrices, and write B and B + for the Borel subgroups of lower-and upper-triangular matrices in GL n .Our work aims to extend what is known about the geometry of the B-orbits on matrix space to symmetric and skew-symmetric matrix spaces.
We begin with some classical background.Consider the type A flag variety Fl n = B\GL n .The subgroup B + acts on Fl n with finitely many orbits, which are naturally indexed by permutations w in the symmetric group S n of permutations of {1, 2, . . ., n}.These orbits afford a CW decomposition of Fl n , so the cohomology classes of their closures X w , the Schubert varieties, form a basis for the integral singular cohomology ring H * (Fl n ).Borel's isomorphism explicitly identifies H * (Fl n ) with a quotient of the polynomial ring Z[x 1 , . . ., x n ], and the Schubert polynomials S w ∈ Z[x 1 , . . ., x n ] are (non-unique) representatives for the Schubert classes [X w ] ∈ H * (Fl n ).
The maximal torus T of diagonal matrices in GL n also acts on Fl n , so we can instead consider the equivariant cohomology ring H * T (Fl n ).Via an extension of Borel's isomorphism, this ring is isomorphic to a quotient of Z[x 1 , . . ., x n , y 1 , . . ., y n ].Lascoux and Schützenberger [27] introduced the double Schubert polynomials S w (x, y) to represent the equivariant classes [X w ] T ∈ H * T (Fl n ).These representatives are distinguished in the following sense.
Let Mat n be the set of n × n complex matrices and write ι : GL n ֒→ Mat n for the obvious inclusion.The product group T × T acts on A ∈ Mat n by (t 1 , t 2 ) • A = t 1 At −1 2 .The matrix Schubert variety of a permutation w ∈ S n is MX w = ι(X w ).Since M n is T × T-equivariantly contractible, H * T×T (Mat n ) ∼ = H * T×T (point) ∼ = Z[x 1 , . . ., x n , y 1 , . . ., y n ].The launching point for Knutson and Miller's program is the following theorem: Theorem 1.1 ( [23]).For all w ∈ S n , we have S w (x, y) As mentioned in the historical notes at the end of [36,Chpt. 15], Theorem 1.1 is equivalent to Fulton's characterization of each S w (x, y) as the class of a certain degeneracy locus for vector bundle morphisms [11].
Our results are related to the geometry of certain spherical varieties studied by Richardson and Springer in [40].Specifically, define the orthogonal group O n as the subgroup of GL n preserving a fixed nondegenerate symmetric bilinear form on C n , and when n is even define the symplectic group Sp n as the subgroup of GL n preserving a fixed nondegenerate skew-symmetric bilinear form.
We consider the actions of O n and Sp n (when n is even) on Fl n .The associated orbit closures Xy and XFPF z are indexed by arbitrary involutions y and fixed-point-free involutions z in S n .Let κ(y) denote the number of 2-cycles in an involution y = y −1 ∈ S n .Wyser and Yong [45] constructed certain polynomials Ŝy , ŜFPF z ∈ Z[x 1 , . . ., x n ] and showed that the classes [ Xy ] and [ XFPF z ] are represented in H * (Fl n ) by 2 κ(y) Ŝy and ŜFPF z .We refer to Ŝy and ŜFPF z as involution Schubert polynomials; for their precise definitions, see Section 2.1.
Write SMat n and SSMat n for the sets of symmetric and skew-symmetric n×n complex matrices.Let t ∈ T act on these spaces by t • A = tAt.One can identify the T-equivariant cohomology rings of both spaces with Z[x 1 , . . ., x n ]; see the discussion in Section 2.2.For each involution y ∈ S n , let M Xy = MX y ∩ SMat n .Similarly, for each fixed-point-free involution z ∈ S n , let M XFPF z = MX z ∩ 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 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 [31].Theorem 1.2 was first announced in a conference proceedings before the appearance of the preprint version of [31], which precedes the preprint version of this article.The proof of Theorem 1.2 is a special case of results of [31].
Remark.Another family of varieties in SMat n indexed by permutations in S n has been studied by Fink, Rajchgot and Sullivant [6].However, their varieties are cut out by northeast rank conditions, while M Xy and M XFPF z are cut out by northwest rank conditions (see (2.3) and (2.4) in Section 2.3).The varieties in [6] are closely related to type C Schubert calculus and generally do not coincide with our M Xy varieties.

Three flavors of pipe dreams
If Z is a closed subvariety of SMat n or SSMat n , then its 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 Ŝy and ŜFPF z .
One associates to D a wiring diagram by replacing the cells (i, j) ∈ n by tiles of two types, given either by a crossing of two paths (drawn as a tile) if (i, j) ∈ D or by two paths bending away from each other (drawn as a tile) if (i, j) / ∈ D. Connecting the endpoints of adjacent tiles yields a union of n continuously differentiable paths, which we refer to as "pipes."For example: This condition holds in the example (1.1).Pipe dreams as described here were introduced by Bergeron and Billey [1], inspired by related diagrams of Fomin and Kirillov [9].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 ∈ S n in the following way.Label the left endpoints of the pipes in D's wiring diagram by 1, 2, . . ., n from top to bottom, and the top endpoints by 1, 2, . . ., n from left to right.Then the associated permutation w ∈ 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 ∈ S 4 .Let PD(w) denote the set of all reduced pipe dreams associated to w ∈ S n .
Pipe dreams are of interest for their role in formulas for S w and S w (x, y).Lascoux and Schützenberger's original definition of these Schubert polynomials in [28] is recursive in terms of divided difference operators.However, by results of Fomin and Stanley [10, §4] we also have The first identity is the Billey-Jockusch-Stanley formula for Schubert polynomials from [3].
There are analogues of this formula for the involution Schubert polynomials Ŝy and ŜFPF z , which involve the following new classes of pipe dreams.A reduced pipe dream D ⊆ n is symmetric if (i, j) ∈ D implies (j, i) ∈ D, and almost-symmetric if both of the following properties hold: • If (i, j) ∈ D where i < j then (j, i) ∈ D.
• If (j, i) ∈ D where i < j but (i, j) / ∈ 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 I n = {w ∈ S n : w = w −1 } and write I FPF n for the subset of fixed-point-free elements of I n .Note that n must be even for I FPF n to be non-empty.Also let Definition 1.4.The set of involution pipe dreams for y ∈ I n is The set of fpf-involution pipe dreams for z ∈ I 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 can now state our second main result, which will reappear as Theorems 4.25 and 4.36.
where δ ij denotes the usual Kronecker delta function.
Example 1.6.The involution y = 1432 = (2, 4) ∈ I 4 has five reduced pipe dreams: Only the last two of these are almost-symmetric, so |ID(y)| = 2 and Theorem 1.5 reduces to the formula Ŝy = ( 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 Ŝy fewer terms. Remark.There is an alternate path towards 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 [11].Knutson and Miller showed that Fulton's generators form a Gröbner basis with respect to any anti-diagonal term order [23].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 ID(y) and 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 [22] along with certain transition equations for Ŝy and ŜFPF z given in [14].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 [1].In Section 6, finally, we describe several related open problems and conjectures.

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 Z >0 = {1, 2, 3, . . .} fixing all i > n, so that there is an automatic inclusion S n ⊂ S n+1 .In this section, we present some relevant background on involution Schubert polynomials and equivariant cohomology, and then prove Theorem 1.2.

Involution Schubert polynomials
To start, we provide a succinct definition of Ŝy and ŜFPF z in terms of the ordinary Schubert polynomials S w given by (1.2).Let denote the set of reduced words for w.The length ℓ(w) of w ∈ S n is the length of any word in R(w).One has ℓ(ws i ) = ℓ(w) + 1 > ℓ(w) if and only if w(i) < w(i + 1).

An involution word for y
(2.1) Note that we could replace An atom for y ∈ I n is a minimal-length permutation w ∈ S n with y = w −1 • w.Let R(y) be the set of involution words for y ∈ I n and let A(y) be the set of atoms for y.The associativity of the Demazure product implies that R(y) = w∈A(y) R(w).
One can show that I n = {w −1 • w : w ∈ S n }, so R(y) and A(y) are nonempty for all y ∈ I n .Involution words are a special case of a more general construction of Richardson and Springer [40], and have been studied by various authors [5,15,17,18].Our notation follows [12,13].
Definition 2.3.The involution Schubert polynomial of y ∈ I n is Ŝy = w∈A(y) S w .
Wyser and Yong [45] originally defined these polynomials recursively using divided difference operators; work of Brion [4] implies that our definition agrees with theirs.For a detailed explanation of the equivalence among these definitions, see [13].
Assume n is even, so that } is nonempty, and let This formulation avoids the Demazure product, but there is an equivalent definition that more closely parallels (2.1).Namely, by [12,Cor. 2.6], an fpf-involution word for z ∈ I FPF n is also a minimal-length word An fpf-atom for z ∈ I FPF n is a minimal length permutation w ∈ S n with z = w −1 1 FPF n w.Let A FPF (z) be the set of fpf-atoms for z, and let RFPF (z) be the set of fpf-involution words for z.The basic properties of reduced words imply that RFPF (z) = w∈A FPF (z) R(w).
Fpf-involution words are special cases of reduced words for quasiparabolic sets [39].Since I FPF n is a single S n -conjugacy class, each z ∈ I FPF n has at least one fpf-involution word and fpf-atom.

Torus-equivariant cohomology
Suppose V is a finite-dimensional rational representation of a torus T ≃ (C × ) n .A character λ ∈ Hom(T, C × ) is a weight of V if the weight space V λ = {v ∈ V : tv = λ(t)v for all t ∈ T} is nonzero.Any nonzero v ∈ V λ is a weight vector, and V has a basis of weight vectors.Let wt(V ) denote the set of weights of V .After fixing an isomorphism T ≃ (C × ) n , we identify the character (t 1 , . . ., t n ) → t a 1 1 • • • t an n with the linear polynomial a The equivariant cohomology ring H T (V ) is isomorphic to Z[x 1 , . . ., x n ], an identification we make without comment from now on.Each T-invariant subscheme X ⊆ V has an associated class [X] ∈ H T (V ), which we describe following [36,Chpt. 8].
First, if X is a linear subspace then we define [X] = λ∈wt(X) λ, where we identify each character λ with a linear polynomial as above.More generally, fix a basis of weight vectors of V , and let z 1 , . . ., z n ∈ V * be the dual basis; this determines an isomorphism Choose a term order on monomials in z 1 , . . ., z n , and let init(I) denote the ideal generated by the leading terms of all members of a given set I ⊆ C[V ].Given that init(I) is a monomial ideal, one can show that each of its associated primes p is also a monomial ideal, and hence of the form z i 1 , . . ., z ir .The corresponding subscheme Z(p) is a T-invariant linear subspace of V .Now define where I(X) is the ideal of X and p runs over the associated primes of init I(X).

Classes of involution matrix Schubert varieties
The matrix Schubert varieties in Theorem 1.1 can be described in terms of rank conditions, namely: where Mat n is the variety of n × n matrices, A [i][j] denotes the upper-left i × j corner of A ∈ Mat n , and we identify w ∈ S n with the n × n permutation matrix having 1's in positions (i, w(i)).
The varieties M Xy and M XFPF z from Theorem 1.2 can be reformulated in a similar way.Specifically, we define the involution matrix Schubert variety of y ∈ I n by where SMat n is the subvariety of symmetric matrices in Mat n .When n is even, we define the fpf-involution matrix Schubert variety of z ∈ I FPF n by where SSMat n is the subvariety of skew-symmetric matrices in Mat n .
Example 2.8.Suppose y = 132 = Let T ⊆ GL n be the usual torus of invertible diagonal matrices.Recall that κ(y) = |{i : y(i) < i}| for y ∈ I n , and that T acts on matrices in Mat n by t • A = tA and on symmetric matrices in SMat n by t • A = tAt.We can now prove Theorem 1.2, which states that if y ∈ I n and z Remark.It is possible, though a little cumbersome, to derive Theorem 1.2 from [31, Thm.2.17 and Lem.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 [31].However, as the argument below is similar to the proofs of the results in [31], we will be somewhat curt here in our presentation of the details.
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 Because T acts freely on GL n and since Fix y ∈ I n and define σ : GL n → SMat n by σ(g) = gg T .Let ι : GL n ֒→ M n be the obvious inclusion and consider the diagram Realize O n as the group {g ∈ GL n : gg T = 1}.The map σ is flat because it is the composition , where the second map sends gO n → gg T and may be identified with the open inclusion GL n ∩ SMat n ֒→ SMat n .For fixed i ∈ [n], one checks using the prescription of §2.2 that 2x i represents both the class of for fixed y and varying m have a common generating set.It is clear from §2.2 that this means that the polynomial [M Xy×1 m ] is independent of m.
The proof for the skew-symmetric case is the same, replacing O n by Sp n and the map σ : g → gg T by g → gΩg T , where Ω ∈ GL n is the nondegenerate skew-symmetric form preserved by Sp n .
Corollary 2.10.The polynomial 2 κ(y) Ŝy (respectively, ŜFPF z ) is a positive integer linear combination of products of terms Proof.The weights of T acting on SMat n are x i + x j for 1 ≤ i ≤ j ≤ n, while the weights of SSMat n are the same with the added restriction i < j.The expression (2.2) makes clear that the classes [M Xy ] and [M XFPF z ] are positive integer linear combinations of products of these weights.
Since the second factor of T × S acts trivially on SMat n , the polynomial 2 κ(y) Ŝy still represents the class [M Xy ] ∈ H T×S (SMat n ).It follows as in the proof of Theorem 1.2 that 2 κ(y) Ŝy also represents the class [ Xy ] S ∈ H S (Fl n ).The latter fact was proven by Wyser and Yong [45], but our approach gives an explanation for the surprising existence of a representative for [ Xy ] S not involving the S-weights.Similar remarks apply in the skew-symmetric case.

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.
If one also records the row indices of the positions α i as a second word, then the resulting words uniquely determine D and are the same data as a compatible sequence for word(D) (see [3, (1)]).We introduce a more general class of reading words.Suppose ω : For us, a linear extension of a finite poset (P, ) with size m = |P | is a bijection ω : 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. .We claim that one can pass from ω to υ by composing ω with a sequence of simple transpositions obeying the condition just described.To check this, we induct on the number of inversions in the permutation υω −1 ∈ S n 2 .If υω −1 is not the identity, then there exists p with υ(ω −1 (p)) > υ(ω −1 (p + 1)).Since υ and ω are both linear extensions of ≤ NE , we can have neither , so the cells ω −1 (p) and ω −1 (p + 1) are not in adjacent antidiagonals.Therefore word(D, ω) and word(D, s p ω) are in the same Coxeter commutation class, which by induction also includes word(D, υ).
Each diagonal is an antichain for ≤ NE , so if ω first lists the elements on diagonal −(n − 1) in any order, then lists the elements on diagonal −(n − 2), and so on, then ω is a reading order.

Pipe dreams
Recall the definitions of the sets of reduced words R(w), involution words R(y), and fpf-involution words RFPF (z) for w ∈ S n , y ∈ I n , and z ∈ I FPF n from Section 2.1.For the standard reading word, the following theorem is well-known from [1].The main new results of this section are versions of this theorem for involution pipe dreams and fpf-involution pipe dreams.Remark.Although this theorem implies that ID(z) ⊆ w∈A(z) PD(w), it is possible for an atom w ∈ A(z) to have no reduced pipe dreams contained in n , in which case ID(z) and PD(w) are disjoint.See Example 3.10 for an illustration of this.
Proof.Recall that R(z) is the disjoint union of the sets R(w), running over all atoms w ∈ A(z).The equivalences (a) ⇔ (b) ⇔ (c) are clear from Lemma 3.5 and Theorem 3.7.Assume D ⊆ n .To prove the final assertion, it suffices to show that D ∈ 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 udiag(D) = a 1 a 2 • • • a m .We construct a sequence w 0 , w 1 , w 2 , . . ., w m of involutions as follows: start by setting w 0 = 1, and for each i ∈ [m] define w i = s a i w i−1 s a i if we have w i−1 s a i = 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 1 a 2 a 3 a 4 a 5 = 13235 then this sequence has 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 1 a 2 a 3 a 4 a 5 = 13235, we have l = 2 and b 2 b 1 = a 4 a 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 ⊔ {(q i , p i ) : .
From this, it is straightforward to show that udiag(D) ∈ R(z) if and only if udiag(E) ∈ R(z); this also follows from the results in [12, §2].Given the previous paragraph, we conclude that udiag(D) ∈ R(z) if and only if D = E ∩ n is an involution pipe dream for z.We turn to the fixed-point-free case.(c) The set D is a reduced pipe dream for some fpf-atom of z.
Moreover, if D ⊆ = n then D ∈ FD(z) if and only if these equivalent conditions hold.
Proof.Recall that RFPF (z) is the disjoint union of the sets R(w), running over all fpf-atoms w ∈ A FPF (z).Properties (a), (b), and (c) are again equivalent by Lemma 3.5 and Theorem 3.7.Assume D ⊆ = n .To prove the final assertion, it suffices to check that D is an fpf-involution pipe dream for z if and only if udiag(D) ∈ RFPF (z).
To this end, first suppose D = E ∩ = n where E = E T ∈ PD(z).Then E is also almostsymmetric, so Theorem 3.9 implies that E ∩ n ∈ ID(z).This combined with Lemma 3.11 implies that udiag(E Conversely, suppose every reading word of D is an fpf-involution word for z, so that udiag(D) ∈ RFPF (z).The set so 3413543 and 5413545 are reduced words for z.These words are the unimodal-diagonal reading words of the symmetric reduced pipe dreams and 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.

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 [22] 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 [25, §4].
Similar recurrences for Ŝy and ŜFPF z appear in [14].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.

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 [22, §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 ⊂ P such that if x ∈ P , y ∈ L, and x < y, then x ∈ L. Let ≤ NW be the partial order on Z >0 × Z >0 with (i, j) ≤ NW (i ′ , j ′ ) if i ≤ i ′ and j ≤ j ′ , i.e., if (i, j) is northwest of (i ′ , j ′ ) in matrix coordinates.
Equivalently, the set dom(D) consists of all pairs (i, j) ∈ D such that whenever (i For distinct i, j ∈ [n], let t ij ∈ S n be the transposition interchanging i and j. Lemma 4.2.Suppose w ∈ S n and (i, j) is an outer corner of some D ∈ PD(w).Then w(i) = j and D ⊔ {(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] × [j] at (i, j), where it avoids the other pipe.Therefore, we have D ⊔ {(i, j)} ∈ PD(w ′ ) for w ′ := wt ip = t jq w ∈ S n , and it holds that ℓ(w) < ℓ(w ′ ) as i < p and w(i) < w(p).then in the notation of the proof, we have p = 3, q = 6, and w ′ = 462135.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.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 new dominant permutation.
Write ≤ for the Bruhat order on S n .Since v ≤ w if and only if some (equivalently, every) reduced word for w has a subword that is a reduced word for v [20, §5.10], Theorem 3.7 implies: Lemma 4.8.If v, w ∈ S n then v ≤ 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 ∈ S n with v dominant.Then v ≤ w if and only if dom(v) ⊆ D for some (equivalently, every) D ∈ PD(w).
Proof.This holds since a dominant permutation has only one reduced pipe dream.
For each i ∈ [n] let c i (w) = |{j : (i, j) ∈ D(w)}|.The code of w ∈ S n is the integer sequence c(w) = (c 1 (w), . . ., c n (w)).The bottom pipe dream of w ∈ S n is the set obtained by left-justifying D(w).It is not obvious that D bot (w) ∈ PD(w), but this holds by results in [1]; see also Theorem 5.2 below.
Example 4.10.If w = 35142 ∈ S 5 , then D(w) is the set of +'s below: Proof.For each D ∈ PD(w) there exists a dominant permutation v ∈ S n with dom(v) = dom(D) and v ≤ w, in which case dom(D) ⊆ dom(E) for all E ∈ PD(w) by Corollary 4.9.This can only hold if dom(D) = dom(E) for all E ∈ PD(w).
To finish the proof, it suffices to show that dom(w) = dom(D bot (w)).It is clear by definition that dom(w) ⊆ dom(D bot (w)).Conversely, each outer corner of dom(w) has the form (i, w(i)) for some i ∈ [n] but no such cell is in dom(D bot (w)), so we cannot have dom(w) dom(D bot (w)).
In the next sections, we define an outer corner of w ∈ S n to be an outer corner of dom(w).

Involution pipe dream formulas
which is formed from D λ by moving the boxes in row i to the right by i − 1 columns.Since dom(z) is a Ferrers diagram, the set shdom(z) is the transpose of the shifted Ferrers diagram of some strict partition.A pair (j, i) ∈ Z >0 × Z >0 with i ≤ j is an outer corner of z if and only if the transpose of shdom(z) ∪ {(j, i)} is a shifted Ferrers diagram, in which case z(j) = i.Proof.This holds since z = z −1 implies that dom(z) = dom(z) T .Corollary 4.14.If z ∈ I n then shdom(z) is the union of all lower sets of ( n , ≤ NW ) that are contained in some (equivalently, every) D ∈ 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: If y, z ∈ I n , then y ≤ 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 [40,Cor. 8.10] with [41], or [19,Thm. 2.8]).The following is an immediate corollary of this property and Theorem 3.9.Lemma 4.17.Let y, z ∈ I n .Then y ≤ z if and only if some (equivalently, every) involution pipe dream for z has a subset that is an involution pipe dream for y.Proof.This is clear since if y ∈ I n is dominant then |ID(y)| = 1.
We need to mention the following technical property of the Demazure product from [24].
Proof.This is clear from Lemma 4.19 given the subword property of ≤.
It is well-known and not hard to check that if w ∈ S n then ℓ(wt ij ) = ℓ(w) + 1 if and only if w(i) < w(j) and no i < e < j has w(i) < w(e) < w(j).
Given y ∈ I n and 1 ≤ i < j ≤ n, let A ij (y) = {wt ij : w ∈ A(y), ℓ(wt ij ) = ℓ(w) + 1}.Each covering relation in (S n , ≤) arises as the image of right multiplication by some transposition t ij .The following theorem characterizes certain operators τ ij which play an analogous role for (I n , ≤).This result has an extension for affine symmetric groups; see [30,34].
Remark.The operators τ ij , which first appeared in [21], can be given a more explicit definition; see [14, Table 1].However, our present applications only require the properties in the theorem.
For y ∈ I n , let l(y) denote the common value of ℓ(w) for any w ∈ A(y).This is also the size of any D ∈ ID(y).By Lemma 4.17, if y, z ∈ I n and y < z then l(y) < l(z).Let Ψ(y, j) = z ∈ I n+1 : z = τ js (y) and l(z) = l(y) + 1 for some s > j for y ∈ I n and j ∈ [n].Since S n ⊂ S n+1 and I n ⊂ I n+1 , this set is well-defined.Theorem 4.23.Let (j, i) be an outer corner of y ∈ I n with i ≤ j.
Proof.We have y(j) = i and y(i) = j by Lemma 4.2.Suppose v ∈ A(y) and D ∈ PD(v) ∩ ID(y).By considering the pipes crossing at position (j, i) in the wiring diagram of D, as in the proof of Lemma 4.2, it follows that D ⊔ {(j, i)} is a reduced pipe dream for a permutation w that belongs to A js (y) for some j < s ≤ n.Set z = w −1 • w ∈ I n .We wish to show that w ∈ A(z), since if this holds then D ⊔ {(j, i)} ∈ ID(z) and Theorem 4.22 implies that z ∈ Ψ(y, j).
Thus, the map in part (a) at least has the desired codomain and is clearly injective.To show that it is also surjective, suppose E ∈ ID(z) for some z ∈ Ψ(y, j).Lemma 4.17 implies some (l, k) ∈ E has E \ {(l, k)} ∈ ID(y).Let E ′ ∈ PD(z) be the almost-symmetric reduced pipe dream with E = E ′ ∩ n .If (j, i) = (l, k) then, since dom(y) = shdom(y) ∪ shdom(y) T ⊂ E ′ , it would follow by considering the wiring diagram of E ′ that z(j) = i = y(j), contradicting the last assertion in Theorem 4.22.Thus (j, i) = (l, k) so the map in part (a) is surjective.Part (b) holds because an involution belongs to I n if any of its involution pipe dreams is contained in {(j, i) : i ≤ j and i + j ≤ n}.
Proof.We abbreviate by setting x (i,j) = 2 −δ ij (x i + x j ), so that x (i,j) = x i if i = j and otherwise x (i,j) = x i + x j .It follows from [14,Thm. 3.30] On the other hand, results of Wyser and Yong [45] (see [13,Thm. 1.3]) show that

Fixed-point-free involution pipe dream formulas
In this section, we assume n is even.
with z(i) = j and each + indicating a position in D(z).The relevant dominant components are dom(z) = As we see in this example, if z ∈ I FPF n is any fixed-point-free involution, then (shdom = (z)) ↑T is the shifted Ferrers diagram of some strict partition.Moreover, a pair (j, i) ∈ = n is an outer corner of z if and only if (shdom = (D) ⊔ {(j, i)}) ↑T is again a shifted Ferrers diagram, in which case z(j) = i by Lemma 4.2.The unique outer corner of z = 465132 in = 6 is (4, 1).As predicted by the theorem with (j, i) = (3, 1), both elements of Ψ FPF (y, 3) are fpf-dominant since .
We may now prove the second half of Theorem 1.5, concerning the polynomials ŜFPF z .

Generating pipe dreams
Bergeron and Billey [1] proved that the set PD(w) is generated by applying simple transformations to a unique "bottom" pipe dream.Here, we derive versions of this result for the sets of involution pipe dreams ID(y) and FD(z).This leads to algorithms for computing the sets ID(y) and FD(z) that are much more efficient than the naive methods suggested by our original definitions.

Ladder moves
Let D and E be subsets of Z >0 × Z >0 , depicted as positions marked by "+" in a matrix.If E is obtained from D by replacing a subset of the form then we say that E is obtained from D by a ladder move and write D ⋖ PD E.More formally: Definition 5.1.We write D ⋖ PD E if for some integers i < j and k the following holds: • It holds that (j, k) ∈ D but (i, k), (i, k + 1), (j, k + 1) / ∈ D.
One can have i + 1 = j in this definition, in which case the first condition holds vacuously.Let < PD be the transitive closure of ⋖ PD .This relation is a strict partial order.Let ∼ PD denote the symmetric closure of the partial order ≤ PD .
Recall the definition of the bottom pipe dream D bot (w) from ( Thus PD(w) is an upper and lower set of ≤ PD , with unique minimum D bot (w).
Define ≤ chute PD to be the partial order with D ≤ chute PD E if and only if E T ≤ PD D T , and let D top (w) = D bot (w −1 ) T for w ∈ S n .Then PD(w) = E : E ≤ chute PD D top (w) by Corollary 3.8 and Theorem 5.2.Bergeron and Billey [1] refer to the covering relation in ≤ chute PD as a chute move.In the next sections, we will see that there are natural versions of ≤ PD and D bot (w) for (fpf-)involution pipe dreams.There do not seem to be good involution analogues of ≤ chute PD or D top (w), however.

Involution ladder moves
To prove an analogue of Theorem 5.2 for involution pipe dreams, we need to introduce a more general partial order < ID on subsets of Z >0 × Z >0 .Again let D and E be subsets of Z >0 × Z >0 .Informally, we define < ID to be the transitive closure of ⋖ PD and the relation that has D ⋖ ID E whenever E is obtained from D by replacing a subset of the form where the upper parts of the antidiagonals with ր are required to be empty.For example, since the relevant antidiagonals in (5.1) are not empty.The precise definition of ⋖ ID is below: Definition 5.3.We write D ⋖ ID E if for some integers i < j and k the following holds: • It holds that (i, k), (j, k) ∈ D but (i, k + 1), (i, k + 2), (j, k + 1) / ∈ D.
• The set D contains no positions strictly northeast of and in the same antidiagonal as (i, k −1), (i, k), (i, k + 1), or (i, k + 2).
One may again have i + 1 = j, in which case the first condition holds vacuously.We define < ID to be the transitive closure of ⋖ PD and ⋖ ID , and write ∼ ID for the symmetric closure of ≤ ID .
Our goal is to show that < ID defines a partial order on ID(z); for an example of this poset, see Figure 1.To proceed, we must recall a few nontrivial properties of the set A(z) from Section 2.1.Fix z ∈ I n .The involution code of z is ĉ(z) = (ĉ 1 (z), ĉ2 (z), . . ., ĉn (z)) with ĉi (z) the number of integers j > i with z(i) > z(j) and i ≥ z(j).Note that we always have ĉi (z) ≤ i.
Suppose  Consider the reading order ω that lists the positions (i, j) ∈ [n] × [n] such that (−j, i) increases lexicographically, i.e., the order that goes down column n, then down column n − 1, and so on.In view of Theorem 3.9, we may assume without loss of generality that columns 1, 2, . . ., k − 1 of D and E are both empty, since omitting these positions has the effect of truncating the same final sequence of letters from word(D, ω) and word(E, ω).
Suppose E ∈ PD(w) ⊆ ID + (z) for some permutation w ∈ A(z).To show that D ∈ ID + (z), it suffices by Theorem 5.5 to check that D ∈ PD(v) for a permutation v ≺ A w.
Consider the wiring diagram of E and let m, m + 1 and m + 2 be the top indices of the wires in the antidiagonals containing the cells (i, k), (i, k + 1), and (i + 1, k + 1), respectively.Since the northeast parts of these antidiagonals are empty, it follows that as one goes from northeast to southwest, wire m of E enters the top of the + in cell (i, k), wire m + 1 enters the top of the + in cell (i, k + 1), and wire m + 2 enters the right of the + in cell (i, k + 1).Tracing these wires through the wiring diagram of E, we see that they exit column k on the left in relative order m + 2, m, m + 1.Since we assume columns 1, 2, . . ., k − 1 are empty, the wires must arrive at the far left in the same relative order.This means that there are numbers a < b < c such that w −1 (m)w −1 (m + 1)w −1 (m + 2) = bca.
Moving the + in cell (i, k + 1) of E to (j, k) gives D by assumption.This transformation only alters the trajectories of wires m, m + 1 and m + 2 and causes no pair of wires to cross more than once, so D is a reduced pipe dream for some v ∈ S n .By examining the wiring diagram of D, we see that v −1 (m)v −1 (m + 1)v −1 (m + 2) = cab, so v ≺ A w and D ∈ ID + (z) as needed.The same considerations show that if D ∈ PD(v) for some v ∈ A(z) then E ∈ PD(w) for a permutation w with v ≺ A w.In this case, it follows that w ∈ A(z) by Theorem 5.5 so E ∈ ID + (z).
We define the bottom involution pipe dream of z ∈ I n to be the set (5.2) Since ĉ(z) = c(α min (z)), it follows by Theorem 3.9 that Dbot (z) = D bot (α min (z)) ∈ ID(z).Proof.Both sets are contained in ID + (z) by Lemma 5.6.Note that ID + (z) is finite since A(z) is finite and each set PD(w) is finite.Suppose Dbot (z) = E = D bot (w) for some w ∈ A(z).In view of Theorem 5.2, we need only show that there exists a subset D ⊂ Z >0 × Z >0 with D ⋖ ID E.
As we assume w = α min (z), it follows from Theorem 5.5 that there exists some p ∈ [n − 2] with w −1 (p+2) < w −1 (p) < w −1 (p+1).Set i = w −1 (p+2), and choose p to minimize i.We claim that if h < i then w(h) < p.To show this, we argue by contradiction.Suppose there exists 1 ≤ h < i with w(h) ≥ p. Choose h with this property so that w(h) is as small as possible.Then w(h) > p + 2 ≥ 3, and by the minimality of w(h), the values w(h)−1 and w(h)−2 appear after position h in the word w(1)w(2) • • • w(n).Therefore, by Lemma 5.4, the one-line representation of w must have the form This contradicts the minimality of i, so no such h can exist.
Let j > i be minimal with w(j) < w(i) and define k = c i (w) − 1.It is evident from the definition of i that such an index j exists and that k is positive.Now consider Definition 5.3 applied to these values of i < j and k.It follows from the claim in the previous paragraph if h < i then c h (w) − c i (w) ≤ i − h − 3. Therefore, we see that the required antidiagonals are empty.The minimality of j implies that c m (w) ≥ c i (w) for all i < m < j, and since we must have j ≤ w −1 (p), it follows that c j (w) < c i (w) − 1.We conclude that replacing position (i, k + 1) in E by (j, k) produces a subset D with D ⋖ ID E, as we needed to show.
Proof.This is clear from Theorem 5.7 since n is a lower set under ≤ ID .

Fixed-point-free involution ladder moves
In this subsection, we assume n is a positive even integer.Our goal is to replicate the results in Section 5.2 for fixed-point-free involutions.To this end, we introduce a third partial order < F D .Again let D and E be subsets of Z >0 × Z >0 .We define < F D as the transitive closure of ⋖ PD and the relation that has D ⋖ F D E whenever E is obtained from D by replacing a subset of the form Here, all positions containing " • " should be empty, including the five antidiagonals extending upwards beyond each ր.For example, since the relevant antidiagonals in (5.3) are not empty.The precise definition of ⋖ F D is as follows: Definition 5.9.We write D ⋖ F D E if for some integers 0 < i < j and k ≥ 2 the following holds: • The set D contains no positions strictly northeast of and in the same antidiagonal as (i, k −2), (i, k − 1), (i, k), (i, k + 1), or (i, k + 2).
When i + 1 = j, the first condition holds vacuously; see the lower dashed arrow in Figure 2. Define < F D to be the transitive closure of ⋖ PD and ⋖ F D .Write ∼ F D for the symmetric closure of ≤ F D .
We will soon show that < F D defines a partial order on FD(z), as one can see in the example shown in Figure 2.For this, we will need a lemma from [5] concerning the set A FPF (z).The involution code and partial order ≺ A both have fixed-point-free versions.Fix z ∈ I FPF n .The fpf-involution code of z is the integer sequence where ĉFPF i (z) is the number of integers j > i with z(i) > z(j) and i > z(j).It always holds that ĉFPF For example, if z = 632541 ∈ I FPF Thus A FPF (z) is an upper and lower set of A FPF , with unique minimum α FPF min (z). For Lemma 5.12.Let z ∈ I FPF n .Suppose D and E are subsets of Z >0 × Z >0 with D < F D E. Then D ∈ FD + (z) if and only if E ∈ FD + (z).
Proof.If D ⋖ PD E then the result follows by Theorem 5.2.Assume D ⋖ F D E and let i < j and k be as in Definition 5.9.
As in the proof of Theorem 5.6, consider the reading order ω that lists the positions (i, j) ∈ [n] × [n] such that (−j, i) increases lexicographically.In view of Theorem 3.12, we may assume without loss of generality that columns 1, 2, . . ., k − 2, as well as all positions below row i in column k − 1, are empty in both of D and E. This follows since omitting these positions has the effect of truncating the same final sequence of letters from word(D, ω) and word(E, ω).
Assume E ∈ PD(v) ⊆ FD + (z) for some w ∈ A FPF (z).To show that D ∈ FD + (z), we will check that D ∈ PD(v) for some v ∈ S n with v ≺ A FPF w.
Consider the wiring diagram of E and let m, m + 1, m + 2, and m + 3 be the top indices of the wires in the antidiagonals containing the cells (i, k − 1), (i, k), (i, k + 1), and (i, k + 2), respectively.Since the northeast parts of these antidiagonals are empty, it follows that as one goes from northeast to southwest, wire m of E enters the top of the + in cell (i, k − 1), wire m + 1 enters the top of the + is cell (i, k), wire m + 2 enters the right of the + in cell (i, k), and wire m + 3 enters the top of cell (i + 1, k + 1), which contains a + if i + 1 < j.Tracing these wires through the wiring diagram of E, we see that they exit column k − 1 on the left in relative order m + 2, m, m + 1, m + 3. Since we assume that D and E contain no positions in the rectangle weakly southwest of (i + 1, k − 1), the wires must arrive at the far left in the same relative order.This means that w −1 (m)w −1 (m + 1)w −1 (m + 2)w −1 (m + 3) = bcad for some numbers a < b < c < d.
Moving the + in cell (i, k − 1) of E to (j, k) gives D by assumption.This transformation only alters the trajectories of wires m, m + 1, m + 2, and m + 3 and causes no pair of wires to cross more than once, so D is a reduced pipe dream for some v ∈ S n .By examining the wiring diagram of D, it is easy to check that v −1 (m)v −1 (m + 1)v −1 (m + 2)v −1 (m + 3) = adbc so v ≺ A FPF w as needed.
If instead D ∈ PD(v) ⊂ FD + (z) for some v ∈ A FPF (z), then a similar argument shows that E ∈ PD(w) for some w ∈ S n with v ≺ A FPF w, which implies that E ∈ FD + (z) by Theorem 5.11.
We define the bottom fpf-involution pipe dream of z ∈ I FPF Proof.Both sets are contained in FD + (z) by Lemma 5.12, and the set FD + (z) is clearly finite.Suppose DFPF bot (z) = E = D bot (w) for some w ∈ A FPF (z).As in the proof of Theorem 5.7, it suffices to show that there exists a subset D ⊂ Z >0 × Z >0 with D ⋖ F D E.
Since w = α FPF min (z), Lemma 5.10 and Theorem 5.11 imply that there exists an odd integer p ∈ [n−3] such that w −1 (p)w −1 (p+1)w −1 (p+2)w −1 (p+3) = bcad for some numbers a < b < c < d.Choose p such that a is as small as possible.We claim that a < w −1 (q) for all q with p + 3 < q ≤ n.To show this, let a 0 = a and b 0 = d and suppose a i and b i are the integers such that Part (1) of Lemma 5.10 implies that a i < b i = z(a i ) for all i, so it suffices to show that a 0 < a i for i ∈ [k].This holds since if i ∈ [k] were minimal with a i < a 0 , then it would follow from part (2) of Lemma 5.10 that a i < a i−1 < b i−1 < b i , contradicting the minimality of a. Now, to match Definition 5.9, let i = a = w −1 (p + 2), define j > i to be minimal with w(j) < w(i), and set k = c i (w).It is clear from the definition of i that such an index j exists and that k ≥ 2. The claim in the previous paragraph shows that if 1 ≤ h < i then h must appear before position p in the one-line representation of w −1 , which means that w(h) < p and therefore c h (w) − c i (w) ≤ i − h − 4. The antidiagonals described in Definition 5.9 are thus empty as needed.Since j ≤ b = w −1 (p), it follows that c j (w) < c i (w); moreover, if i < m < j then w(m) > w(d) = p + 3 so c m (w) ≥ c i (w) + 1. Collecting these observations, we conclude that replacing (i, k − 1) in E with (j, k) gives a subset D with D ⋖ ID E, as we needed to show.Proof.This is clear from Theorem 5.13 since = n is a lower set under ≤ F D .
6 Future directions In this final section we discuss some related identities and open problems.

Enumerating involution pipe dreams
Choose w ∈ S n and let p = ℓ(w).Macdonald [29, (6.11)] proved that the following specialization of a Schubert polynomial gives an exact formula for the number of reduced pipe dreams for w:  (a 1 ,a 2 ,...,ap)∈ RFPF (z) a 1 a 2 • • • a p .
Proof.In both parts, the first equality is immediate from Theorem 1.5 and the second equality is a consequence of (6.1), via Definitions 2.3 and 2.6.
Billey, Holroyd, and Young gave the first bijective proof of (6.1) (and of a more general qanalogue) in the recent paper [2].This follow-up problem is natural: Problem 6.2.Find bijective proofs of the identities in Corollary 6.1.
For some permutations, better formulas than 6.1 are available.A reverse plane partition of shape D ⊂ Z >0 × Z >0 is a map T : D → Z ≥0 such that T (i, j) ≤ T (i + 1, j) and T (i, j) ≤ T (i, j + 1) for all relevant (i, j) ∈ D. If λ is a partition, then let RPP λ (k) be the set of reverse plane partitions of Ferrers shape D λ = {(i, j) ∈ Z >0 × Z >0 : j ≤ λ i } with entries in {0, 1, . . ., k}.
Fomin and Kirillov [7] introduce these objects in order to state this formula for the (generalized) Grothendieck polynomial G w of a permutation w ∈ S n : β |D|−ℓ(w) (i,j)∈D x i ∈ Z[β][x 1 , x 2 , . . ., x n ]. (6.5)This identity is nontrivial to derive from Lascoux and Schützenberger's original definition of Grothendieck polynomials in terms of isobaric divided difference operators [26,28].
Grothendieck polynomials becomes Schubert polynomials on setting β = 0.In [31,32], continuing work of Wyser and Yong [45], the second two authors studied orthogonal and symplectic Grothendieck polynomials G O y and G Sp z indexed by y ∈ I n and z ∈ I FPF n .These polynomials likewise recover the involution Schubert polynomials Ŝy and ŜFPF z on setting β = 0, and it would be interesting to know if they have analogous pipe dream formulas.
The symplectic case of this question is more tractable.The polynomials G Sp z have a formulation in terms of isobaric divided difference operators due to Wyser and Yong [45], which suggests a natural K-theoretic variant of the set FD(z).A formula for G Sp z involving these objects appears in [33, §4].By contrast, no simple algebraic formula is known for the polynomials G O y .It is a nontrivial problem even to identify the correct K-theoretic generalization of ID(y).Problem 6.9.Find a pipe dream formula for the polynomials G O y involving an appropriate "Ktheoretic" generalization of the sets of involution pipe dreams ID(y).

Definition 1 . 3 .
A subset D ⊆ n is a reduced pipe dream if no two pipes in the associated wiring diagram cross more than once.

Theorem 1 . 5 .
If y ∈ I n and z ∈ I FPF n then Ŝy = D∈ID(y) (i,j)∈D
so that the path through the upper-left corner of (2.5) sends the polynomial [M Xy ] to [ Xy ].The variety Xy is the closure of an O n -orbit on Fl n [44, §2.1.2].The path through the lower-right corner of (2.5) is simply the classical Borel map Z[x 1 , . . ., x n ] → H * (Fl n ).We claim [M Xy×1 m ] is constant for fixed y and varying m.Since [M Xy ] is a representative for [ Xy ], the result then follows by Theorem 2.9.For y = 1 ∈ S n , define maxdes(y) = max{i ∈ Z ≥0 : y(i) > y(i + 1)}.Replacing [n] in the definition (2.3) by [maxdes(y)] yields exactly the same variety M Xy .Since maxdes
Fix a set D ⊆ [n] × [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 [43, Lem.3.3].Proof.Let s p ∈ S n 2 be the simple transposition interchanging p and p + 1, and choose a reading order ω on [n] × [n].The sequence word(D, s p ω) is equal to word(D, ω) when {p, p + 1} ⊂ ω(D), and otherwise is obtained by interchanging two adjacent letters in word(D, ω).In the latter case, if ω −1 (p) = (i, j) and ω −1 (p + 1) = (i ′ , j ′ ) are not in adjacent antidiagonals, then word(D, ω) and word(D, s p ω) are in the same Coxeter commutation class.Now suppose υ is a second reading order on [n] × [n]
Theorem 3.7.A subset D ⊆ [n] × [n] is a reduced pipe dream for w ∈ S n if and only if some (equivalently, every) reading word of D is a reduced word for w.Proof.Fix D ⊆ [n] × [n] and w ∈ S n .The set R(w) is a union of Coxeter commutation classes, so word(D) ∈ R(w) if and only every reading word of D belongs to R(w) by Lemma 3.5.Saying that D is a reduced pipe dream for w if and only if word(D) ∈ R(w) is Bergeron and Billey's original definition of an rc-graph in [1, §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 (
[1, Lem.3.2]).If D is a reduced pipe dream for w ∈ S n then D T is a reduced pipe dream for w −1 .Recall that the set ID(z) of involution pipe dreams for z ∈ I n consists of all intersections D ∩ n where D is a reduced pipe dream for z that is almost-symmetric and n = {(j, i)∈ [n]×[n] : i ≤ j}.Theorem 3.9.Suppose z ∈ I n and D ⊆ [n] × [n].The following are equivalent: (a) Some reading word of D is an involution word for z.(b) Every reading word of D is an involution word for z.(c) The set D is a reduced pipe dream for some atom of z.Moreover, if D ⊆ n then D ∈ ID(z) if and only if these equivalent conditions hold.

Lemma 3 . 11 .Theorem 3 . 12 .
Assume n is even.Suppose z ∈ I n is an involution with a symmetric reduced pipe dream D = D T .Then z ∈ I FPF n if and only if {(i, i) : i ∈ [n/2]} ⊆ D. Proof.In fact, a stronger statement holds: for symmetric D and i ∈ [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) / ∈ 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) ∈ D then z(a) = b (hence z(b) = a), and if (i, i) / ∈ D then z(a) = a and z(b) = b.Recall that the set FD(z) of fpf-involution pipe dreams for z ∈ I FPF n consists of all intersections D ∩ = n where D is a reduced pipe dream for z that is symmetric and = n = {(j, i) ∈ [n]× [n] : i < j}.Suppose n is even, z ∈ I FPF n , and D ⊆ [n] × [n].The following are equivalent: (a) Some reading word of D is an fpf-involution word for z.(b) Every reading word of D is an fpf-involution word for z.

Definition 4 . 4 .
The dominant component of a permutation w ∈ S n is dom(w) = dom(D(w)).We say that permutation w ∈ S n is dominant if dom(w) ∈ PD(w).

Lemma 4 . 5 .
A permutation w ∈ S n is dominant if and only if it holds that PD(w) = {dom(w)}, in which case dom(w) = D(w).Proof.If w ∈ S n is dominant then PD(w) = {dom(w)} = {D(w)} since all reduced pipe dreams for w have size ℓ(w) = |D(w)| and contain dom(w) ⊆ D(w).

Definition 4 . 12 .
The shifted dominant component of z ∈ I n is the set shdom(z) = dom(z) ∩ n .Fix z ∈ I n .By Proposition 4.11, we have shdom(z) = dom(D) ∩ n for all D ∈ PD(z).Recall that the shifted Ferrers diagram of a strict partition λ

Corollary 4 . 18 .
Let y, z ∈ I n with y dominant.Then y ≤ z if and only if shdom(y) ⊆ D for some (equivalently, every) D ∈ ID(z).

Theorem 4 .
22 (See[14,  §3]).For each pair of integers 1 ≤ i < j ≤ n, there are unique maps τ ij : I n → I n with the following properties:(a) If y ∈ I n and A ij (y) ∩ A(z) = ∅ for some z ∈ I n then τ ij (y) = z.(b) If y ∈ I n and A ij (y) ∩ A(z) = ∅ for all z ∈ I n then τ ij (y) = y.Moreover, if y ∈ I n and y = τ ij (y) = z, then y(i) = z(i) and y(j) = z(j).
the integers a ∈ [n] with a ≤ z(a) and set b i = z(a i ).Define α min (z) ∈ S n to be the permutation whose inverse is given in one-line notation by removing all repeated letters from b 1 a 1 b 2 a 2 • • • b l a l .For example, if z = 4231 ∈ I 4 then the latter word is 412233 and α min (z) = (4123) −1 = 2341 ∈ S 4 .Additionally, ĉ(z) = c(α min (z)) [13, Lem.3.8].Finally, let ≺ A be the transitive closure of the relation on S n that has v ≺ A w whenever the inverses of v, w ∈ S n have the same one-line representations outside of three consecutive positions where v −1 = • • • cab • • • and w −1 = • • • bca • • • for some integers a < b < c.The relation ≺ A is a strict partial order.Let ∼ A denote the symmetric closure of the partial order A .Theorem 5.5 ([12, §6.1]).Let z ∈ I n .Then A(z) = {w ∈ S n : α min (z) A w} = {w ∈ S n : α min (z) ∼ A w}.Thus A(z) is an upper and lower set of A , with unique minimum α min (z).

ForLemma 5 . 6 .
z ∈ I n , let ID + (z) = w∈A(z) PD(w), so that ID(z) = {D ∈ ID + (z) : D ⊆ n }.Let z ∈ I n .Suppose D and E are subsets of Z >0 × Z >0 with D < ID E. Then D ∈ ID + (z) if and only if E ∈ ID + (z).Proof.If D ⋖ PD E then we have D ∈ ID + (z) if and only if E ∈ ID + (z) by Theorem 5.2.Assume D ⋖ ID E and let i < j and k be as in Definition 5.3.

6
then we have α FPF min (z) = (162345) −1 = 134562 ∈ S 6 .One can check that ĉFPF (z) = c(α FPF min (z)) [13, Lem.3.8].Define ≺ A FPF to be the transitive closure of the relation in S n that has v ≺ A FPF w whenever the inverses of v, w ∈ S n have the same one-line representations outside of four consecutive positions where v −1 = • • • adbc • • • and w −1 = • • • bcad • • • for some integers a < b < c < d.This is a strict partial order on S n .Let ∼ A FPF denote the symmetric closure of the partial order A FPF .Theorem 5.11 ([12, §6.2]).Let z ∈ I FPF n .Then