Shifted insertion algorithms for primed words

This article studies some new insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed. When primes are ignored in the input word these algorithms reduce to known correspondences, namely, a shifted form of Edelman-Greene insertion, Sagan-Worley insertion, and Haiman's shifted mixed insertion. These maps have the property that when the input word varies such that one output tableau is fixed, the other output tableau ranges over all (semi)standard tableaux of a given shape with no primed diagonal entries. Our algorithms have the same feature, but now with primes allowed on the main diagonal. One application of this is to give another Littlewood-Richardson rule for products of Schur $Q$-functions. It is hoped that there will exist set-valued generalizations of our bijections that can be used to understand products of $K$-theoretic Schur $Q$-functions.


Introduction
This article studies some new insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed. We summarize our main results below, and then discuss the problems that motivate our constructions.

Outline
Let S Z be the group of permutations of the integers with finite support, and set s i := (i, i + 1) ∈ S Z for i ∈ Z. A reduced word for w ∈ S Z is a minimal length integer sequence a 1 a 2 · · · a n with σ = s a 1 s a 2 · · · s an . Write R(σ) for the set of reduced words for σ ∈ S Z . Suppose a = a 1 a 2 · · · a n is a reduced word for an element of S Z . Then there is a unique subword a i 1 a i 2 · · · a im of maximal length such thatâ := a im · · · a i 2 a i 1 a 1 a 2 · · · a n is also a reduced word. We refer to the indices in {1, 2, . . . , n} \ {i 1 , i 2 , . . . , i m } as the commutations in a.
One can show thatâ is always a reduced word for an involution z = z −1 ∈ S Z . If a is of minimal length such thatâ ∈ R(z), then a is an involution word for z. Write R inv (z) for the set of involution words for z = z −1 ∈ S Z . For a more constructive definition of this set, see Section 2.1.
Involution words have been studied previously in different forms and under various names, for example, in [5,11,15,17,20,34]. We are concerned here with the following slight generalization. A primed involution word for z = z −1 ∈ S Z is any word formed by adding primes to the entries indexed by a subset of commutations in some a ∈ R inv (z). Such a word is a sequence of letters in the primed alphabet {· · · < 1 ′ < 1 < 2 ′ < 2 < . . . }. Write R + inv (z) for the set of primed involution words for z. The number κ of nontrivial cycles of z is also the number of commutations in any a ∈ R inv (z), so |R + inv (z)| = 2 κ |R inv (z)|. For example, if z = 321 ∈ S 3 ⊂ S Z , then R(z) = {121, 212}, R inv (z) = {12, 21}, and R + inv (z) = {12, 1 ′ 2, 21, 2 ′ 1}. For any word a, let Incr ∞ (a) denote the set of sequences (a 1 , a 2 , a 3 , . . . ) where each a i is a weakly increasing word such that a = a 1 a 2 a 3 · · · . For a set of words A, let Incr ∞ (A) = a∈A Incr ∞ (A). Our first main result is the following theorem, which is explained in greater detail in Section 3.1.
Theorem (See Theorem 3.6 and Corollary 3.9). Let z = z −1 ∈ S Z . There is a bijection from R + inv (z) (respectively, Incr ∞ (R + inv (z))) to the set of pairs (P, Q) where P is a shifted tableau with increasing rows and columns and no primed entries on the diagonal whose row reading word is in R + inv (z), and Q is a standard (respectively, semistandard) shifted tableau of the same shape. Restricting a → (P O EG (a), Q O EG (a)) to unprimed words gives the map called involution Coxeter-Knuth insertion in [12,28] and orthogonal Edelman-Greene insertion in [29]. The latter, in turn, is a special case of the shifted Hecke insertion algorithm from [14,33]. As explained in Remark 2.3, the correspondence a → (P O EG (a), Q O EG (a)) is the "orthogonal" counterpart to a "symplectic" shifted insertion algorithm studied in [16,28,29].
It is an open problem to find a "primed" generalization of shifted Hecke insertion that extends our bijection a → (P O EG (a), Q O EG (a)). The image of such a map should consist of pairs of shifted tableaux (P, Q) of the same shape, in which P has increasing rows and columns with no primed entries on the diagonal, and Q is an arbitrary (semistandard) set-valued shifted tableau in the sense of [23, §9.1]. It is less clear what superset of R + inv (z) should be the domain of such a correspondence. As discussed in the next subsection, generalizing shifted Hecke insertion in this way would have interesting applications. Our main theorem is a first step towards this goal.
Besides constructing the map a → (P O EG (a), Q O EG (a)), we also seek to understand how a can vary when P O EG (a) is held constant, and how such changes affect Q O EG (a). Our second set of results, sketched below and explained more thoroughly in Section 3.2, fully solves this problem.
Theorem (See Theorem 3.16 and Corollary 3.17). There are explicit operators ock i on primed words which act by changing at most three consecutive letters, along with operators d i on standard shifted tableaux which act by changing at most three consecutive entries, such that if a is a primed involution word then Q O EG (ock i (a)) = d i (Q O EG (a)), and if a and b are both primed involution words then P O EG (a) = P O EG (b) if and only a = ock i 1 ock i 2 · · · ock i k (b) for some sequence i 1 , i 2 , . . . , i k .
Proving this result is unexpectedly difficult. Our derivation occupies all of Section 4, which takes up a significant part of this rather long document.
We prove some additional results. In Section 3.3, we describe a variation of Sagan-Worley insertion from [35,38] whose domain is the set of all primed biwords, which coincides with a → (P O EG (a), Q O EG (a)) on words whose unprimed forms have distinct letters. Section 3.3 investigates a related "primed" extension of Haiman's shifted mixed insertion algorithm from [9].
Finally, there is a notion of primed involution words for any Coxeter system, which we discuss in Section 5. There, generalizing some results of Hansson and Hultman [15], we derive a minimal set of "primed braid relations" that span all such words associated to a given element.

Motivation
The Schur P -function of a strict partition λ is the generating function P λ = T x T for all semistandard shifted tableaux T of shape λ with no primed entries on the diagonal. The Schur Q-function Q λ is defined in the same way but without excluding primes from the diagonal, or more directly as the scalar multiple Q λ = 2 ℓ(λ) P λ . It is well-known that both power series are symmetric functions that are Schur positive, and that the set of all P λ 's (respectively, all Q λ 's) is a Z-basis for a ring with nonnegative integer structure constants [37].
Ikeda and Naruse introduce K-theoretic analogues GP λ and GQ λ for the Schur P -functions and Q-functions in [23]. These power series are also symmetric, and may be defined similarly as the generating functions for all semistandard set-valued shifted tableaux of a given shape, where for GP λ primed entries are again prohibited from appearing in diagonal positions. 1 One recovers P λ and Q λ by taking the homogeneous terms of lowest degree in GP λ and GQ λ , respectively.
It is conjectured in [23] that the set of all GP λ 's (respectively, all GQ λ 's) is a basis for a ring with nonnegative structure constants. For the GP λ 's this follows from results in [6]. For the GQ λ 's, surprisingly, Ikeda and Naruse's conjecture is technically still unresolved, though only just. It follows from [23] that each product GQ λ GQ µ is a formal sum of GQ ν 's. However, in general, it remains to show that this expansion has finitely many terms and to give an interpretation of its (positive) coefficients. 2 These difficulties have to do with the fact that GQ λ is no longer a scalar multiple of GP λ . In fact, it is an open problem to describe the expansion of GQ λ in terms of GP µ 's; see [25,Conj. 5.15].
There is a bijective approach to proving that the K-theoretic Schur P -functions generate a ring, which we sketch below. The results in this article are a first step toward extending this strategy to handle the K-theoretic Schur Q-functions.
For each even integer n > 0, let I fpf n denote the set of fixed-point-free involutions in the symmetric group S n := s 1 , s 2 , . . . , s n−1 . Each element z ∈ I fpf n has an associated set of symplectic Hecke words H Sp (z) defined in [28, §1.3]. This set is infinite unless z is 1 fpf := (1, 2)(3, 4) · · · (n − 1, n). Each word in H Sp (z) is a finite integer sequence that does not begin with an odd letter. The shortest words in H Sp (z) are the minimal length sequences a 1 a 2 · · · a k with z = s a k · · · s a 2 s a 1 1 fpf s i 1 s a 2 · · · s a k .
Given z ∈ I fpf n and a strict partition λ, define KP z := φ∈Incr∞(H Sp (z)) x φ where x φ := i x ℓ(a i ) i for φ = (a 1 , a 2 , a 3 , . . . ) and let KP λ := T x T where the sum is over all (semistandard) weak set-valued shifted tableaux of shape λ with no primed entries on the diagonal, as described in [14,Def. 3.1] (and with x T as defined in the same place). These power series are related to GP λ by the identity GP λ = ω(KP λ ), where ω is the automorphism of the algebra of symmetric functions sending each Schur function s λ → s λ ⊤ [25,Cor. 6.6]. In turn, each KP z is related to KP λ by: Theorem (See [28,Thm. 4.5]). Let z ∈ I fpf n . There is a bijection φ → (P Sp (φ), Q Sp (φ)) from Incr ∞ (H Sp (z)) to the set of pairs (P, Q) where P is a shifted tableau with increasing rows and columns whose row reading word is in H Sp (z), and Q is a weak set-valued shifted tableau of the same shape with no primed entries on the diagonal. Moreover, it always holds that x φ = x Q Sp (φ) . This bijection is called symplectic Hecke insertion in [28]. If a = a 1 a 2 · · · a k ∈ H Sp (z) then we set P Sp (a) = P Sp (φ) and Q Sp (a) = Q Sp (φ) for φ = (a 1 , a 2 , . . . , a k , ∅, ∅, . . . ). The value of P Sp (φ) depends only on the underlying word, but not on its division into weakly increasing factors. All letters in a symplectic Hecke word for z ∈ I fpf n are in {1, 2, . . . , n−1}, so there are only finitely many shifted tableaux with increasing rows and columns that can have row reading words in H Sp (z). It follows that KP z is the finite sum T ∈{P Sp (a):a∈H Sp (z)} KP shape(T ) .
It is clear from the results about symplectic Hecke words in [28, §1.3] that (φ, ψ) → φ ⊕ ψ is a bijection Incr ∞ (H Sp (y)) × Incr ∞ (H Sp (z)) ∼ − → Incr ∞ (H Sp (y × z)). Therefore KP y KP z = KP y×z . In turn, if the largest part of λ is less than n − 1, then there exists z fpf λ ∈ I fpf n (with an explicit formula) such that KP λ = KP z fpf λ [31,Thm. 4.17]. Since ω is an algebra automorphism, we have where e ν λµ is the number of tableaux in P Sp (a) : a ∈ H Sp z fpf λ × z fpf

Acknowledgments
This work was partially supported by Hong Kong RGC grants ECS 26305218 and GRF 16306120. I am especially grateful to Travis Scrimshaw for many useful conversations, and for hosting a productive visit to the University of Queensland. I also thank Zach Hamaker, Joel Lewis, and Brendan Pawlowski for helpful discussions.

Preliminaries
In this section we review some basic facts and background material. Throughout, we write Z for the set of integers. When n ∈ Z is nonnegative, we let [n] := {i ∈ Z : 0 < i ≤ n}.

Involution words
We use the term word to mean a finite sequence of integers a = a 1 a 2 · · · a n . We write ℓ(a) := n for the length of a word. Recall from the introduction that R(σ) denotes the set of reduced words for a permutation σ ∈ S Z := s i : i ∈ Z . Let ≡ be the equivalence relation on words that has aX(X + 1)Xb ≡ a(X + 1)X(X + 1)b and aXY b ≡ aY Xb for all words a, b and all X, Y ∈ Z with |X − Y | > 1. The following is well-known: . Each set R(σ) for σ ∈ S Z is an equivalence class under ≡. An arbitrary word is a reduced word for some permutation in S Z if and only if its ≡-equivalence class contains no words with equal adjacent letters.
There is a unique associative product • : S Z × S Z → S Z such that σ • s i = σ if σ(i) > σ(i + 1) and σ • s i = σs i if σ(i) < σ(i + 1) for each i ∈ Z [22,Thm. 7.1]. A reduced word for σ ∈ S Z is thus a sequence a 1 a 2 · · · a n of shortest possible length such that σ = 1 • s a 1 • s a 2 • · · · • s an . Analogously, an involution word for z ∈ S Z is a word a 1 a 2 · · · a n of shortest possible length such that z = s an • · · · • s a 2 • s a 1 • 1 • s a 1 • s a 2 • · · · • s an . 4 This definition is equivalent to the one given in the introduction. Any involution word for z ∈ S Z is itself a reduced word for some (usually shorter) permutation.
Let I Z := {σ ∈ S Z : σ = σ −1 } and I n := S n ∩ I Z when 0 < n ∈ Z. If z ∈ I Z and i ∈ Z then s i • z • s i = z when z(i) > z(i + 1), while s i • z • s i = zs i = s i z when i and i + 1 are fixed points of z, and otherwise s i • z • s i = s i zs i . This implies that z ∈ S Z has an involution word if and only if z ∈ I Z . If z = s an • · · · • s a 2 • s a 1 • 1 • s a 1 • s a 2 • · · · • s an then a 1 a 2 · · · a n ∈ R inv (z) if and only if for each i ∈ [n]. As in the introduction, we denote the set of involution words for z ∈ I Z by R inv (z). For example, we have R inv (3412) = {132, 312} and R inv (4231) = {123, 231, 213, 321}.
Let≡ be the transitive closure of ≡ and the relation with XY a≡ Y Xa for all words a and X, Y ∈ Z. Hu and Zhang derive the following analogue of Proposition 2.1 in [17]: . Each set R inv (z) for z ∈ I Z is an equivalence class under≡. An arbitrary word is an involution word for some element of I Z if and only if its≡-equivalence class contains no words with equal adjacent letters.
This result is also a special case of [15,Thm. 4.1], which we review in Section 5.2.
Suppose z ∈ I Z . An index i is a commutation for a word a 1 a 2 · · · a n ∈ R inv (z) if both a i and 1 + a i are fixed points of s a i−1 • · · · • s a 2 • s a 1 • 1 • s a 1 • s a 2 • · · · • s a i−1 . A primed involution word for z is a primed word whose unprimed form is in R inv (z) and whose primed letters occur only at indices that are commutations. We denote the set of such words by R + inv (z). For example, The number of commutations in each involution word for an element z ∈ I Z is the absolute length ℓ abs (z) := |{i ∈ Z : i < z(i)}|. Thus |R + inv (z)| = 2 ℓ abs (z) |R inv (z)|. Remark 2.3. As explained in [39, §2.2-2.3] or [10, §8.1], the set I n ⊂ S n indexes the orbits of the orthogonal group O n (C) acting on the type A n−1 flag variety Fl n := GL n (C)/B. In [3], Brion derives a formula for the cohomology classes of the closures of these orbits, involving a certain directed graph on the set of orbits. The directed paths that arise in Brion's cohomology formula (from the orbit indexed by z to the unique dense orbit) are in bijection with R + inv (z). This is one motivation for studying these sets. This is also why we will often include the adjective "orthogonal" with constructions involving R + inv (z). There is a parallel "symplectic" story for a different analogue of reduced words corresponding to the orbits of Sp 2n (C) acting on Fl 2n (see, e.g., [13,28,31,40]).
The following is a special case of Lemma 5.5. It is also easy to check directly. Proposition 2.4. Suppose a 1 a 2 · · · a n ∈ R + inv (z) for some z ∈ I Z . (a) If ⌈a i ⌉ = ⌈a i+1 ⌉ ± 1 for i ∈ [n − 1] then at most one of a i or a i+1 is in Z ′ .
Write≡ for the transitive closure of the relation with aXY b≡ aY Xb (2.1) for all primed words a, b and letters X, Y ∈ Z ⊔ Z ′ such that |⌈X⌉ − ⌈Y ⌉| = 1, as well as with for all primed words a, b and unprimed letters X, Y ∈ Z such that |X − Y | = 1, and finally with Xa≡ X ′ a and XY a≡ Y Xa for all primed words a and unprimed letters X, Y ∈ Z. For example, we have This extends our earlier definition of the symbol≡. The following is a special case of Theorem 5.7.
Proposition 2.5. Each set R + inv (z) for z ∈ I Z is an equivalence class under≡.

Tableaux
A partition of an integer n ≥ 0 is a finite sequence of integers λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ k > 0) that sum to n. In this event we set ℓ(λ) := k, λ i := 0 for i > ℓ(λ), and |λ| := i λ i = n. A partition is strict if the parts λ i are all distinct. The diagram of a partition λ is the set of positions The shifted diagram of a strict partition µ is the set In this article, a tableau of shape λ means an arbitrary map D λ → Z and a shifted tableau of shape µ means an arbitrary map SD µ → Z ⊔ Z ′ . If T is a (shifted) tableau then we write T ij for the value assigned to some position (i, j) in its domain. We draw tableaux in French notation, so that row indices increase from bottom to top and column indices increase from left to right. If S = 3 5 7 1 2 4 6 and T = then S is a tableau and T is a shifted tableau of shape λ = (4, 3), and S 23 = 7 while T 23 = 5 ′ . The (main) diagonal of a shifted tableau is the set of positions (i, j) in its domain with i = j. We sometimes refer to positions (i, j) in the domain of a tableau as its boxes.
A (shifted) tableau is increasing if its rows and columns are strictly increasing. An increasing (shifted) tableau of shape λ is standard if it contains an entry equal to i or i ′ for each i ∈ [|λ|].
In our examples S and T are both standard. A (shifted) tableau is semistandard if its entries are all positive and its rows and columns are weakly increasing, such that no primed entry is repeated in any row and no unprimed entry is repeated in any column. It is sometimes required that a (semi)standard shifted tableau not have primed entries in diagonal positions, but we do not adopt this convention.
Suppose T is a map from a finite subset of Z × Z to a totally ordered set. The row reading word of T is the sequence row(T ) formed by listing the values T ij as (i, j) ranges over the domain of T in the order that makes (−i, j) increase lexicographically. The column reading word of T is the sequence col(T ) formed by listing the values T ij as (i, j) ranges over the elements of the domain of T in the order that makes (j, −i) increase lexicographically. Above, we have row(S) = 3571246, col(S) = 3152746, row(T ) = 35 ′ 71 ′ 2 ′ 4 ′ 6, and col(T ) = 1 ′ 32 ′ 5 ′ 4 ′ 76.
When T is a shifted tableau, we form unprime(T ) by removing all primes from T 's entries.
Proposition 2.6. Suppose T is a shifted tableau such that row(T ) is a primed involution word for an element of I Z . Then T is increasing if and only unprime(T ) is increasing.
Proof. It is clear that if unprime(T ) is increasing then T is also increasing. Assume conversely T that is increasing. Since row(unprime(T )) is an (unprimed) involution word and therefore a reduced word, no row of T can contain both x ∈ Z and x ′ ∈ Z ′ . It remains to show that this property also applies to the columns of T . Arguing by contradiction, suppose that x ′ and x occur as consecutive entries in some column of T . After possibly omitting some of the initial rows of T , we may assume without loss of generality that all such entries appear in the first and second rows. Let j ≥ 2 be the first column with T 1j = x ′ and T 2j = x for some x ∈ Z.
If position (2, j + 1) is occupied in T , then we cannot have ⌈T 2,j+1 ⌉ = x + 1, as then it would be necessary to have T 1j = x ′ and T 1,j+1 = x ′ + 1, which is impossible by Proposition 2.4. We must therefore have ⌈T 1,j−1 ⌉ = x−1 as otherwise row(unprime(T )) would be equivalent under≡ to a word with adjacent letters both equal to x, contradicting Proposition 2.2. If j = 2 then row(unprime(T )) would be equivalent under≡ to a word starting with x(x − 1)x, contradicting Proposition 2.2 since x(x − 1)x≡(x − 1)xx. But if j > 2, then the inequalities T 1,j−1 < T 2,j−1 < T 2j would force us to have T 1,j−1 = x ′ − 1 and T 1j = x ′ , which is again impossible by Proposition 2.4.

Insertion algorithms
This section contains our main results, which are organized around three insertion algorithms a → (P, Q) that assign pairs of shifted tableaux of the same shape to certain sets of primed (bi)words. In this regard, our maps are similar to various super-RSK correspondences that exist in the literature (see, e.g., [2,24,32,36]). The first of our algorithms, to which we will devote the most attention, is a shifted version of Edelman-Greene insertion [7]. The second is a variant of Sagan-Worley insertion [35,38], while the third is an extension of shifted mixed insertion [9].

Shifted Edelman-Greene insertion
The insertion algorithm described below is central to what follows. For this reason, we include a number of running examples in our definitions, which incorporate some auxiliary data that will not be used until Section 4.
Definition 3.1. Suppose T is an increasing shifted tableau and u ∈ Z ⊔ Z ′ is such that row(T )u is a primed involution word for some element of I Z . We define sequences of pairs (x 1 , y 1 ), (x 1 ,ỹ 1 ), (x 2 , y 2 ), (x 2 ,ỹ 2 ), . . . , (x m , y m ), (x m ,ỹ m ) ∈ Z × Z, according to the following inductive procedure: (1) Assume i ≥ 1 and that T i−1 and u i−1 are given. On the ith iteration, the number u i−1 is inserted into either the ith row or ith column of T i−1 . We insert into the ith column if any of (x 1 , y 1 ), . . . , (x i−1 , y i−1 ) belong to {(x, x) : x ∈ Z}, and otherwise we insert into the ith row.
(2) If ⌈u i−1 ⌉ is not less than any entry in the current (possibly empty) row or column, then we set m := i and let (x i , y i ) = (x i ,ỹ i ) be the first position (x, y) ∈ Z × Z with 1 ≤ x ≤ y that is unoccupied in the current row or column. We then form T i from T i−1 by filling this position by when inserting into an empty row with u i−1 = 7 ′ , or when inserting into a nonempty row with u i−1 = 7 ′ . We say that our insertion process ends in column insertion if on this final iteration we are inserting into a column, or if x i = y i and u i−1 is primed. Otherwise, we that the process ends in row insertion.
(3) Suppose instead that ⌈u i−1 ⌉ is less than some entry in the current row or column. Let v i and w i be the smallest entries in the current row or column with ⌈u i−1 ⌉ ≤ ⌈v i ⌉ and ⌈u i−1 ⌉ < ⌈w i ⌉.
Define (x i , y i ) and (x i ,ỹ i ) to be the respective positions of v i and w i .
(a) Suppose (x i , y i ) = (x i ,ỹ i ). Then we set u i := u i−1 + 1. If v i and w i are both primed or both unprimed, then we define T i := T i−1 . Otherwise, we form T i from T i−1 by reversing the primes on v i and w i . For example, we could have when inserting into a row with u i−1 = 4 ′ , v i = 4, w i = 5, and u i = 5 ′ , or . We refer to these sequences as the weak and strict bumping paths that result from inserting u into T .
The corresponding bumping paths are Proposition 3.14 will show that if row(T )u is a primed involution word then so is row(T O ← − u). We can therefore iterate the above insertion process as follows: Definition 3.3. Given a primed involution word a = a 1 a 2 · · · a n for some element of ← − a n and let Q O EG (a) be the standard shifted tableau with the same shape as P O EG (a) that has i (respectively, i ′ ) in the box added by inserting a i into P O EG (a 1 a 2 · · · a i−1 ) when this ends in row insertion (respectively, column insertion).
We refer to a → (P O EG (a), Q O EG (a)) as orthogonal Edelman-Greene insertion.
The original Edelman-Greene correspondence a → (P EG (a), Q EG (a)) from [7], sending reduced words a ∈ R(σ) for σ ∈ S n to pairs of (unshifted) tableaux, may be embedded in Definition 3.3 in the following way. Take any involution word b for z = (0, n)(−1, n−1)(−2, n−2) · · · (−n+1, 1) ∈ I Z . Then a → ba is an injective map R(σ) ֒→ R inv (σ −1 zσ) and we recover P EG (a) from P O EG (ba) by omitting the first n columns, while Q EG (a) is given by omitting the same columns of Q O EG (ba) and subtracting ℓ(b) from the remaining entries, which are all unprimed numbers.
As noted in the introduction, a → (P O EG (a), Q O EG (a)) restricted to unprimed involution words reduces to a map previously studied in [12,28,29]. Our inclusion of primes may seem like a minor generalization. However, there seems to be no simple way to derive our main results as corollaries of what is known about the unprimed form of orthogonal Edelman-Greene insertion. • Each of T = T 0 , T 1 , . . . , T m = T O ← − u is increasing with no primes on the diagonal.
• It always holds that u i−1 < u i and ⌈u i ⌉ = ⌈w i ⌉.
Our first result about orthogonal Edelman-Greene insertion is the following theorem.
Theorem 3.6. For each z ∈ I Z , the map a → (P O EG (a), Q O EG (a)) is a bijection from the set of primed involution words R + inv (z) to the set of pairs (P, Q) of shifted tableaux of the same shape, in which P is increasing with no primes on the diagonal, Q is standard, and row(P ) ∈ R + inv (z).
The theorem remains true on replacing the two instances of R + inv (z) by R inv (z) if we further require Q to have no primes on the diagonal [12,Thm. 5.19]. It is routine, following [28, §3.3] or [33, §5.3], to describe a reverse insertion algorithm that gives the inverse map (P O EG (a), Q O EG (a)) → a. However, we will end up deriving Theorem 3.6 by another method at the end of Section 4.2.
If T is a shifted tableau, then we construct unprime(T ) from T by removing the primes from all entries, and we form unprime diag (T ) by removing the primes from all diagonal entries. Clearly for any primed involution word a. An integer i is a descent of a standard shifted tableau T if either (a) i and i + 1 both appear in T with i + 1 in a row strictly after i, (b) i ′ and i ′ + 1 both appear in T with i ′ + 1 in a column strictly after i ′ , or (c) i and i ′ + 1 both appear in T . Let Des(T ) denote the set of descents of T . If T is as in (2.4), then Des(T ) = {1, 3, 6}. If a = a 1 a 2 · · · a n is a primed word then let Des(a) := {i ∈ [n − 1] : a i > a i+1 }. These descent sets are related as follows: Proposition 3.7. Let a ∈ R + inv (z) for some z ∈ I Z . Then Des(a) = Des(Q O EG (a)).
When a is a word in a totally ordered alphabet and N is a nonnegative integer, we let Incr N (a) denote the set of N -tuples of weakly increasing, possibly empty subwords (a 1 , a 2 , · · · , a N ) such that a = a 1 a 2 · · · a N . We again define Incr ∞ (a) to be the set of infinite sequences (a 1 , a 2 , · · · ) of weakly increasing words such that a = a 1 a 2 · · · ; here, all but finitely many a i must be empty. If A is a set of words and N ∈ {0, 1, 2, . . . } ⊔ {∞} then we let Incr N (A) = a∈A Incr N (a).
We discuss an application mentioned in Section 1.2. Let x i for i ∈ Z be commuting indeterminates. Given a shifted tableau T , let x T := i∈Z x c i i where c i is the number of entries in T equal to i or i ′ . The Schur Q-function of a strict partition λ is the formal power series where T ranges over all semistandard shifted tableaux of shape λ. It is well-known that the Schur Q-functions are symmetric in the x i variables and linearly independent [37]. We present another proof that they span a ring with nonnegative integer structure coefficients.
Lemma 3.10. If λ is a strict partition, then P O EG (a) for all a ∈ R + inv (z λ ) is equal to the increasing shifted tableau T of shape λ with T ij = i + j − 1 for all (i, j) ∈ SD λ .
Proof. One can check that row(T ) ∈ R inv (z λ ) directly, or this is equivalent to [11,Prop. 3.9]. [29,Lem. 5.5] assserts that P O EG (a) = T for all a ∈ R inv (z λ ) ⊂ R + inv (z λ ), so |R inv (z λ )| is the number of standard shifted tableau of shape λ with no primes on the diagonal. Thus, by Theorem 3.6 and (3.1), the number of primed words a ∈ R + inv (z λ ) with P O EG (a) = T must be 2 ℓ(λ) |R inv (z λ )|. But this is the size of R + inv (z λ ) since ℓ(λ) = ℓ abs (z λ ) is the number of nontrivial cycles in z λ .  inv (z)} Q shape(T ) and Q z λ = Q λ . As in the introduction, given elements v ∈ S m and w ∈ S n , let v ×w ∈ S m+n be the permutation mapping i → v(i) for i ∈ [m] and m + j → m + w(j) for j ∈ [n].
Corollary 3.12. If λ and µ are strict partitions then Q λ Q µ = ν g ν λµ Q ν where the sum is over strict partitions ν and g ν λµ is the number of elements in P O EG (a) : a ∈ R + inv (z λ × z µ ) of shape ν.
Proof. Let y ∈ I Z ∩ S m and z ∈ I Z ∩ S n . It is straightforward to deduce from Proposition 2.5 that Incr ∞ (R + inv (y × z)) is in bijection with the product Incr ∞ (R + inv (y)) × Incr ∞ (R + inv (z)) via the map ((a 1 , a 2 , . . . ), (b 1 , b 2 , . . . )) → (a 1 c 1 , a 2 c 2 , . . . ) where c i is formed by adding m to each letter of b i . This implies that Q y Q z = Q y×z , and so the result follows from Corollary 3.11.

Orthogonal Coxeter-Knuth equivalence
An essential property of orthogonal Edelman-Greene insertion is that the fibers of a → P O EG (a) are equivalence classes for a simple relation on primed words, which we define below. Let ock denote the operator that acts on 1-and 2-letter primed words by interchanging for all X, Y ∈ Z. In addition, let ock act on 3-letter primed words as the involution interchanging for all X, Y ∈ Z and all A, B, C ∈ Z ⊔ Z ′ with ⌈A⌉ < ⌈B⌉ < ⌈C⌉, while fixing any 3-letter words not of these forms. Given a primed word a = a 1 a 2 a 3 · · · a n and i ∈ [n − 2], we define ock −1 (a) := ock(a 1 )a 2 a 3 · · · a n , ock 0 (a) := ock(a 1 a 2 )a 3 · · · a n , ock i (a) := a 1 · · · a i−1 ock(a i a i+1 a i+2 )a i+3 · · · a n ,  Proof. This is clear unless i ∈ [ℓ(a) − 2] and ⌈a i ⌉ = ⌈a i+2 ⌉, but if this happens then Proposition 2.4 tells us that i + 1 is not a commutation in a and that at most one of i or i + 2 is a commutation.
The transitive closure of the relation on unprimed words with a ∼ ock i (a) for all i > 0 is often called Coxeter-Knuth equivalence [7,Def. 6.19]. We define orthogonal Coxeter-Knuth equivalence O ∼ to be the transitive closure of the relation on primed words with a Proposition 3.14. Suppose T is an increasing shifted tableau and u ∈ Z ⊔ Z ′ is such that row(T )u is a primed involution word for an element of I Z . Then a is a primed involution word then a O ∼ row(P O EG (a)).
If there is no index p with x p = y p , then it is easy to see that Suppose p is the first index with x p = y p . Then it is also easy to see that row There are two cases to consider, according to whether y p andỹ p are equal. First assume that Since row(T p−1 ) = row(U ) and col(T p ) = col(V ), it suffices to show that then row(U ) O ∼ col(V ). Form the northeast (respectively, southwest) diagonal reading words of U (and similarly for V ) by listing the entries U ij in the order that makes (j − i, i) (respectively, (j − i, −i)) increase lexicographically. In our example, these words for U are 1 ′ 3547 ′ and 531 ′ 47 ′ , respectively. Then defineŨ andṼ by removing the main diagonals from U and V . It is easy to see thatŨ =Ṽ .
We now appeal to [28, §2.2], which contains several general lemmas that relate our various reading words. These lemmas are stated in terms of tableaux without primed entries, but may be applied to our situation by Proposition 2.4. In particular, [28,Lem. 2.8] implies that row(U ) is related under Coxeter-Knuth equivalence to the southwest diagonal reading word of U , [28, Lem. 2.9] implies that col(V ) is related under Coxeter-Knuth equivalence to the northeast diagonal reading word of V , and [28, Cor. 2.10] implies that the two diagonal reading words ofŨ =Ṽ are Coxeter-Knuth equivalent. It is therefore enough to show that reading the diagonal of U in the southwest diagonal reading order gives a word that is equivalent under O ∼ to the word given by reading the diagonal of V in the northeast diagonal reading order. This is straightforward since both words have at most one primed letter; for example, we have 531 We are left to consider the second case in which y p =ỹ p . By Remark 3.5 this can only occur when Moreover, it is easy to see that the index of T pp in row(T p−1 ) is a commutation, so it follows from Proposition 2.4 that u p−1 must be unprimed. Define Then row(T p−1 ) O ∼ row(U ) and col(T p ) = col(U ), so it suffices to show that row(U ) O ∼ col(U ). This follows by repeating the argument in the previous paragraph V replaced by U . In this setting, the key claim is that reading the main diagonal of U in either diagonal reading order gives words that are equivalent under O ∼. This holds since the diagonal has no primed entries.
The converse of the preceding result also holds. The proof of this fact is more difficult, and requires us to understand precisely how the operators then shword(T ) = 4521376, for example. One can check that i ∈ Des(T ) if and only if i + 1 appears before i in the sequence shword(T ).
Let n be the size of the domain of T . For each i ∈ [n], write i for the unique position of T containing i or i ′ , and define s i (T ) to be the shifted tableau formed from T as follows: • If i and i+1 are in the same row or column then reverse the primes on the entries of whichever of these positions is off the diagonal; then, if both i−1 and i+1 (respectively, i and i+2 ) are on the diagonal when i − 1 ∈ [n] (respectively, i + 2 ∈ [n]), and their entries are not both primed or both unprimed, also reverse the primes on these entries.
• Otherwise, swap i with i + 1 and i ′ with i ′ + 1.
If T is the shifted tableau in (3.2), then , and s 4 (T ) = Next, for each i ∈ Z, we construct a shifted tableau d i (T ) from T as follows. If i ∈ {−1, 0} and i + 2 ∈ [n] then we form d i (T ) from T by swapping i + 2 with i ′ + 2. If i ∈ [n − 2] then we set If i is an integer with i + 2 / ∈ [n], then we set d i (T ) := T . When T is again as in (3.2), this gives Restricted to standard shifted tableaux with no primes on the diagonal, the operators d i for i > 0 coincide with the maps ψ i+1 defined in [1, §6]. The definitions of d i and ψ i+1 diverge for tableaux with primed entries on the diagonal. However, [1, Thm. 6.3] (stating that {ψ i } 1<i<n is a dual equivalence for standard shifted tableaux) should still be true if one replaces ψ i by d i−1 .
Before relating d i , ock i , P O EG , and Q O EG , we note a few useful properties. Recall that unprime diag (T ) is formed by removing all primes from the diagonal of T . Let primes(T ) be the total number of primed positions in T and set primes diag (T ) := primes(T ) − primes(unprime diag (T )).
Proposition 3.15. Suppose T is a standard shifted tableau with n boxes. Let j for j ∈ [n] denote the unique box of T containing j or j ′ . For each i ∈ Z, the following properties hold: In addition, for each i ∈ [n − 2], the following properties hold: . Assume i + 2 is between i and i + 1 in shword(T ). If i and i+1 are not in the same row or column, then shword(s i (T )) is formed from shword(T ) by reversing the positions of i and i + 1, so i + 2 is also between i and i + 1 in shword(s i (T )) and we have d i (d i (T )) = s i (s i (T )) = T . If i and i+1 are both in the same row or column, but neither box is on the diagonal, then one box must be primed and the other must be unprimed for i+ 2 to appear between i and i+ 1 in shword(T ). In this case, reversing the primes on these boxes preserves the property that i + 2 appears between i and i + 1 in the shifted reading word, so again d i (d i (T )) = s i (s i (T )) = T .
Assume further that i and i+1 appear in the same row or column and at least one of the boxes is on the diagonal. Then i , i+1 , and i+2 must be arranged in T as i + 2 for i + 2 to appear between i and i + 1 in shword(T ). Applying d i interchanges the first two of these cases, and in the second two cases we have where T has n boxes. For each j ∈ [n], let j be the unique box of T containing j or j ′ . Since d i applied to T can only change the entries of i , i+1 , and i+2 , the identity is also clear if none of these boxes is on the diagonal.
If exactly one of the boxes j for j ∈ {i, i+1, i+2} is on the diagonal, then it is straightforward to check that j comes before the other two boxes in both the row reading word order and column reading row order. In this case, whether or not j is between the other two elements of {i, i + 1, i + 2} in shword(T ) therefore does not depend on whether the entry in j is primed, If two of i , i+1 , or i+2 are on the diagonal, then we must for some q > 0, and it is easy to verify unprime diag (d i (T )) = d i (unprime diag (T )) directly. This exhausts the cases to consider.
(c) It suffices to observe that if i + 2 is between i and i + 1 in shword(T ), then i−1 and i+1 cannot both be on the diagonal, and that if i is between i + 1 and i + 2 in shword(T ) then i+1 and i+3 cannot both be on the diagonal. Now assume i ∈ [n − 1].
(d) This holds since applying d i to T either preserves which diagonal positions are primed or reverses the primes on exactly two diagonal boxes that are not both primed or both unprimed.
(e) Note that if i + 2 is between i and i + 1 in shword(T ) (respectively, if i + 2 is between i and i + 1 in shword(T )) but i and i+2 are not both on the diagonal, then either i and i+1 (respectively, i+1 and i+2 ) are not in the same row or column or exactly one of these boxes has a primed entry, so primes(T ) = primes(d i (T )).
(f) Observe that if i and i+2 are both on the diagonal, then we must have i = (q − 1, q − 1), i+1 = (q − 1, q), and i+2 = (q, q) for some q > 1. No matter how the entries in these boxes are primed, we have d i (T ) = s i (T ) = s i+1 (T ). This tableau is formed from T by reversing the prime on the entry in position (q − 1, q), while possibly interchanging the primes on the entries in positions (q − 1, q − 1) and (q, q), so primes(T ) = primes(d i (T )) ± 1.
This completes the proof of the proposition.
Our proof of the following theorem occupies all of Section 4. Theorem 3.16. If i ∈ Z and a is a primed involution word for an element of I Z then . When a = unprime(a), this theorem is equivalent to results in [29]; see Lemma 4.1. Extending these identities to primed involution words is surprisingly involved. The proof of the unprimed version of Theorem 3.16 in [29] relies heavily on the involution Little map, which gives a family of bijections z∈X R inv (z) ↔ z∈Y R inv (z) for certain finite subsets X, Y ⊂ I Z . Describing a "primed involution Little map" does not appear to be straightforward; one difficulty is that with primes allowed, the unions z∈X R + inv (z) and z∈Y R + inv (z) often have different cardinalities. As such, proving Theorem 3.16 requires a quite different strategy compared to [29]. Proof. This is immediate from Proposition 3.14 and Theorem 3.16.

Modified Sagan-Worley insertion
Before embarking on the proof of Theorem 3.16, we discuss two related insertion algorithms sending primed (bi)words to pairs of shifted tableaux. A biword is a two-line array of positive integers where the entries in the top row are weakly increasing such that if i j = i j+1 then a j ≤ a j+1 . A primed biword is a two-line array satisfying the same conditions, but where we allow entries in the bottom row to have 0 < a j ∈ Z ⊔ Z ′ as long as no column i a with a ∈ Z ′ is repeated.
We identify (primed) words with the (primed) biwords whose first rows are 1, 2, 3, . . . , n. If a = a 1 a 2 · · · a n is fixed, then the elements of Incr N (a) for N ∈ {0, 1, 2 . . . } ⊔ {∞} give the same data as (primed) biwords of the form Definition 3.18. Suppose φ is a primed biword of the form (3.3). We construct a sequence of shifted tableaux ∅ = P 0 , P 1 , . . . , P n in which P j is formed from P j−1 as follows: (1) On each iteration, an entry u ∈ Z ⊔ Z ′ is inserted into a row or column of a shifted tableau; the process begins with a j inserted into the first row of P j−1 .
(2) If inserting into a row when u ∈ Z, or into a column when u ∈ Z ′ , locate the first entry v in the row or column such that u < v; otherwise, locate the first entry v such that u ≤ v. When such an entry exists, we say that u "bumps" v from its position.
(3) If no such v exists then u is added to the end of the row or column to form P j . If u is primed and the added position is on the diagonal, then we change its value to ⌈u⌉ and say that the insertion process ends in column insertion. We also say that the process ends in column insertion if we are inserting into a column; otherwise, the process ends in row insertion.
(4) If v is off the diagonal, replace v by u and insert v into the next row (respectively, column).
(5) Assume v is on the diagonal. If ⌈u⌉ = ⌈v⌉, then continue by inserting this unprimed integer into the next column. If ⌈u⌉ = ⌈v⌉, then replace v byũ and insertṽ into the next column, where if u and v are both unprimed or both primed thenũ := u andṽ := v, and otherwiseũ andṽ are given by reversing the primes on u and v, respectively. Now define P SW (φ) := P n and let Q SW (φ) be the shifted tableau with the same shape whose entry in the unique box of P j that is not in P j−1 is either i j (when adding a j to P j−1 ends in row insertion) or i ′ j (when adding a j to P j−1 ends in column insertion).
This slightly modifies the definition of Sagan-Worley insertion in [35, §8] or [38, §6.1]. The latter map, which we will denote by φ → (P classical SW (φ), Q classical SW (φ)), is given by repeating Definition 3.18 with two changes: first, in step (3) we do not remove the prime from a newly added diagonal entry and we say that the insertion process ends in column insertion only if the last step inserts into a column; second, in step (5) we always takeũ := u andṽ := v. Primes can occur on the diagonal of P classical SW (φ) but not Q classical SW (φ), while the opposite is true for P SW (φ) and Q SW (φ).
Example 3.19. Suppose φ = 1 1 2 2 2 4 5 ′ 2 ′ 3 7 ′ . Then in the notation of Definition 3.18 On the other hand, one can check that Here we evaluate P SW (c) and Q SW (c) by converting c to a primed biword as explained above.
Our variant of Sagan-Worley insertion is related to orthogonal Edelman-Greene insertion by the next proposition. Given a primed word a = a 1 a 2 · · · a n , form double(a) by applying the map with i → 2i and i ′ → 2i ′ for i ∈ Z to the letters of a. If φ is a primed biword then define double(φ) by applying double to its second row. From a shifted tableau T , construct halve(T ) by applying the map with i → ⌊i/2⌋ and i ′ → ⌊i/2⌋ ′ for i ∈ Z to the entries of T .
We say that a primed word a is a partial signed permutation if unprime(a) has all distinct letters. 5 Define a primed biword to be value-strict if its second row is a partial signed permutation.
Proof. Let φ be as in (3.3). The first claim holds since unprime(double(a 1 a 2 · · · a n )) is an involution word in which every index is a commutation by Proposition 2.2. This ensures that halve are defined, and it is easy to see that the first tableau coincides with Let a = a 1 a 2 · · · a n be a primed word, and define P SW (a) := P SW 1 2 . . . n a 1 a 2 . . . a n .
For each j ∈ [n], consider the shifted tableaux P SW (a 1 a 2 · · · a j−1 ) and P SW (a 1 a 2 · · · a j ). If these tableaux have different numbers of rows or the same entries in all diagonal positions, then define τ SW j (a) to be the identity permutation of Z. Otherwise, there is a unique diagonal position that contains different entries in the two tableaux, and we let τ SW j (a) be the transposition of Z that interchanges these entries. If a = 45 ′ 2 ′ 37 ′ as in Example 3.19, then τ SW 3 (a) = (2, 4) and We say that a position (i, j) in a semistandard shifted tableau T is free if ⌈T xy ⌉ = ⌈T ij ⌉ whenever x > i or y < j. Every diagonal position is free. One can freely add or remove primes from free positions without changing whether a shifted tableau is semistandard. Moreover, if u and v are the entries in distinct free positions of a semistandard shifted tableau, then ⌈u⌉ = ⌈v⌉. Given a semistandard shifted tableau T , form unprime free (T ) from T by removing the primes from the entries in all free positions.
Finally, we say that u ∈ Z is initially primed (respectively, unprimed) in a primed word if u ′ (respectively, u) appears in the word and there is no earlier letter equal to u (respectively u ′ ).
Let (i, j) be a free position in P classical SW (φ) and let u ∈ Z be the value in this position with its prime removed. The entry of P classical SW (φ) in this position is primed if and only if u is initially primed in the second row of φ. If i = j (respectively, i = j), then the entry of P SW (φ) (respectively, Q SW (φ)) in position (i, j) is primed if and only if τ SW (φ)(u) is initially primed in the second row of φ.
Proof. In the insertion process that defines P classical SW (φ), whenever a free position with entry v is bumped by a number u, the position that v subsequently bumps (or the new position added to the tableau when v is placed at the end of a row or column) only depends on ⌈v⌉. The latter position is also free unless v is on the diagonal and ⌈u⌉ = ⌈v⌉, in which case the diagonal free entry is unchanged. If u = ⌈a j ⌉ and T = P classical SW (a 1 a 2 · · · a j−1 ) has no entries equal to u or u ′ , then the position bumped by a i in the first row of T is free in P classical SW (a 1 a 2 · · · a j ) and contains a j . Given these observations, it follows by induction on the number of columns of φ that P classical SW (φ) contains u ′ in a free position for some u ∈ Z if and only if u is initially primed in the second row of φ. Moreover, it is easy to see that P SW (φ) is formed from P classical SW (φ) by toggling the primes on certain free positions, and that the identities in the displayed equation hold. Thus P SW (φ) is semistandard, since P classical SW (φ) is known to be semistandard [35,Thm. 8.1]. For the last part of the lemma, consider a semistandard shifted tableau T and let u for u ∈ Z denote the free position of T containing u or u ′ , if this exists. If u and v are both defined, then let (u, v) ∈ S Z act on T by reversing the primes on the entries in these positions if they are not both primed or both unprimed, and otherwise leaves T unchanged. This operation extends to an action of the group of permutations of the entries of unprime(T ).
Let a = a 1 a 2 · · · a n be the second row of φ. Form P SW (a) from P SW (a) by adding primes to all diagonal positions that are primed in Q SW (a). Then P SW (a) is constructed by the same insertion process as the one that defines P classical SW (a), except that whenever an inserted number u is about to bump a diagonal entry v with ⌈u⌉ < ⌈v⌉ and {u, v} ⊂ Z and {u, v} ⊂ Z ′ , we reverse the primes on u and v. In the exceptional case τ SW j (a) is the transposition exchanging ⌈u⌉ and ⌈v⌉, and outside this case τ SW j (a) = 1. Thus, with respect to the action defined in the previous paragraph, it follows that τ SW (a) : P SW (a) → P classical SW (a). This implies the rest of the lemma.
We may represent a primed biword φ as the matrix A whose entry in position (i, j) is the This gives a bijection between primed biwords and {0 < 1 ′ < 1 < 2 ′ < 2 < . . . }-valued matrices with finitely many nonzero entries. Following [35], we call the latter circled matrices. For example, For a circled matrix A with primed biword φ, we set P SW (A) = P SW (φ) and Q SW (A) = Q SW (φ) as well as P classical to pairs (P, Q) of semistandard shifted tableaux of the same shape, where P has no primes on the diagonal and where the number of times that j or j ′ (for any j ∈ Z) appear in P and in Q are the respective sums i ⌈A ij ⌉ and k ⌈A jk ⌉.
The same statement holds for the map A → (P classical Proof. Let φ be the primed biword associated to a circled matrix A. Toggling whether a given number in the second row of φ is initially primed or not has no effect on τ SW (φ) by Lemma  Finally, we discuss a conjectural analogue of Theorem 3.16. Let shk denote the operator that acts on 1-and 2-letter primed words by interchanging for all distinct X, Y ∈ Z. Let shk act on 3-letter primed words as the involution interchanging 2 and X + 1 2 ≤ ⌈Y ⌉ ≤ Z, while fixing any 3-letter words not of these forms. For a primed word a = a 1 a 2 · · · a n and i ∈ [n − 2], define shk −1 (a) := shk(a 1 )a 2 a 3 · · · a n , shk 0 (a) := shk(a 1 a 2 )a 3 · · · a n , shk i (a) := a 1 · · · a i−1 shk(a i a i+1 a i+2 )a i+3 · · · a n , while setting shk i (a) := a for i ∈ Z with i + 2 / ∈ [ℓ(a)]. These shifted Knuth operators coincide with ock i on partial signed permutations.
Conjecture 3.23. If i ∈ Z and a is any primed word then P SW (shk i (a)) = P SW (a) and Q SW (shk i (a)) = d i (Q SW (a)).
It is trivial to verify these identities when i ∈ {−1, 0}. As with Theorem 3.16, the difficulty lies in the case when 1 ≤ i ∈ ℓ(a) − 2. Let shK ∼ denote the transitive closure of the relation on primed words with a ∼ shk i (a) for all i ∈ Z. Checking the following is also straightforward:  For partial signed permutations, we can derive all of these conjectural results from Section 3.2:

Shifted mixed insertion
There is also a relevant extension of Haiman's shifted mixed insertion algorithm [9, Def. 6.6] to primed words. Define a primed biword to be index-strict if its first row is strictly increasing, so that the associated circled matrix has at most one nonzero entry in each row.
Definition 3.30. Suppose φ is an index-strict primed biword of the form (3.3). We construct a sequence of shifted tableaux ∅ = U 0 , U 1 , . . . , U n = U in which U j is formed from U j−1 as follows: Define α ∈ {±} × Z to be (+, ⌈a j ⌉) if a j ∈ Z or (−, ⌈a j ⌉) if a j ∈ Z ′ , and insert this pair into the first row of U j−1 according to the following procedure. At each stage, a pair β 1 = (ǫ 1 , u 1 ) with u 1 ∈ Z ⊔ Z ′ is inserted into a row or column. Let β 2 = (ǫ 2 , u 2 ) be the first pair in the current row or column with u 1 < u 2 . If no such pair exists then β 1 is added to the end of the row or column. If β 2 is on the diagonal, then necessarily u 2 ∈ Z, and we continue by replacing β 2 by β 1 and inserting (ǫ 2 , u ′ 2 ) into the next column. Otherwise, replace β 2 by (ǫ 2 , u 1 ) and then insert (ǫ 1 , u 2 ) into the next row if u 2 ∈ Z or into the next column if u 2 ∈ Z ′ .
Form P HM (φ) from U by replacing each diagonal entry (ǫ, x) where ǫ = − by x ′ , and all other entries (ǫ, x) by x. Let Q HM (φ) be the shifted tableau with the same shape whose entry in the unique box of U j that is not in U j−1 is either i j or i ′ j , with a primed number occurring precisely when this box is off the diagonal and its entry in U j has the form (ǫ, x) where ǫ = −. Example 3.31. Suppose φ = 2 3 4 5 7 2 ′ 2 1 1 ′ 2 ′ . Then, writing ±x in place of (±, x), we have There is a transpose operation φ → φ ⊤ on primed biwords given as follows: first move the primes from any primed elements in the second row of φ to the entries directly above them, then interchange the two rows and reorder the columns to be lexicographically increasing. If for example. In terms of the associated circled matrices, this operation is just the matrix transpose, so it interchanges index-strict biwords and value-strict biwords.
The following result, which relates our primed forms of shifted mixed insertion and Sagan-Worley insertion via the biword transpose, generalizes [9, Thm. 6.10].
To observe this property, compare Examples 3.19 and 3.31. There may be a way to extend Definition 3.30 so that this theorem holds all primed biwords, similar to what is done in [36, §3.4] for (unshifted) mixed insertion. We will not pursue this here, however.
Note that (φ| i ) ⊤ is formed from φ ⊤ by omitting all columns whose entries in the second row are greater than i. Thus, the entry of P SW ((φ| i ) ⊤ ) in position 1 is the same as in P SW (φ ⊤ ). Moreover, if any diagonal entry of unprime diag (Q SW ((φ| i−1 ) ⊤ )) differs from the corresponding entry of unprime diag (Q SW ((φ| i ) ⊤ )), and the last such entry occurs in position 2 , then the entry of P SW ((φ| i−1 ) ⊤ ) in position 2 must bump i from the same position in P SW ((φ| i ) ⊤ ). In this case, it follows that i ′ appears in P SW (φ ⊤ ) if and only if the entry of Q SW ((φ| i−1 ) ⊤ ) in position 2 is primed. On the other hand, if no such position 2 exists then i or i ′ never reaches the diagonal as the successive entries of the second row of φ ⊤ are inserted to form P SW (φ ⊤ ), so i ′ appears in this shifted tableau if and only if i ′ appears in the second row of φ ⊤ , or equivalently if a i is primed.
We may assume by induction that P HM (φ| i−1 ) = Q SW ((φ| i−1 ) ⊤ ). Thus, it follows in view of (3.6) that the conditions in the previous paragraph for i ′ to appear in P SW (φ ⊤ ) are equivalent to the conditions in the one before it for i ′ to appear in Q HM (φ). We conclude that Q HM (φ) = P SW (φ ⊤ ). Now suppose that the position 1 is on the diagonal. Denote the value of unprime diag (P HM (φ)) = unprime diag (Q SW (φ ⊤ )) in this position by u ∈ Z. To show that P HM (φ) = Q SW (φ ⊤ ), it suffices to show that u ′ appears in P HM (φ) if and only if u ′ appears in Q SW (φ ⊤ ). Let j ∈ [n] be the first index such that u appears in position 1 of unprime diag (P HM (φ| j )). Observe that a j ≤ u and that if ψ is the primed biword formed from φ by removing all columns k a k with u < a k , then P HM (ψ) has u or u ′ in position 1 and Q HM (ψ) has j in position 1 . Similar to above, if any diagonal entry of unprime diag (P HM (φ| j−1 )) differs from the corresponding entry of unprime diag (P HM (φ| j )), and the last such entry occurs in position 2 , then u ′ appears in P HM (φ) if and only if the entry of P HM (φ| j−1 ) in position 2 is primed. Otherwise u ′ appears in P HM (φ) if and only if a j is primed.
Likewise, if any diagonal entry of unprime diag (Q SW ((φ| j−1 ) ⊤ )) differs from the corresponding entry of unprime diag (Q SW ((φ| j ) ⊤ )), and the last such entry occurs in position 2 , then u ′ appears in Q SW (φ ⊤ ) if and only if the entry of Q SW ((φ| j−1 ) ⊤ ) in position 2 is primed. Assume no such position 2 exists. Since ψ ⊤ consists of the first u columns of φ ⊤ , as the successive entries of the second row of φ ⊤ are inserted to form P SW (φ ⊤ ), the first entry to reach position 1 in P SW (φ ⊤ ) will be j or j ′ , and this will result from inserting the entry from column u of φ ⊤ . Thus u ′ appears in Q SW (φ ⊤ ) if and only if j ′ appears in the second row of φ ⊤ , or equivalently if a j is primed.
As in the first case of our argument, the conditions in the last two paragraphs are equivalent given (3.6) since we may assume that P HM (φ| i−1 ) = Q SW ((φ| i−1 ) ⊤ ) by induction. Thus, we also have P HM (φ) = Q SW (φ ⊤ ). This proves the theorem when (3.5) holds, and it is easy to see that the result still holds as long as |{i 1 < i 2 < · · · < i n }| = |{⌈a 1 ⌉, ⌈a 2 ⌉, . . . , ⌈a n ⌉}| = n.
To finish the proof, suppose φ is any index-strict primed biword with n columns. Form ψ from φ by taking its transpose, then replacing the first row by the consecutive numbers 1 < 2 < · · · < n, and then taking the transpose again. For example, if It is clear that P SW (φ ⊤ ) = P SW (ψ ⊤ ) and also not hard to see that Q HM (φ) = Q HM (ψ). Let is the entry in the first row of φ ⊤ in column i. Then φ is formed by applying F to the second row of ψ, and we have F(Q SW (ψ ⊤ )) = Q SW (φ ⊤ ) and F(P HM (ψ)) = P HM (φ). As we already know that Q HM (ψ) = P SW (ψ ⊤ ) and P HM (ψ) = Q SW (ψ ⊤ ), the theorem follows.
Recall that we identify a word a = a 1 a 2 · · · a n with the biword 1 2 . . . n a 1 a 2 . . . a n .
Corollary 3.33. The map a → (P HM (a), Q HM (a)) is a bijection from primed words with positive letters to the set of pairs (P, Q) of shifted tableaux of the same shape, in which P is semistandard and Q is standard with no primed entries on the main diagonal.
Proof. Primed words with positive letters correspond to circled matrices with exactly one nonzero entry, given by 1 or 1 ′ , in each of the first ℓ(a) rows, and no other nonzero rows. By Theorem 3.22, the map A → (P SW (A), Q SW (A)) is a bijection from the transposes of such matrices to the set of pairs (P, Q) of shifted tableaux of the same shape in which P is standard with no primes on the diagonal and Q is semistandard. The result therefore holds by Theorem 3.32.

Remaining proofs
Our main object in this section is to prove Theorem 3.16, though we will also derive Theorem 3.6 at the end of Section 4.2. Underpinning all of our results along the way to these proofs is the following lemma, which says that Theorem 3.16 already holds for unprimed words: In view of the lemma, to prove Theorem 3.16 we just need to precisely understand the relationship between the indices of the primed letters in a and the locations of the primed entries in P O EG (a) and on the main diagonal of Q O EG (a). To this end, we will first prove an analogue of Lemma 3.21, showing that the positions of the relevant primes are controlled by a permutation τ (a) that can be read off from the successive tableaux P O EG (a 1 a 2 · · · a i ) for i ∈ [ℓ(a)]. Then, in Sections 4.3 and 4.4, we will prove a series of lemmas clarifying the relationship between τ (a) and τ (ock i (a)).

Bumping paths
We start by listing some properties of the bumping paths in Definition 3.1. For (x, y) ∈ Z × Z, let (x, y) := {(i, j) ∈ Z × Z : x ≥ i and y ≥ j} and (x, y) := {(i, j) ∈ Z × Z : x ≤ i and y ≤ j}. In this subsection, let T be an increasing shifted tableau with no primes on the diagonal and let u ∈ Z ⊔ Z ′ be such that row(T )u is a primed involution word for an element of I Z . Write (4.1) for the bumping paths specified in Definition 3.3. The following observations are straightforward to derive from the definitions and Remark 3.5. We omit a detailed proof. (a) If 1 ≤ i ≤ q then x i =x i = i andỹ i ∈ {y i , y i + 1}, while y 1 ≥ y 2 ≥ · · · ≥ y q andỹ 1 ≥ỹ 2 ≥ · · · ≥ỹ q .
We sometimes treat path ≤ (T, u) and path < (T, u) as sets. This practice is justified as Proposition 4.2 shows that the positions in each path are distinct and their order is uniquely determined.
Define q as above to be the index of the row containing the unique diagonal position in path ≤ (T, u) or set q = m if no such row exists. Write  If We think of these subsequences as the "row-bumping paths" and "column-bumping paths" from inserting u into T . Finally, if a = a 1 a 2 · · · a n is a primed involution word and i ∈ [n], then we let path ≤ i (a) := path ≤ (T, a i ) and path < i (a) := path < (T, a i ).
with T = P O EG (a 1 a 2 · · · a i−1 ). We define rpath ≤ i (a), cpath ≤ i (a), rpath < i (a), and cpath < i (a) analogously. Proof. Both parts are straightforward to check directly, using Remark 3.5 and Proposition 4.2.

Cycle sequences
A set of distinct integers {j, k} is a cycle of an element z ∈ I Z if j = z(j) = k. We denote the set of these pairs by cyc(z). For an unprimed word a = a 1 a 2 · · · a n ∈ R inv (z) and i ∈ [n], define  Now suppose T is a shifted tableau and b is a word with letters in Z ⊔ Z ′ such that row(T )b is an (unprimed) involution word. For each position (i, j) ∈ Z × Z, define where in the second case a := row(T )b and k := 1 + j − i + λ j+1 + λ j+2 + · · · + λ q is the index of the letter in a corresponding to the entry of T in box (i, j). We also let γ ij (T ) := γ ij (T, ∅). In the next lemma, we assume that T is an increasing shifted tableau and b is a word such that row(T )b ∈ R inv (z) for some z ∈ I Z . Assume b is nonempty with first letter u, and let c be the subword formed by removing this letter. Denote the weak and strict bumping paths resulting from inserting u into T as in (4.1). Write u i for the entry of T in position (x i ,ỹ i ) for i ∈ [m − 1] and set u 0 = u. Then define θ 0 := γ |T |+1 (row(T )b) where |T | is the size of the domain of T and let if (x, y) = (i + 1, i + 1).

Proof. Suppose a is any involution word and
Given these observations and Proposition 2.4, it is straightforward to derive the lemma by following the proof of Proposition 3.14. The explicit details of this argument are left as an exercise.
Continuing our notation from above, define p to be the index of the unique diagonal position in path ≤ (T, u) or set p = m if not such index exists. Define the sequence for i ∈ [p − 1]. For any involution word a = a 1 a 2 · · · a n and j ∈ [n], define ∆ bump j (a) := ∆ bump (T, b) where T = P O EG (a 1 a 2 · · · a j−1 ) and b = a j a j+1 · · · a n . Assume T is a shifted tableau with q rows and b is a word such that row(T )b is a (unprimed) involution word. We define the cycle sequence cseq(T, b) to be the two-line array For involution words a = a 1 a 2 · · · a n and 0 ≤ i ≤ n, we define cseq i (a) := cseq(T, b) where T = P O EG (a 1 a 2 · · · a i ) and b = a i+1 a i+2 · · · a n . If T = P O EG (51324) and b = 3154 as in Example 4.5 then When T is increasing and b is a word such that row(T )b ∈ R inv (z) for some z ∈ I Z , the second row of cseq(T, b) is strictly increasing and the elements in the first row are distinct cycles of z, since the index of T ii in row(T )b is a commutation for all diagonal positions (i, i) in T .
Exactly one of the following cases applies: (a) The sequence path ≤ j (a) ends before reaching the diagonal if and only p < y p . In this case i appears in Q O EG (a) in an off-diagonal position and cseq j (a) = cseq j−1 (a).
(b) The sequence path ≤ j (a) terminates on the diagonal if and only if p = y p =ỹ p = q + 1. In this case i appears in Q O EG (a) in position (q + 1, q + 1) and cseq j (a) = γ 1 γ 2 . . . γ q θ q c 1 c 2 . . . c q u q .
(c) The sequences path ≤ j (a) and path < j (a) reach (but do not terminate on) the diagonal in the same row if and only if p = y p =ỹ p ≤ q. In this case i ′ appears in Q O EG (a) and we have where η := γ p if u p−1 + 1 = c p and η := θ p−1 if u p−1 + 1 < c p .
(d) The sequences path ≤ j (a) and path < j (a) reach the diagonal in different rows if and only if p = y p <ỹ p = p + 1 ≤ q. In this case i ′ appears in Q O EG (a) and we have Proof. The assertion that exactly one of these cases applies follows from Proposition 4. Putting all of this together, we associate a permutation of Z 2 := {{i, j} : i, j ∈ Z, i < j} to each involution word. This is analogous to the definition of τ SW (a) : Z → Z from Section 3.3. Let a = a 1 a 2 . . . a n be an involution word for some z ∈ I Z . For each i ∈ [n], let τ i (a) be the following permutation of Z 2 with support in cyc(z). If cseq i−1 (a) and cseq i (a) are equal or have different lengths then τ i (a) := 1. Otherwise, writing there is either a unique index j ∈ [q] with d j < c j , or a unique index j ∈ [q − 1] with γ j+1 = η j = γ j = η j+1 , and in both cases we define τ i (a) to be the transposition of Z 2 that swaps η j and γ j while fixing all other elements. We then let τ (a) := τ 1 (a)τ 2 (a) · · · τ n (a). , and the corresponding values of γ xy (T, b) for T = P O EG (a 1 a 2 · · · a i ) and b = a i+1 a i+2 · · · a 9 are 34    ) has θ ∈ cyc(z) and τ (a)(θ) ∈ marked(a). Define T j := P O EG (a 1 a 2 · · · a j ) and b j := a j+1 a j+2 · · · a n for 0 ≤ j ≤ n, and abbreviate by writing marked(T j , b j ) := marked(row(T j )b j ). It suffices to check that marked(T j , b j ) = τ j (a)(θ) : θ ∈ marked(T j−1 , b j−1 ) for all j ∈ [n], since this will imply that marked(row( P O EG (a))) = {θ : τ (a)(θ) ∈ marked(a)}. Let ∼ be the transitive closure of the relation on primed involution words that has w ∼ ock i (w) for all i ∈ Z such that marked(w) = marked(ock i (w)). In Lemma 4.8, if we are in case (a), (b), or (c) with η = γ p , then τ j (a) = 1 and it follows by tracing through the proof of Proposition 3.14 that we have row(T j−1 )b j−1 ∼ row(T j )b j as needed.
If we are in case (c) of Lemma 4.8 with η = γ p , then τ j (a) is the transposition of cyc(z) interchanging η ↔ γ p , and it follows similarly that marked(T j , b j ) is formed by applying this transposition to all elements of marked(T j−1 , b j−1 ). Finally, suppose we are in case (d) of Lemma 4.8, so that τ j (a) = (γ p ↔ γ p+1 ). Form U j from T j by reversing the primes on the diagonal entries in positions (p, p) and (p + 1, p + 1) if these entries are not both primed or both unprimed, and otherwise set U j := T j . Then, again following the proof of Proposition 3.14, one can check that row(T j−1 )b j−1 ∼ row(U j )b j where ∼ is the relation defined in the paragraph above. It follows that marked(T j−1 , b j−1 ) = marked(U j , b j ) = τ j (a)(θ) : θ ∈ marked(T j−1 , b j−1 ) as desired.
We may now prove Theorem 3.6 from Section 3.1.
Proof of Theorem 3.6. Remark 3.5 and Proposition 3.14 imply that if a ∈ R + inv (z) for some z ∈ I Z , then P O EG (a) is an increasing shifted tableau with no primes on the diagonal whose row reading word is in R + inv (z); it follows by definition that Q O EG (a) is a standard shifted tableau of the same shape. Let (P, Q) be an arbitrary pair of shifted tableaux with these properties. The unprimed form [12,Thm. 5.19] of the result to prove asserts that there is a unique unprimed word a ∈ R inv (z) with P O EG (a) = unprime(P ) and Q O EG (a) = unprime diag (Q). Since we have γ ii (P ) ∈ cyc(z) for all diagonal positions (i, i) in P , Lemma 4.10 implies that there is a unique way to assign primes to the commutations in a to obtain a primed word b ∈ R + inv (z) with P O EG (b) = P and Q O EG (b) = Q.

Some reductions
In this section we prove three technical results constraining the values of cseq i (a) and τ i (a). Let entries(T ) ⊂ Z ⊔ Z ′ denote the set of entries in a shifted tableau T . Let diag(T ) denote the subset of entries appearing on the main diagonal of T . Our first result is the following:  with γ p = η p , or a unique p ∈ [q −1] with γ p+1 = η p = γ p = η p+1 , and in either case τ i+1 (a)τ i+2 (a) = τ j+1 (b)τ j+2 (b) is the permutation of Z 2 swapping γ p and η p . Next suppose r = s = 1. Consider the weak bumping paths path ≤ i+1 (a) and path ≤ i+2 (a) that result from inserting a i+1 and a i+2 successively into P O EG (a 1 a 2 · · · a i ). Since a i+1 < a i+2 , it follows from Proposition 4.3 that path ≤ i+2 (a) terminates at a diagonal position (q + 1, q + 1) and path ≤ i+1 (a) contains a unique non-terminal diagonal position (p, p) for some p ∈ [q]. Denote cseq i (a) = cseq i (b) as in (4.9). There are four possibilities for cseq i+2 (a) = cseq j+2 (b), namely: where η p , η q+1 / ∈ {γ 1 , γ 2 , . . . , γ q } and d p < c p − 1. In each case, it is straightforward to work out the unique possibility for cseq i+1 (a) from Lemma 4.8.
The values for cseq j+1 (b) are constrained by Lemma 4.8 and the fact that cseq i (a) = cseq j (b) = cseq j+1 (b) = cseq j+2 (b) = cseq i+2 (a). In cases (1)-(3) there are two possibilities for cseq j+1 (b) but for either one τ j+1 (b) and τ j+2 (b) commute and τ i+1 (a)τ i+2 (a) = τ j+1 (b)τ j+2 (b). In case (4), In case (6), we must have We conclude in each situation that Here is our second technical lemma: Let T := P O EG (a) and assume rpath < (T, u) ∩ rpath < (T, v) contains an off-diagonal position. Then cseq n+1 (auvb) = cseq n+1 (avuc), and n + 1 is on the diagonal in Q O EG (auvb) if and only if n + 1 is on the diagonal in Q O EG (avuc), while n ′ + 1 is in Q O EG (auvc) if and only if n ′ + 1 is in Q O EG (avuc). Proof. Suppose rpath < (T, u) ∩ rpath < (T, v) is nonempty and the first position in this intersection is (j, k). We claim that (j, k) also belongs to rpath ≤ (T, u) ∩ rpath ≤ (T, v). To check this, write u 0 := u < v 0 := v and let u i and v i be the entries of T in the ith positions of path < (T, u) and path < (T, v) respectively. Then u j−1 < v j−1 and the smallest entry of T in row j that is greater than both of these numbers is u j = v j by definition. This means that row j of T cannot contain any entry w with u j−1 < w ≤ v j−1 , so by Remark 3.5, row j of T also cannot contain u j−1 . Hence (j, k) ∈ rpath ≤ (T, u) ∩ rpath ≤ (T, v) as desired.
It is clear from Definition 3.1 that rpath < (T, u) and rpath < (T, v) coincide after their first j − 1 positions, and it follows by our claim that rpath ≤ (T, u) and rpath ≤ (T, v) also coincide after their first j −1 positions. If j = k, then all of these paths continue after row j, and we have γ xy (T, uvb) = γ xy (T, vuc) for all positions (x, y) since uvb and vuc are reduced words for the same permutation. Given these observations, the result follows from Lemma 4.7.
Our last result in this section requires a longer argument. Lemma 4.13. Suppose a, b are unprimed words and u, v ∈ Z are such that u + 1 < v and auvb is an involution word for an element of I Z . Let T = P O EG (a) and n = ℓ(a), and assume rpath ≤ (T, u) and rpath ≤ (T, v) are disjoint. Then cseq n+2 (auvb) = cseq n+2 (avub), and for each ǫ ∈ {0, 1}, the number n + 1 + ǫ is on the diagonal in Q O EG (auvb) if and only if n + 2 − ǫ is on the diagonal in Q O EG (avub), while n ′ + 1 + ǫ appears in Q O EG (auvb) if and only if n ′ + 2 − ǫ appears in Q O EG (avub). Proof. Again write u 0 := u < v 0 := v and let u i and v i be the entries of T in the ith positions of path < (T, u) and path < (T, v) respectively. Suppose rpath ≤ (T, u) and rpath ≤ (T, v) are disjoint. The first paragraph of the proof of Lemma 4.12 shows that rpath < (T, u) and rpath < (T, v) must also be disjoint. We argue that since u + 1 < v, it must further hold that rpath < (T, u) and rpath ≤ (T, v) are disjoint. To see this, note that if u i = v i − 1 in some row i > 0 of T occupied by both rpath < (T, u) and rpath < (T, v), then this row of T must also contain u i − 1 and we must have u i−1 = u i − 1 and v i−1 = u i , since otherwise rpath ≤ (T, u) and rpath ≤ (T, v) would intersect in the position of u i in row i. But this means that if u i = v i − 1 for any row i > 0 then we also have u 0 = v 0 − 1, which is a contradiction since u 0 = u and v 0 = v.
Then each position in cpath ≤ (T, u) ∪ cpath < (T, u) belongs to cpath ≤ (T, u, i) ∪ cpath < (T, u, i) for a unique value of i, and every position in cpath ≤ (T, u, i) ∪ cpath < (T, u, i) occurs in a column strictly to the left of every position in cpath ≤ (T, u, i + 1) ∪ cpath < (T, u, i + 1) by Proposition 4.2.
Let i be minimal such that cpath ≤ (T, u, i) ∪ cpath < (T, u, i) and rpath ≤ (T, v) ∪ rpath < (T, v) intersect. Assume the leftmost position in cpath ≤ (T, u, i) ∪ cpath < (T, u, i) is in column j + 1 while |cpath ≤ (T, u, i)| = l and |cpath < (T, u, i)| = k + l for some integers k, l ≥ 0 with k + l > 0. If i = 1 then we must have l = 0 and j + k − 1 must be the length of the first row of T . If i > 1 then we must have v j+k+t = v j+k + t for t ∈ [l]. Finally, all positions in cpath ≤ (T, u, i) ∪ cpath < (T, u, i) must be occupied in T , except that when l = 0 the single position (i, j + k) may be outside the domain of T .
First assume all positions in cpath ≤ (T, u, i) ∪ cpath < (T, u, i) are occupied in T . Then we must have i > 1, so the entries of T in positions {i − 1, i} × {j + 1, j + 2, . . . , j + k + l} are In this case one of the following holds: (1) i = j and T ii = u j , (2) i = j + 1 and k = 0 and (3) i < j and u j appears in column j of T above row i.
So we may assume that (i, j + δ) ∈ rpath < (T, v)∩ cpath < (T, u, i) for some δ ∈ [k + l]. If k < δ ≤ l then we also have (i − 1, j + δ) ∈ rpath < (T, v) ∩ cpath ≤ (T, u, i). In view of the minimality of i, apart from these one or two positions there are no other elements in the intersection of rpath < (T, v) and cpath ≤ (T, u) ∪ cpath < (T, u), since rpath < (T, v) contains at most one position in each row, and since all positions of rpath < (T, v) above row i contain entries of T that are greater than u j+δ while all positions cpath ≤ (T, u) ∪ cpath < (T, u) above row i contain entries of T that are at most u j . To proceed, we divide our analysis into six subcases: (a) If k + 1 < δ ≤ l then Lemma 4.7 implies ∆ bump n+1 (avub) = ∆ bump n+2 (auvb) which suffices.
(d) We claim that the case δ = k > 1 cannot occur. In this event, it would follow in view of Proposition 4.2 that (i − 1, j + k) and (i, j + k) are both in rpath < (T, v) with u j+k−1 ≤ T i−1,j+k < u j+k , which contradicts the fact that T i−1,j+k < u j+k−1 as (i − 1, j + k) / ∈ rpath < (T, u).
(f) Next suppose k > 0 and δ = 1. If u j < v i−1 then the argument in subcase (e) still applies.
Assume v i−1 ≤ u j . Then we cannot be in cases (1) or (2) without contradicting rpath ≤ (T, u)∩ rpath ≤ (T, v) = ∅, so u j appears in column j of T above row i and position (i + 1, j) in T contains an entry that is at most u j . The entry in position (i, j) of T cannot be greater than v i−1 since (i, j + 1) ∈ rpath < (T, v), and this entry must also not be equal to v i−1 since then we would have u j+1 = v i−1 + 1 which can only hold if u j = v i−1 , in which case column j of T would have two equal entries, contradicting the fact that all columns of T are strictly increasing. Thus position (i, j) in T contains an entry that is less than v j−1 .
This completes our argument if all positions in cpath ≤ (T, u, i) ∪ cpath < (T, u, i) are occupied in T . When this does not occur, we must have l = 0 and (i, j + k) / ∈ T . In this case row i of T is Here, cases (1) or (3) from above must apply. We cannot have (i, j +k) ∈ rpath ≤ (T, v)\rpath < (T, v) if (i, j +k) / ∈ T , so again (i, j +δ) ∈ rpath < (T, v)∩cpath < (T, u, i) for some δ ∈ [k]. By the minimality of i, this position is the unique element in both rpath < (T, v) and cpath ≤ (T, u) ∪ cpath < (T, u), since rpath < (T, v) contains at most one position in each row, and since all positions of rpath < (T, v) above row i contain entries greater than u j+δ while all positions of rpath ≤ (T, v) ∪ rpath < (T, v) above row i contain entries that are at most u j . We are left with two further subcases: (g) If u j < v i−1 , then it follows from Lemma 4.7 as in subcase (e) that ∆ bump n+1 (avub) and ∆ bump n+2 (auvb) differ only in their ith term, where if this term of ∆ bump n+1 (avub) is (y,ỹ, d, η) then the corresponding term of ∆ bump n+2 (auvb) is (1 + y, 1 +ỹ, d, η). In this event, both sequences have more than i terms unless y =ỹ = j + k. Since (j, j + k) is not a diagonal position, we conclude that the lemma holds holds either way.
(h) Assume v i−1 ≤ u j . Then we cannot be in case (1) without contradicting rpath ≤ (T, u) ∩ rpath ≤ (T, v) = ∅, so i < j and u j appears in column j of T above row i. If δ < k then we can repeat the argument given in subcase (f) to deduce our result. If δ = k then we must have k = 1 and v i−1 < u j . In this situation, ∆ bump n+1 (avub) has only i terms and ends with a term of the form (j + 1, j + 1, v i−1 , θ) for some cycle θ, and it is easy to see that ∆ bump n+2 (auvb) is formed from ∆ bump n+1 (avub) by appending the tuple (j + 1, j + 1, u j , φ) for some cycle φ. Since neither (i, j + 1) nor (i + 1, j + 1) is a diagonal position, this shows that rpath ≤ (T, v) and

Final arguments
Combining the preceding results, we can now explain how to derive Theorem 3.16. There are three main steps in our argument, corresponding to the different cases in which we can have ock i (a) = a for a primed involution word.
Lemma 4.14. Suppose a = a 1 a 2 · · · a n is a primed involution word. Write i for i ∈ [n] to denote the box of Q O EG (a) containing i or i ′ . Assume that i ∈ {−1, 0}, or that i ∈ [n − 2] and i and i+2 are both on the main diagonal. Then P O EG (ock i (a)) = P O EG (a) and Q O EG (ock i (a)) = d i (Q O EG (a)). Proof. If i ∈ {−1, 0}, then the desired identities follows easily from the definitions. Assume i ∈ [n − 2] and i and i+2 are both on the main diagonal. Write i = (q − 1, q − 1) and Q = Q O EG (a). Then we must have i+1 = (q − 1, q) and i+2 = (q, q), and consequently d i (Q) = s i (Q) = s i+1 (Q) is formed from Q by swapping i + 1 and i ′ + 1, and then reversing the primes on the entries in the diagonal boxes (q − 1, q − 1) and (q, q) if these entries are not both primed or both unprimed.
After possibly invoking Proposition 3.15 to interchange Q with d i (Q), we may assume that the entry in position (q − 1, q) of Q is i + 1 rather than i ′ + 1. Then it is evident from Lemma 4.8 that τ i (a) = τ i+1 (a) = τ i+2 (a) = 1. Write b := ock i (a). Since we know from Lemma 4.1 that Q O EG (unprime(b)) is formed by applying d i to Q O EG (unprime(a)) = unprime diag (Q), which adds a prime to position (q − 1, q), it is also clear from Lemma 4.8 that τ i (b) = τ i+2 (b) = 1.
To this end, write j for j ∈ [n] to denote the box of Q O EG (a) containing j or j ′ . We first check that i and i+2 are not both on the main diagonal. Arguing by contradiction, we observe that these positions could only both be on the diagonal if the weak bumping paths path ≤ i (a), path ≤ i+1 (a), and path ≤ i+2 (a) that result from inserting a i , a i+1 , and a i+2 successively into P O EG (a 1 a 2 · · · a i−1 ) respectively terminate at (q − 1, q − 1), (q − 1, q), and (q, q) for some q > 0. Assume this is the case, so that we have path ≤ i (a) = rpath ≤ i (a) and path ≤ i+1 (a) = rpath ≤ i+2 (a). Since a i > a i+1 , Proposition 4.3 implies that the positions in rpath ≤ i+1 (a) are all weakly to the left of the corresponding positions in rpath ≤ i (a). The second to last position in path ≤ i+1 (a) must therefore be (q − 1, q − 1), so the entry in position (q − 1, q) of P O EG (a 1 a 2 · · · a i+1 ) is the same as the entry in position (q − 1, q − 1) of P O EG (a 1 a 2 · · · a i ). Since a i+1 < a i < a i+2 , it is easy to check that the first q − 1 positions in path ≤ i+2 (a) are strictly to the right of the corresponding positions in path ≤ i (a), and that if path ≤ i+2 (a) reaches row q then its position in that row must be strictly to the right of (q − 1, q). But this makes it impossible for path ≤ i+2 (a) to terminate at (q, q). Thus i and i+2 are not both on the diagonal. By Lemma 4.1 we have P O EG (a 1 a 2 · · · a j ) = ) and i and i+2 are not both on the diagonal, it follows from Proposition 3.15 that conditions (b) and (c) in Lemma 4.11 also hold, so that result implies that τ i+1 (a)τ i+2 (a) = τ i+1 (b)τ i+2 (b).
The next lemma has a significantly longer proof, though much of this is routine case analysis. Lemma 4.16. Let a = a 1 a 2 · · · a n ∈ R + inv (z) for some z ∈ I Z . Write j for j ∈ [n] to denote the box of Q O EG (a) containing j or j ′ . Suppose i ∈ [n − 2] is such that ⌈a i ⌉ ≤ ⌈a i+2 ⌉ < ⌈a i+1 ⌉, but i and i+1 are not both on the main diagonal. Then τ (ock i (a)) = τ (a) and P O EG (ock i (a)) = P O EG (a).
Proof. Define b = ock i (a) and assume a = unprime(a) has no primed letters. As in the proof of Lemma 4.15, it suffices by Lemmas 4.10 and 4.1 to check that τ (a) = τ (b). We have either a i < a i+2 < a i+1 and b = a 1 · · · a i+1 a i a i+2 · · · a n , or a i = a i+2 < a i+1 and b = a 1 · · · a i+1 a i a i+1 · · · a n . In either case, Lemma 4.1 implies that P O EG (a 1 a 2 · · · a j ) = P O Suppose the intersection of rpath < i (a) and rpath < i (b) includes a position off the diagonal. Then Lemma 4.12 implies that cseq i (a) = cseq i (b) so τ i (a) = τ i (b). Because s(a) = s(b) and r(a) = r(b), we can then apply Lemma 4.11 Instead suppose a i < a i+2 < a i+1 and rpath ≤ i (a) and rpath ≤ i (b) are disjoint. Then Lemma 4.13 implies that cseq i+1 (a) = cseq i+1 (b) so τ i+2 (a) = τ i+2 (b). Because s(a) = s(b) and r(a) = r(b), we can again apply Lemma 4.11 Thus, we may assume that rpath < i (a) and rpath < i (b) intersect in at most one position, which is on the main diagonal, and that if a i < a i+2 < a i+1 then rpath ≤ i (a) and rpath ≤ i (b) intersect in at least one position.
Assume that if a i < a i+2 < a i+1 then the first position in the intersection of rpath ≤ i (a) and rpath ≤ i (b) is off the diagonal in row j > 0. This position cannot belong to rpath < i (a) ∩ rpath < i (b), so it must be occupied by some entry u in T . If instead a i = a i+2 < a i+1 , then we set j := 0 and u := a i . Assume the last position in rpath ≤ i (a) is in row k. Then j < k and the following claims are easy to deduce from our assumption that rpath < i (a) and rpath < i (b) do not intersect off the diagonal: (A1) Suppose t ∈ {1, 2, . . . , k − j − 1} or t = 0 < j. Then row j + t of T contains both u + t and u + t + 1, and the positions of u + t and u + t (A3) The first k − 1 terms of rpath < i (a) and rpath < i+1 (b) coincide, as do the first k − 1 terms of rpath < i (b) and rpath < i+1 (a), as do the first k − 1 terms of rpath ≤ i (a) and rpath ≤ i+1 (b).
(A4) The first k − 1 terms of rpath ≤ i (b) and rpath ≤ i+1 (a) are the same except in the rows j + t where T does not contain u+t−1, for t ∈ {1, 2, . . . , k −j −1} or t = 0 < j. In these rows, rpath ≤ i+1 (a) contains the position of u + t + 1 in T , rather than the position of u + t which is in rpath ≤ i (b).
(A5) The first j terms of path < i+2 (a) and path < i+2 (b) coincide, as do the first j terms of path ≤ i+2 (a) and path ≤ i+2 (b). If j > 0 then term j of all four paths is the position of u + 1 in row j of T .
(A6) If t ∈ [k − j − 1], then the (j + t)th terms of path ≤ i+2 (a), path < i+2 (a), path ≤ i+2 (b), and path < i+2 (b) are either the respective positions in row j + t of T of u + t − 1, u + t, u + t, and u + t + 1 when row j + t of T contains the entry u + t − 1, or the respective positions of u + t, u + t + 1, u + t + 1, and u + t + 1 when the same row does not contain u + t − 1.
Combining the preceding observations, we arrive at the following key claim: (A7) Let v = u + k − j − 1 and assume k > 1. Likewise, the entries of the shifted tableaux T , This last property still makes sense when j = 0 and k = 1 if we define the entries in the "0th position" of rpath < m (a) and rpath < m (b) to be a m and b m , respectively. We need just one other observation. Let U be the shifted tableau formed from T by omitting the first k − 1 rows. Using Proposition 3.14 and property (A7), it is easy to check that a is equivalent under O ∼ to a word that begins with row(U ) v (v + 1) v. If U were empty or if all entries in U were greater than v + 2 then this word is an involution equivalent under≡ to v (v + 1) v row(U ) which is impossible by Proposition 2.5. Thus: (A8) The entry of T in position (k, k) is occupied by v, v + 1, or v + 2.
We can now reason precisely about the possibilities for τ i (a), τ i+1 (a), τ i+2 (a), τ i (b), τ i+1 (b), and τ i+2 (b) under our assumptions that rpath < i (a) and rpath < i (b) do not intersect off the diagonal and that if a i < a i+2 < a i+1 then rpath ≤ i (a) and rpath ≤ i (b) do intersect off the diagonal. Below, we will refer to the entries of the shifted tableaux arranged in the diagram bi ai ai+1 ai+2 bi+1 bi+2 (4.13) where in this picture, an arrow u − − → connects two tableaux if inserting u into the first tableau according to Definition 3.1 gives the second. Write (4.14) As noted above, there are three possibilities for the entry of T in position (k, k).
Suppose next that the entry of T in position (k, k) is v + 1. Then, again in view of Remark 3.5, the entries of T in positions {k, k + 1} × {k, k + 1} must be T k+i,k+j = v + i + j for all 0 ≤ i ≤ j ≤ 1. Assume k > 1. Then row k − 1 of T contains v and v + 1 in off-diagonal positions, so the entry in position (k − 1, k + 1) of T is at most v + 1. If equality holds, then the entries of the six tableaux in (4.13) .
On the other hand, if the entry in position (k − 1, k + 1) of T is less than v + 1 then position (k − 1, k + 2) of T must have an entry less than v + 2. When this happens or when k = 1, the entries in the six tableaux in (4.13) in positions {k, k + 1} × {k, k + 1, k + 2} must instead be v + 3 ? v where ? denotes a position that may be unoccupied. In both cases, it follows using Lemmas 4.7 that the values of γ xy applied to the six tableaux in (4.13) in positions {k, k + 1} × {k, k + 1} are Thus, it follows by Lemma 4.
Finally, suppose the entry of T in position (k, k) is v + 2. If k > 1 then row k − 1 of T contains v and v + 1 off the diagonal, so the entry in position (k − 1, k + 1) of T must be less than v + 2. There are two subcases depending on the entry in position (k − 1, k + 2) of T . If k > 1 and this position contains a number less than v + 2, or if k = 1, then the entries in the six tableaux in (4.13) in positions {k, k + 1} × {k, k + 1, k + 2} are ? ?
If k > 1 and position (k − 1, k + 2) of T is unoccupied or contains a number greater than or equal to v + 2, then positions (k − 1, k) and (k − 1, k + 1) of T must contain the numbers v and v + 1. In this case the entries in the six tableaux in (4.13) in positions {k − 1, k, k + 1} × {k, k + 1} are ? v + 2 ?
v v + 2 Write η k and η k+1 for the entries in the first row of cseq i+2 (a) in columns k and k + 1. The following assertions apply equally to both of the cases above. First, since cseq i−1 (a) = cseq i−1 (b) and cseq i+2 (a) = cseq i+2 (b), one can check using Lemmas 4.7 and 4.8 that γ k = η k . If cseq i−1 (a) has only k columns, then it follows similarly that the values of γ xy applied to the six tableaux in (4.13) in positions {k, k + 1} × {k, k + 1} are where we set β := ∅ in the first subcase above and β := η k+1 in the second. Thus τ i (a) = τ i+2 (a) = (γ k , η k+1 ) and τ i+1 (a) = τ where the entry β has the same definition as before. Thus Lemma 4.8 gives τ i (a) = τ i+2 (a) = (γ k , η k+1 ) and τ i+1 (a) = (γ k , γ k+1 ) while τ i (b) = τ i+1 (b) = 1 and τ i+2 (b) = (γ k+1 , η k+1 ), so This completes the proof of the lemma under our first set of hypotheses. It remains to consider the case when a i < a i+2 < a i+1 and rpath < i (a) and rpath < i (b) do not intersect off the diagonal, but rpath ≤ i (a) and rpath ≤ i (b) intersect in a unique position which is on the diagonal. Suppose this position is (k, k). This position must be occupied in T , since otherwise it is straightforward to check using Remark 3.5 that both i and i + 2 would be on the diagonal of Q O EG (a). The reasoning we used to justify (A3) lets us similarly derive the following claims: (B1) The first k − 1 terms of path < i (a) and path < i+1 (b) coincide, as do the first k − 1 terms of path < i+1 (a) and path < i (b). Each of the first k − 1 terms of the first two paths is strictly to the right of the main diagonal and strictly to the left of the corresponding term in the second two paths. The same statements hold for the corresponding weak bumping paths.
(B2) The first k − 1 terms of path < i+2 (a) and path < i+2 (b) coincide. Each of the first k − 1 terms of these paths is strictly to the right of the corresponding term in path < i (a) or path < i+1 (b), and weakly to the left of corresponding term in path < i+1 (a) or path < i (b). The same statements hold for the corresponding weak bumping paths.
If k = 1 then let u := a i = b i+1 < v := a i+2 = b i+2 < w := a i+1 = b i . If k > 1 then define u, v, and w to be the entries of T , i+2 (a), and path < i+1 (a) respectively. It follows from (B1) and (B2) that: (B3) Assume k > 1. Then u is also the entry of T In turn, w is also the entry of T in position k − 1 of path < i (b), and u < v < w.
(B4) The entry of T in position (k, k) is at least w since (k, k) ∈ rpath ≤ i (b).
First suppose the entry in position (k, k) of T is w. Then, in view of Remark 3.5, the entries of T in positions {k, k + 1} × {k, k + 1} must be T k+i,k+j = w + i + j for all 0 ≤ i ≤ j ≤ 1. If k > 1, then row k − 1 of T contains both u and w in positions off the main diagonal, so the entry in position (k − 1, k + 1) of T is at most w. If k > 1 and this entry is equal to w, then the entries of the six tableaux in (4.13) Alternatively, if k > 1 and the entry in position (k − 1, k + 1) of T is less than w, then the entry of T in position (k − 1, k + 2) must be occupied by a number less than w + 1. In this case, or if k = 1, the entries of the six tableaux in (4.13) in positions {k, k + 1} × {k, k + 1, k + 2} are w + 2 ?
Finally suppose that the entry in position (k, k) of T is x > w + 1. If k > 1 then the entry in position (k − 1, k + 1) of T is at most w, so the entries of the six tableaux in (4.13) in positions {k, k + 1} × {k, k + 1} are ?
Finally, we arrive at the proof of Theorem 3.16.

Primed involution words
So far we have been exclusively concerned with primed involution words for elements of S Z . There is a notion of such words for any Coxeter system, however, along with an analogue of Proposition 2.5.
To conclude this article, we explain how to derive this from results in [15].

Twisted Coxeter systems
Let (W, S) be a Coxeter system with length function ℓ : W → N. Let w → w * be a self-inverse automorphism of the group W that preserves S. We refer to (W, S, * ) as a twisted Coxeter system. There is a unique associative product • : W × W → W satisfying v • w = vw for all v, w ∈ W with ℓ(vw) = ℓ(v) + ℓ(w) and s • s = s for all s ∈ S [22, Thm. 7.1]. An involution word for z ∈ W is a minimal length sequence (s 1 , s 2 , . . . , s n ) with s i ∈ S and z = s * n • · · · • s * 2 • s * 1 • s 1 • s 2 • · · · • s n . When W = S Z , S = {s i : i ∈ Z}, and * = id, this reduces to the definition in Section 2.1, except that now we write words as "(s i 1 , s i 2 , . . . , s in )" rather than "i 1 i 2 · · · i n ." If z belongs to the set of twisted involutions I * (W ) := {w ∈ W : w −1 = w * } then if ℓ(z) < ℓ(zs) and zs = s * z s * zs if ℓ(z) < ℓ(zs) and zs = s * z for all s ∈ S (5.1) by [20,Lemma 3.4]. It follows that z ∈ W has an involution word if and only if z ∈ I * (W ). We write R inv, * (z) for the set of all involution words for z ∈ I * (W ). Because • is associative, the set R inv, * (z) is preserved by the usual braid relations (· · · , s, t, s, t, . . . , mst factors · · · ) ∼ (· · · , t, s, t, s, . . . , mst factors · · · ) for any s, t ∈ S such that |st| = m st < ∞, where |x| is the order of x ∈ W . These are not sufficient to span R inv, * (z). For example, if |st| = 3 so that S 3 ∼ = s, t , then both (s, t) and (t, s) are involution words for tst = t • s • s • t = s • t • t • s = sts.
Example 5.1. The system (W J , J, * ) has type 2 A 3 if J = {r, s, t} where r * = t, s * = s, t * = r, and |rs| = |st| = 3 > |rt| = 2. A corresponding initial relation is (s, r, t, s, -) ∼ (s, r, s, t, -). When m = 2, this is the relation (s, -) ∼ (t, -) that lets us replace the single letter s by t at the beginning of a word. This is not an ordinary braid relation for any m. The following theorem of Hansson and Hultman extends earlier results in [17,18,19,27]. There is a more explicit form of this result (see [15,Thm. 4.1]), similar to Theorem 5.6 below.

Primed spanning relations
An index i is a commutation in an involution word (s 1 , s 2 , . . . , s l ) if it holds that s * i y = ys i for y = s * i−1 • · · · • s * 2 • s * 1 • s 1 • s 2 • · · · • s i−1 . The number of commutations is the same in every involution word for a fixed z ∈ I * (W ) [21,Prop. 2.5]. A primed involution word for z ∈ I * (W ) is a word formed by taking an involution word for z by adding primes to letters indexed by a subset of commutations. The letters in such a word to belong to the set S ⊔ S ′ where S ′ = {s ′ : s ∈ S} is a duplicate set of formal symbols. Let R + inv, * (z) be the set of primed involution words for z. Lemma 5.5. Let z ∈ I * (W ) and a = (a 1 , a 2 , . . . , a n ) ∈ R inv, * (z). Fix integers m ≥ 2 and 0 ≤ i ≤ n − m and suppose (a i+1 , a i+2 , . . . , a i+m ) = (s, t, s, t, . . . ) for s, t ∈ S with |st| = m. Then:  (c) Replace (a i+1 , a i+2 , . . . , a i+m ) in a by (t, s, t, s, . . . ) to form b = (b 1 , b 2 , . . . , b n ) ∈ R inv, * (z).
Then the index i + 1 (respectively, i + m) is a commutation in a if and only if i + m (respectively, i + 1) is a commutation in b.
Theorem 5.6. Let z ∈ I * (W ). Then R + inv, * (z) is a single equivalence class under the transitive closure of the primed braid relations plus the primed initial relations of types A 1 , 2 A 3 , BC 3 , D 4 , H 3 , I 2 (m), or 2 I 2 (m) for 2 ≤ m < ∞.
Proof. Fix a primed involution word α = (s 1 , s 2 , . . . , s l ) ∈ R + inv, * (z). It suffices by Theorem 5.4 to show that this word is equivalent under the given relation to a word in R inv, * (z) ⊂ R + inv, * (z). It is enough to check this when s 1 , s 2 , . . . , s l−1 ∈ S and s l = s ′ ∈ S for some s ∈ S. Note that in this case zs = s * z.
If l = 0 then z = s = s * and α = (s ′ ) ∼ (s) using the primed initial relation of type A 1 . If l > 0 then s l−1 is another right descent of z, so z has at least one other involution word ending in s l−1 . Theorem 5.4 implies we may apply a sequence of ordinary braid relations and initial relations to transform (s 1 , s 2 , . . . , s l−1 , s) to this word.
Consider what happens if we try to apply this sequence to the word α, with primed braid relations in place of ordinary braid relations but still using unprimed initial relations. One of two cases must occur. Either some relation in this sequence moves the single prime from s l to an earlier letter, or we reach a point where we wish to apply an initial relation of length l. (If neither case occurs, then the relations would not change the last letter of our word.) In the first case, we may assume by induction (on the position of the primed letter) that some sequence of primed braid relations and primed initial relations turns α into an element of R inv, * (z) as needed. In the second case, we just substitute the initial relation we want to apply with a primed initial relation that removes all primes from our word, and we again get an element of R inv, * (z).
We do not need to include all primed initial relations to span R + inv, * (w). The following theorem describes a minimal set that is sufficient. In this statement, we assume the Coxeter diagram of (W J , J) in types A 1 , A 3 , BC 3 , D 4 , H 3 , snd I 2 (m), respectively are labeled as follows: In types 2 I 2 (m) and 2 A 3 we assume * acts to interchange a and b.
Theorem 5.7. If z ∈ I * (W ), then R + inv, * (z) is an equivalence class under the transitive closure of the primed braid relations and the symmetric relation with the following properties for each J = J * ⊆ S labeled as above, for any A ∈ {a, a ′ }, B ∈ {b, b ′ }, C ∈ {c, c ′ }, and X ∈ {x, x ′ }: