Highest weight crystals for Schur Q-functions

Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra $\mathfrak{q}_n$. Such $\mathfrak{q}_n$-crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur $P$-polynomials. This article studies a modified form of this category, whose connected normal objects again have unique highest weight elements but now possess characters that are Schur $Q$-polynomials. The crystals in this category have some interesting features not present for ordinary $\mathfrak{q}_n$-crystals. For example, there is an extra crystal operator, a different tensor product, and an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of $\mathfrak{q}_n$-crystal structures on certain families of shifted tableaux and factorized reduced words. We describe extended forms of these structures that give similar examples in our new category.


Overview
Crystals are an abstraction for the crystal bases of quantum group representations. Invented by Kashiwara [24,25] and Lusztig [27,28] in the 1990s, crystals may be viewed concretely as directed acyclic graphs with labeled edges, along with a map assigning weight vectors to each vertex, satisfying certain axioms. Isomorphisms of crystals correspond to weight-preserving graph isomorphisms, while subcrystals correspond to unions of weakly connected graph components.
For each finite-dimensional Lie superalgebra g there is a category of (abstract) g-crystals. The structure of g imposes different requirements for the weight map and edge labels. These categories have some common features. There is always a direct sum operation ⊕ for crystals corresponding to the disjoint union of directed graphs. There is also a more subtle notion of a crystal tensor product ⊗. There is a character map ch assigning to each finite crystal its weight-generating function. Finally, there is a standard crystal B corresponding to the vector representation of an associated quantum group U q (g).
These ingredients are enough to define a full subcategory of normal g-crystals: this consists of the g-crystals whose connected components are each isomorphic to a subcrystal of B ⊗m for some m ≥ 0. Such crystals form the smallest monoidal subcategory containing the standard crystal that is closed under isomorphisms, direct sums, and passage to subcrystals.
Defined in this way, the normal g-crystals are typically the abstract g-crystals that correspond directly to crystal bases of finite-dimensional integrable U q (g)-modules. This connection implies some desirable properties: for example, that each connected normal crystal has a unique highest weight element whose weight determines the crystal's isomorphism class. In such cases, the character map usually identifies the split Grothendieck group of the category of normal g-crystals with a familiar algebra of symmetric polynomials.
The next section reviews how this works in two cases that have been well-studied, when g = gl n is the complex general linear Lie algebra and when g = q n is the queer Lie superalgebra. Section 1.3 outlines our main results, which establish similar formal properties of a new category of what we call q + n -crystals. It is desirable to find proofs of crystal properties using only the relevant combinatorial axioms rather than any connection to quantum groups, and this will be our approach throughout.

Crystals for Schur functions and Schur P -functions
Let n be a positive integer. When g = gl n the edges in each crystal graph are labeled by indices in {1, 2, . . . , n − 1} and the weight map takes values in Z n . The standard gl n -crystal is where the weight of i is the standard basis vector e i ∈ Z n . We review the precise definition of gl n -crystals and their tensor product in Section 3.1.
A gl n -crystal is normal if its connected components are each isomorphic to a subcrystal of a tensor power of the standard gl n -crystal. A remarkable property of normal gl n -crystals is that they are characterized by a set of local conditions known as the Stembridge axioms [39]. For this reason such crystals are sometimes called Stembridge crystals 1 .
A vector λ ∈ Z n is a partition if λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 0. A gl n -highest weight element of a gl n -crystal is any vertex in the associated crystal graph with no incoming edges. The claims in the following theorem are well-known, and serve as a prototype for subsequent results. . If B is a connected normal gl n -crystal, then B has a unique gl n -highest weight element, whose weight λ ∈ Z n is a partition such that ch(B) = s λ (x 1 , x 2 , . . . , x n ). For each partition λ ∈ Z n , there is a connected normal gl n -crystal with highest weight λ, and finite normal gl n -crystals are isomorphic if and only if they have the same character.
Here s λ (x 1 , x 2 , . . . , x n ) denotes the Schur polynomial of a partition λ in n commuting variables. The Schur polynomials indexed by partitions λ ∈ Z n are a Z-basis for the subring Sym(x 1 , x 2 , . . . , x n ) of all symmetric polynomials in Z[x 1 , x 2 , . . . , x n ].
The split Grothendieck group of an additive category C is the abelian group generated by the symbols [ Corollary 1.2. The map assigning a gl n -crystal to its character defines a ring isomorphism from the split Grothendieck group of the category of finite normal gl n -crystals to Sym(x 1 , x 2 , . . . , x n ).
The general linear Lie algebra gl n has two super-analogues given by gl m|n−m (see [3]) and the queer Lie superalgebra q n . Grantcharov et al. develop a theory of crystals for q n in [11,12,13]. In this theory, the edges in each crystal graph are labeled by indices in {1, 1, 2, . . . , n − 1} and the weight map takes values in N n where N := {0, 1, 2, . . . }. The standard q n -crystal is formed by adding a single 1-arrow to the standard gl n -crystal: 1 2 3 · · · n 1 1 2 3 n − 1 (1.2) For the precise definitions of q n -crystals and their tensor product, see Section 3.2. Besides in [11,12,13], these crystals have been studied in [2,6,10,19,20,31], for example. Normal q n -crystals are defined in terms of tensor powers of the standard q n -crystal in the same way as in the gl n -case. The notion of q n -highest weight elements for q n -crystals is slightly different: these are again the vertices with no incoming edges, but now in an extended crystal graph involving additional arrows with labels in {n − 1, . . . , 2, 1, 1, 2, . . . , n − 1}; see Definition 3.9.
A partition λ ∈ N n is strict if it has no repeated nonzero entries. The following q n -analogue of Theorem 1.1 contains several results in [12], especially [12,Thm. 2.5 and Cor. 4.6]. Theorem 1.3 (See [12]). If B is a connected normal q n -crystal, then B has a unique q n -highest weight element, whose weight λ ∈ N n is a strict partition such that ch(B) = P λ (x 1 , x 2 , . . . , x n ). For each strict partition λ ∈ N n , there is a connected normal q n -crystal with highest weight λ, and finite normal q n -crystals are isomorphic if and only if they have the same character.
In this statement, P λ (x 1 , x 2 , . . . , x n ) is the generating function for semistandard shifted tableaux known as a Schur P -polynomial (see Section 2.2 for the definition). The fact that the characters of connected normal q n -crystals are Schur P -polynomials is not explicitly stated in [12] but can be deduced using [37,Thm. 2.17]. The family of Schur P -polynomials indexed by strict partitions λ ∈ N n form a basis for a subring Sym P (x 1 , x 2 , . . . , x n ) ⊂ Sym(x 1 , x 2 , . . . , x n ).
Corollary 1.4. The map assigning a q n -crystal to its character defines a ring isomorphism from the split Grothendieck group of the category of finite normal q n -crystals to Sym P (x 1 , x 2 , . . . , x n ).

Crystals for Schur Q-functions
Our main results concern a generalization of the category of q n -crystals. We call the objects of this new category q + n -crystals. Viewed as directed graphs, these crystals have edges labeled by indices in {1, 0, 1, 2, . . . , n − 1} and weights in N n . The standard q + n -crystal is where both i and i ′ have weight e i ∈ Z n . This is isomorphic to the direct sum of two copies of the standard q n -crystal, with one additional 0-arrow. The q + n -tensor product is slightly unusual and combines features of queer crystals and of gl m|n -crystals from [3] in the degenerate case m = n = 1. For the precise definitions, see Section 3.3.
Normal q + n -crystals are defined in terms of tensor powers of the standard q + n -crystal in the same way as in the gl n -and q n -cases. The q + n -highest weight elements of a q + n -crystal are again the source vertices in a certain extended crystal graph; see Definition 3.19. Our main result is the following extension of Theorem 1.3, which combines Theorem 6.20, Corollary 7.14, and Theorem 7.16. Theorem 1.5. If B is a connected normal q + n -crystal, then B has a unique q + n -highest weight element, whose weight λ ∈ N n is a strict partition such that ch(B) = Q λ (x 1 , x 2 , . . . , x n ). For each strict partition λ ∈ N n , there is a connected normal q + n -crystal with highest weight λ, and finite normal q + n -crystals are isomorphic if and only if they have the same character.
Here Q λ (x 1 , x 2 , . . . , x n ) is the Schur Q-polynomial of a strict partition λ, which is defined to be 2 ℓ(λ) P λ (x 1 , x 2 , . . . , x n ) where ℓ(λ) is the number of nonzero parts of λ. As λ ranges over strict partitions in N n these polynomials are a Z-basis for another subring Sym Q (x 1 , x 2 , . . . , x n ).
Corollary 1.6. The map assigning a q + n -crystal to its character defines a ring isomorphism from the split Grothendieck group of the category of finite normal q + n -crystals to Sym Q (x 1 , x 2 , . . . , x n ).
As an application of Theorem 1.5, we can derive a new shifted Littlewood-Richardson rule for products of Schur Q-polynomials. The classical shifted Littlewood-Richardson rule (see [29, (8.17)(i)] or [38,Thm. 8.3]) expands products of Schur P -functions as N-linear combinations of Schur P -functions. This can be converted to a rule for Schur Q-functions by dividing by appropriate powers of two, but then it is not obvious that the coefficients that appear are all integers. Using q + n -crystals lets us avoid this issue. For each strict partition λ ∈ N n , fix a connected normal q + n -crystal B λ with highest weight λ. Using Theorem 1.5 to decompose the character of B λ ⊗ B µ implies the following: Corollary 1.7. It holds that Q λ (x 1 , x 2 , . . . , x n )Q µ (x 1 , x 2 , . . . , x n ) = ν g ν λµ Q ν (x 1 , x 2 , . . . , x n ) for all strict partitions λ, µ ∈ N n , where the sum is over all strict partitions ν ∈ N n and g ν λµ ∈ N is the number of q + n -highest weight elements in B λ ⊗ B µ of weight ν.
Queer crystals may be used to show that certain power series are Schur P -positive in the sense of being positive linear combinations of Schur P -functions (see [31,Cors. 3.34 and 3.38], for example). A similar application of q + n crystals is to demonstrate Schur Q-positivity (see Corollary 7.19). The latter is a stronger property compared to Schur P -positivity, as is Theorem 1.5 compared to Theorem 1.3. Although there is a commutative diagram forgetful functors { q + n -crystals } { q n -crystals } { gl n -crystals } the horizontal arrow does not take normal q + n -crystals to normal q n -crystals. This means that Theorem 1.3 does not directly imply similar properties of normal q + n -crystals. As such, extending Theorem 1.3 to Theorem 1.5 is nontrivial.
An interesting feature of q + n -crystals concerns an action of the finite Coxeter group W BC n of type BC n . There is an action of the symmetric group S n on the vertices of normal gl n -and q ncrystals. Under this action, the longest permutation w 0 ∈ S n interchanges highest and lowest weight elements; see Proposition 3.11. This property does not hold for normal q + n -crystals. Instead, we show that there is an action of W BC n on the vertices of normal q + n -crystals, and for this action the longest preimage of w 0 under the projection W BC n → S n interchanges highest and lowest weight elements; see Proposition 7.15.
Our strategy for proving Theorem 1.5 has the following outline. In Section 5, we describe a q + n -crystal structure on increasing factorizations of primed involution words, which are certain analogues of reduced words for permutations. This generalizes a q n -crystal identified by Hiroshima in [20]. It is relatively easy to show that every connected normal q + n -crystal may be embedded in one of these objects; this is carried out later in Section 7.1.
Next, we show in Section 6 how to extend a q n -crystal structure on semistandard shifted tableaux studied in [2,17,19] to a q + n -crystal on a larger set. Building on results in [19], we are able to prove that each q + n -crystal of shifted tableaux of a fixed strict partition shape is connected with unique highest and lowest weight elements; see Theorem 6.20.
In Section 7.2 we show how to embed our q + n -crystals of increasing factorizations into q + n -crystals of shifted tableaux. This requires some technical results from [32] about a shifted form of Edelman-Greene insertion. Combining these steps lets us deduce that each connected normal q + n -crystal occurs as a crystal of shifted tableaux and therefore has a Schur Q-polynomial as its character.
Our final task in Section 7.3 is to show that all of our q + n -crystals of shifted tableaux are normal. We can prove this directly for crystals of one-row tableaux. Each Schur Q-polynomial appears as constituent of some product of Schur Q-polynomials indexed by one-row partitions. Using this fact, we deduce that each q + n -crystal of shifted tableaux occurs as a full subcrystal of a tensor product of crystals of one-row tableaux, and is therefore normal; see Theorem 7.16.

Connections to representation theory
We briefly summarize how crystals arise from representation theory. The quantum group U q (gl n ) may be defined as a bialgebra over the field of formal Laurent series C((q)) with generators e i and f i indexed by i ∈ {1, 2, . . . , n − 1}. These generators give rise to operatorsẽ i andf i on U q (gl n )modules M that are integrable in the sense of [25, §1.2]. Each pair e i and f i generates a copy of Kashiwara's results in [24,25] show that every integrable module M has a crystal basis, which consists of a pair (L, B) where L is a free C[[q]]-module with M = C((q)) ⊗ C[[q]] L and B ⊂ L is C-basis of L/qL, subject to several conditions involvingẽ i andf i [25, §2.3]. In particular, one must This means that much of the information in a crystal basis may be recorded in the crystal graph on B with labeled directed edges b i − → c wheneverf i (b) = c. This graph gives an example of a normal gl n -crystal, and every finite normal gl n -crystal arises in this way.
The quantum queer superalgebra U q (q n ) is another bialgebra over C((q)), also with generators e i and f i indexed by i ∈ {1, 2, . . . , n − 1} but now with one extra generator k 1 [13, Def. 1.1]. There is a semisimple category of integral modules for U q (q n ) [13, Def. 1.5] and on such modules M the generators of U q (q n ) give rise to certain operatorsẽ i andf i indexed by i ∈ {1, 1, 2, . . . , n − 1} and an additional operatork 1 [13, §2].
In this context, a crystal basis [13,Def. 2.2] for M also consists of a pair (L, The set B, however, is not a basis for L/qL but instead a set ofk 1 -invariant subspaces that give a direct sum decomposition of L/qL. It is again  Grantcharov et al. prove that every integral U q (q n )-module has a crystal basis and that every finite normal q n -crystal arises from such a basis [13,Thm. 4.6].
Our concept of normal q + n -crystals should correspond in a similar way to crystal bases (L, B) for integrable U q (q n )-modules, but with the following additional information. Namely, one must also specify a refined direct sum decomposition of each subspace b ∈ B, which is compatible with e i andf i , such the action ofk 1 on the summands of this decomposition defines a gl 1|1 -crystal in the sense of [3, §2.3]. We have not yet found a completely satisfactory way of characterizing the data that makes up this kind of extended crystal basis for integrable U q (q n )-modules. For both g = gl n and g = q n , the natural tensor product for integrable U q (g)-modules gives rise to a tensor product for crystal bases, and this informs the definition of the relevant tensor product for g-crystals. We also do not yet fully understand how to motivate the tensor product for q + n -crystals described in Section 3.3 from representation theory. We hope clarify these points in future work.

Comparison with Gillespie-Levinson-Purbhoo crystals
On the way to proving Theorem 1.5, we construct a connected normal q + n -crystal on the set ShTab + n (λ) of semistandard shifted tableaux of a given strict partition shape λ with all entries at most n. The character of this object is the Schur Q-polynomial Q λ (x 1 , x 2 , . . . , x n ). In [8,9] Gillespie, Levinson, Purbhoo study another crystal-like structure on semistandard shifted tableaux. Their objects are also encoded as certain directed acyclic graphs with labeled edges, and have characters that are Schur Q-polynomials.
Several differences offset these formal similarities, and we do not know of any way to derive our crystal constructions from those in [8,9] or vice versa. In particular: • The vertices in Gillespie, Levinson, Purbhoo's crystal graph are a proper subset of ShTab + n (λ), consisting of representatives for a certain equivalence relation; see [8,Def. 2.6].
• There are 2(n − 1) edge labels for the crystal graphs in [8,9] whereas the edge labels for our crystals ShTab + n (λ) come from the (n + 1)-element set {1, 0, 1, . . . , n − 1}. The crystal operators corresponding to these two sets of edges do not seem to be easily related.
• There is an axiomatic definition of Gillespie, Levinson, Purbhoo's crystal graphs in [9], but no notion of a tensor product analogous to the tensor product for q + n -crystals. Independent of this comparison, it is an interesting open problem to give the category of objects in [9] a monoidal structure and to relate this to representation theory.

Outline
Here is a brief outline of the rest of this article. Section 2 explains some notational conventions and preliminaries on symmetric functions. Section 3 gives the precise definitions of the gl n -, q n -and q + n -crystals discussed informally above. In Sections 4, 5, and 6 we construct three families of q + ncrystals-on words, increasing factorizations, and shifted tableaux, respectively. Then in Section 7 we describe several morphisms between these crystals, in order to prove Theorem 1.5.

Preliminaries
Given integers p, q ∈ Z let [p, q] := {i ∈ Z : p ≤ i ≤ q} and [q] := [1, q]. Recall that N := {0, 1, 2, . . . }. For i ∈ Z, we set i ′ := i − 1 2 and Z ′ : We refer to elements of Z ′ as primed numbers. Removing the prime for some i ∈ Z ⊔ Z ′ means to replace i with ⌈i⌉. Adding a prime to a number i ∈ Z ⊔ Z ′ means to replace it with ⌈i⌉ − 1 2 . Throughout, we fix a positive integer n and let x 1 , x 2 , x 3 , . . . be commuting variables.

Shifted tableaux
If T is a (shifted) tableau, then we write (i, j) ∈ T to indicate that (i, j) belong to the domain of T and we let T ij denote the value assigned to this position. We draw tableaux in French notation, so that row indices increase from bottom to top and column indices increase from left to right. If then S is a tableau and T is a shifted tableau of shape λ = (4, 3), and S 23 = 7 while T 23 = 2. The (main) diagonal of a shifted tableau is the set of positions (i, j) in its domain with i = j. 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. The examples in (2.1) are both semistandard. For n ∈ N, we write Tab n (λ) for the set of semistandard tableaux of shape λ with all entries in [n], ShTab + n (µ) for the set of semistandard shifted tableaux of shape µ with all entries in {1 ′ < 1 < · · · < n ′ < n}, and ShTab n (µ) for the subset of elements in ShTab + n (µ) with no primed entries on the diagonal.

Abstract crystals
This section contains the precise definitions of the abstract and normal gl n -, q n -and q + n -crystals discussed in the introduction. Each of these structures will formally consist of a nonempty set B with a weight map wt : B → Z n and a family of raising operators e i : B → B ⊔ {0} and lowering operators f i : The crystal graph associated to this data has vertex set B and labeled edges b i − → c whenever f i (b) = c = 0. This graph determines both the raising and lowering operators, since it will always be required for A subset of B that forms a weakly connected component in the crystal graph inherits its own crystal structure and is called a full subcrystal. Within each family of crystals, an isomorphism will mean a weight-preserving map that defines an isomorphism of the corresponding crystal graphs.

Crystals for general linear Lie algebras
The definition of a gl n -crystal explained below is fairly standard in the literature. In presenting this material we follow the conventions of Bump and Schilling's book [4].
Let B be a nonempty set with a function wt : B → Z n and an auxiliary element 0 / ∈ B. For each i ∈ [n − 1], assume that maps e i , f i : B → B ⊔ {0} are given. We define ε i , ϕ i : B → N ⊔ {∞} by We refer to the ε i 's and ϕ i 's as string lengths. The value of wt(b) is the weight of b ∈ B. Finally, write e 1 , e 2 , . . . , e n for the standard basis of Z n .
More precisely, this is the definition of a seminormal gl n -crystal in [4]. The character of any finite gl n -crystal is a symmetric Laurent polynomial [4, §2.6].
The notion of highest and lowest weight elements for gl n -crystals is straightforward. Namely, if B is a gl n -crystal, then a gl n -highest (respectively, gl n -lowest) weight element b ∈ B is an element with e i (b) = 0 (respectively, f i (b) = 0) for all i ∈ [n − 1].
An essential feature of each of category of crystals is the existence of a nontrivial tensor product. If B and C are nonempty sets, then let B ⊗ C := {b ⊗ c : b ∈ B, c ∈ C} be the set of formal tensors of elements of B with elements of C. The next definition follows the "anti-Kashiwara" convention.
Let ½ be a gl n -crystal with a single element, whose weight is 0 ∈ Z n . The standard gl n -crystal is the crystal in (1.1); we denote this by B n . As in the introduction, a gl n -crystal is normal if each of its full subcrystals is isomorphic to a full subcrystal of B ⊗m n for some m ∈ N, where B 0 n := ½.
Remark 3.3. The following well-known signature rule (discussed, for example, in [4, §2.4]) can be used to compute the crystal operators for B ⊗m n . Suppose w = w 1 ⊗ w 2 ⊗ · · · ⊗ w m ∈ B ⊗m n , and i ∈ [n − 1]. Mark the entries w k = i by a right parenthesis ")" and entries w j = i + 1 by a left parenthesis "(". The i-unpaired indices in w are the indices j ∈ [m] with w j ∈ {i, i + 1} that are not the positions of matching parentheses. Now let k be the last i-unpaired index of w with w k = i. If no such index exists then f i (w) = 0; otherwise f i (w) is formed from w by changing w k to i + 1. Similarly, let j be the first i-unpaired index of w with w j = i + 1. If no such index exists, then e i (w) = 0; otherwise e i (w) is formed from w by changing w j to i. (P3) If wt(b) 1 = wt(b) 2 = 0, then (ε 1 + ϕ 1 )(b) = 0, and otherwise (ε 1 + ϕ 1 )(b) = 1.

Crystals for queer Lie superalgebras
We typically consider q n -crystals when n ≥ 2, but for convenience we also define an (abstract) q 1 -crystal to be a nonempty set B with a weight map wt : B → N.
The definitions in [12,13] omit (P3), which implies that the character of any finite q n -crystal is in Sym P (x 1 , x 2 , . . . , x n ) [31,Prop. 2.5]. In [12,13], abstract q n -crystals are also required to be normal as gl n -crystals, but it is common to omit this condition.
Our description of the tensor product for q n -crystals again follows the "anti-Kashiwara" convention, which is opposite to that of [12,Thm. 1.8] and [13,Thm. 2.7].
Theorem 3.5 (See [12,13]). Suppose B and C are q n -crystals. Then the gl n -crystal B ⊗ C has a unique q n -crystal structure whose queer crystal operators are given by The standard q n -crystal is the crystal (1.2) specified in the introduction; we denote this object by B n . Crystal graphs of q n -crystals have edges labeled by indices in {1, 1, 2, . . . , n − 1}. The 1-element gl n -crystal ½ may be regarded as a q n -crystal. A q n -crystal is normal if each of its full q n -subcrystals is isomorphic to a full q n -subcrystal of B ⊗m n for some m ∈ N, where B 0 n := ½.
It remains to give the formal definition of the q n -highest weight elements that are mentioned in Theorem 1.3. This is more complicated than in the gl n -case and involves the following operators. Let B be a gl n -crystal. For each i ∈ [n − 1] define a map σ i : B → B by When we erase all arrows except those of the form i −→, the crystal graph of B becomes a disjoint union of paths called i-strings, and σ i reverses the order of the elements in each i-string. One has σ 2 i (b) = b, and wt(σ i (b)) is obtained from wt(b) by interchanging wt(b) i and wt(b) i+1 .
Remark 3.7. If B is a normal gl n -crystal, then there is a unique group action of S n on the set B in which the transposition (i, i + 1) acts as σ i [4,Thm. 11.14]. In general there may fail to be such a group action. For example, there is a unique gl 3 -crystal with crystal graph using the convention that σ i (0) = 0.
Example 3.8. In Example 3.6, the operator f 2 acts as which means that f 2 (b) = f 2 (b) for all crystal elements b = 2 ⊗ 2 .
Definition 3.9. Suppose B is a q n -crystal. A q n -highest (respectively, q n -lowest) weight element b ∈ B is an element with e i (b) = 0 for all i ∈ I ⊔ I (respectively, f i (b) = 0 for all i ∈ I ⊔ I ′ ).
The unique q n -highest and q n -lowest weight elements in B n (respectively, B 3 ⊗ B 3 ) are 1 and n (respectively, 1 ⊗ 1 and 3 ⊗ 3 ). Let ≺ be the transitive closure of the relation on Z n that has v ≺ v + e i − e i+1 for all i ∈ [n − 1]. This relation is a strict partial order. Since the operators e i , e i , and e i ′ change the weight of an element by adding e i − e i+1 when they do not act as zero, we have X ⊂ Y ⊂ Z.
All weights for elements of q n -crystals are in N n , so Hence any b ∈ B must have b ≺ c for some c ∈ X, so the set X is nonempty if B is nonempty. This means that if B has a unique q n -highest weight element b then Proposition 3.11. Suppose B is a normal gl n -crystal. Then b ∈ B is a gl n -lowest weight element if and only if σ w 0 (b) ∈ B is a gl n -highest weight element. If B is also a q n -crystal, then b ∈ B is a q n -lowest weight element if and only if σ w 0 (b) ∈ B is a q n -highest weight element.
The characterization of q n -lowest weight elements in this proposition is [12,Def. 1.12].
The tableaux T λ and U λ are interchanged by the Lusztig involution, which also swaps highest and lowest weight elements; see [4,Exercises 5.1 and 5.2] or [26, §2.4]. Since T λ is the unique highest weight element [4,Thm. 3.2], U λ is therefore the unique lowest weight element. The involution σ w 0 therefore interchanges highest and lowest weight elements in normal gl n -crystals.
When B is a q n -crystal that is normal as a gl n -crystal, σ w 0 f i ′ = e n−i σ w 0 for i ∈ [n − 1] by definition, so f i (b) = 0 for all i ∈ I ⊔ I ′ if and only if e i σ w 0 (b) = 0 for all i ∈ I ⊔ I.

Extended queer supercrystals
In this section n is allowed to be any positive integer. Suppose B is a q n -crystal with additional maps e 0 , f 0 : B → B ⊔ {0}. Define ε 0 , ϕ 0 : B → N ⊔ {∞} by the formula (3.1) with i = 0. The following extension of Definition 3.4 is our primary subject. Here is a first link between q + n -crystals and Schur Q-functions: Proof. Let B be a finite q + n -crystal. Since q + n -crystals are q n -crystals, we know that ch(B) ∈ Sym P (x 1 , x 2 , . . . , x n ) [31, Prop. 2.5]. As noted in Section 2.2, an element f ( The following result gives a tensor product for q + n -crystals. This is more complicated than for q n -crystals, but will turn out to have several desirable properties.
Theorem 3.14. Let B and C be q + n -crystals. Then the gl n -crystal B ⊗ C has a unique q + n -crystal structure in which e 0 and f 0 are given by and in which (when n ≥ 2) e 1 and f 1 are given by Proof. When e 0 or f 0 do not act as zero, they do not change the values of ε i or ϕ i for any i = 0 by property (Q1). From this observation, the conditions in Definitions 3.4 and 3.12 are straightforward to derive from the q + n -crystal axioms for B and C, so B ⊗ C is a q + n -crystal. Now suppose B, C, and D are q + n -crystals. The natural bijection (B ⊗ C) ⊗ D ∼ − → B ⊗ (C ⊗ D) commutes with the gl n -crystal operators and also with e 0 and f 0 , while preserving the weight map. It remains to check that this map commutes with e 1 and f 1 . This requires a somewhat involved case analysis. Fix b ∈ B, c ∈ C, and d ∈ D. We check that If wt(c) 1 = 0, f 0 e 1 (c) = 0, and e 0 (d) = 0, then and that in the remaining case , then one checks that The first and last terms are equal by assumption. The second equality holds if wt(b ⊗ c) 1 = 0 as then we must have Then (1) and (2) that This shows that the e 1 operator commutes with the natural bijection . A similar argument shows that f 1 also commutes with this map.
The standard q + n -crystal is the crystal described by (1.3) in the introduction; we denote this object by B + n . Crystal graphs of q + n -crystals have edges labeled by indices in {1, 0, 1, . . . , n − 1}. There are two full q + 2 -subcrystals B and C, which are isomorphic via the map that exchanges the two elements in the middle of the top row with the two elements in the middle of the bottom row, and reflects all other elements across the central vertical axis. The character of B + 2 is the Schur Remark 3.16. We can use Example 3.15 to explain the origin of the tensor product rules in Theorem 3.14. The formulas for e 0 (b ⊗ c) and f 0 (b ⊗ c) are designed so that if only the 0-arrows are retained in the crystal graph, then any normal q + n -crystal becomes a gl 1|1 -crystal in the sense of [3, §2.4]; this idea is explained more fully in Remark 3.21.
It remains to motivate the definitions of e 1 (b ⊗ c) and f 1 (b ⊗ c). Here, we are lead by three principles. First, we want these formulas to agree with the ones in Theorem 3.5 if {b, c} ⊂ B n ⊂ B + n . Second, when i = 0 we want e i and f i to commute with the map unprime : B + n ⊗ B + n → B n ⊗ B n that removes the primes from each factor of i 1 ⊗ i 2 (and sends 0 → 0). Finally, we want the crystal graph of B + 2 ⊗ B + 2 to have two isomorphic connected components. The first principle requires us to have f 1 ( 1 ⊗ 1 ) = 2 ⊗ 1 and f 1 ( 1 ⊗ 2 ) = 2 ⊗ 2 , which already places the 8 elements on the right side of (3.6) in one connected component. Since a vertex in the crystal graph can only be the target node of one arrow with a given label, the other principles then force us to have f 1 ( 1 ′ ⊗ 1 ) = 2 ⊗ 1 ′ and f 1 ( 1 ′ ⊗ 2 ) = 2 ′ ⊗ 2 . The remaining 1-arrows in (3.6) are uniquely determined if we want there to be two isomorphic components. Expressing the resulting cases for e 1 (b ⊗ c) and f 1 (b ⊗ c), just for b, c ∈ B + 2 , solely in terms of the weight map and the values of e 0 , f 0 , e 1 , and f 1 on each factor leads to the formulas in Theorem 3.14.
The 1-element gl n -crystal ½ may be regarded as a q + n -crystal. Following the conventions in the previous sections, we define a q + n -crystal to be normal if each of its full q + n -subcrystals is isomorphic to a full q + n -subcrystal of (B + n ) ⊗m for some m ∈ N, where (B + n ) 0 := ½.
Since there is an isomorphism of gl n -crystals B + n ∼ = B n ⊔ B n , a normal q + n -crystal is normal as a gl n -crystal. However, normal q + n -crystals are not necessarily normal as q n -crystals. This means that results like Theorem 1.3 do not directly imply similar properties of normal q + n -crystals.
Example 3.17. The following crystal graph shows a full q Although this is a normal q + 2 -crystal, it is not a normal q 2 -crystal. The elements 1 ′ 21, 121 ′ , 221 ′ , and 2 ′ 21 make up a full q 2 -subcrystal which is not normal, since it has two q 2 -highest weight elements.
To define highest and lowest weight elements for q + n -crystals, we need a few more operators.
where σ i is defined as in (3.2). This means that e [1] 0 = e 0 and f Example 3.18. In Example 3.15, the operator f [2] 0 acts as The unique q + n -highest and q + n -lowest weight elements in B + n are 1 and n ′ . The q + 2 -highest weight elements in the crystal B + 2 ⊗ B + 2 from Example 3.15 are 1 ⊗ 1 and 1 ⊗ 1 ′ , while the q + 2 -lowest weight elements are 2 ′ ⊗ 2 and 2 ′ ⊗ 2 ′ . Theorem 1.5, which we prove in Section 7, asserts that each connected normal q + n -crystal has a unique q + n -lowest weight element. The following statement is a corollary of this property.
Corollary 3.20. Let B be a normal q + n -crystal. Then there is a unique map ν 0 : B → N such that n this is given by setting ν( a ) := 1 and ν( a ′ ) := 0. If B and C both have such a map, then ν(b ⊗ c) := ν(b) + ν(c) is a map ν : B ⊗ C → N with the desired property. Therefore such a map ν exists for any normal q + n -crystal.
We can use this result to motivate part of the q + n -crystal tensor product.
If we ignore the other operators, then Theorem 3.14 coincides with the tensor product for gl 1|1 -crystals in [3, §2.4].

Crystal operators on words
It is useful to provide a model for the q + n -crystals (B + n ) ⊗m when m ∈ N. Define W + n (m) to be the set of words of length m with letters in which transfers a q + n -crystal structure to W + n (m). We describe this below.

Formulas for crystal operators
Let w = w 1 w 2 · · · w m ∈ W + n (m) and i ∈ [n − 1]. Consider the word formed from w by replacing each letter w k ∈ {i ′ , i} by a right parenthesis ")" and each letter w j ∈ {i + 1, i + 1 ′ } by a left parenthesis "(". The i-unpaired indices in w are the indices j ∈ [m] with w j ∈ {i ′ , i, i + 1 ′ , i + 1} that are not the positions of matching parentheses in this word. For example, if w = 131 ′ 22 ′ 131 ′ 2 and i = 1, then the word with parentheses is )3)(()3)( so the i-unpaired indices are 1, 3, and 9. The following description of the lowering and raising operators f i and e i for W + n (m) is a minor generalization of the signature rule in Remark 3.3.
The definitions of f 0 (w) and e 0 (w) are next given as follows: . If no such j exists or w j = 1, then f 0 (w) := 0. Otherwise f 0 (w) is formed from w by changing w j to 1 ′ . Similarly, if no such j exists or w j = 1 ′ , then e 0 (w) := 0. Otherwise e 0 (w) is formed by changing w j to 1.
The definitions of f 1 (w) and e 1 (w) require a few more cases to state in full.
• If such k exists, then f 1 (w) is formed from w by changing w j to w k + 1 and w k to w j .
• Otherwise, f 1 (w) is formed from w by adding 1 to w j .
• If such k exists, then e 1 (w) is formed from w by changing w j to w k and w k to w j − 1.
• Otherwise, e 1 (w) is formed from w by subtracting 1 from w j .
is evidently a gl n -subcrystal of W + n (m) and it follows from (4.1) that unprime defines a gl n -crystal isomorphism from this set to W n (m). We can reformulate this observation as the following property.
We define a quasi-isomorphism between gl n -, q n -, or q + n -crystals, respectively, to be a map ψ : B → C such that for each full subcrystal X ⊂ B, the image Y := ψ(X ) is a full subcrystal of C and the restricted map ψ : X → Y is a crystal isomorphism.

Weyl group action
On normal q + n -crystals, there is a action of hyperoctahedral group extending (3.2). This is an interesting feature of q + n -crystals not present for q n -crystals. Suppose B is a q + n -crystal. Then the formula (3.2) for σ i : B → B makes sense when i = 0, and gives a self-inverse, weight-preserving bijection σ 0 : n denote the group whose elements are the permutations w of Z satisfying w(−i) = −w(i) for all i ∈ [n] and w(i) = i for all i > n. This is the finite Coxeter group of type BC and rank n. Its simple generators are given by t 0 := (−1, 1) and t i : Then there exists a unique action of W BC n on B in which t 0 and t i for i ∈ [n − 1] act as the operators σ 0 and σ i , respectively.
Proof. It suffices to check that σ 0 , σ 1 , σ 2 , . . . , σ n−1 satisfy the braid relations for type BC. Since B is normal as a gl n -crystal, we already know from Remark 3.7 that all relevant (type A) braid relations among σ 1 , σ 2 , . . . , σ n−1 hold. It is also clear from axioms (Q1) and (Q2) in Definition 3.12 that σ 0 commutes with σ i for all i ∈ [2, n − 1]. Thus it remains only to show that σ 0 σ 1 σ 0 σ 1 = σ 1 σ 0 σ 1 σ 0 as operators B → B. For this, we may assume that B = W + n (m) for some m ≥ 0. Choose an element w = w 1 w 2 · · · w m ∈ W + n (m). The subword of letters in w at 1-unpaired indices has the form a 1 a 2 · · · a r b 1 b 2 · · · b s where each a i ∈ {1 ′ , 1} and each b i ∈ {2 ′ , 2}. One can check that σ 1 acts on w by changing this subword to with primed letters occurring at the same locations as in w. On the other hand, σ 0 acts on w by simply toggling the prime on the first letter equal to 1 ′ or 1, fixing w if there are no such letters. For example, If w contains no letters equal to 1 ′ or 1, then σ 0 σ 1 σ 0 σ 1 (w) and σ 1 σ 0 σ 1 σ 0 (w) are both formed from w by toggling the prime on the first letter equal to 2 ′ or 2, if one exists. Likewise, if r = s = 0, then σ 0 σ 1 σ 0 σ 1 (w) = σ 1 σ 0 σ 1 σ 0 (w) = w. Assume w contains a letter equal to 1 ′ or 1, and let j be the index of the first such letter. Also assume r + s > 0 so that w has at least one 1-unpaired index, and let k be the first such index. If j < k, then σ 0 σ 1 (w) = σ 1 σ 0 (w) so σ 0 σ 1 σ 0 σ 1 (w) = σ 1 σ 0 σ 1 σ 0 (w) = w. The same conclusion holds j = k and s > 0.

Involution words
Let S Z be the group of permutations of Z that fix all but finitely many points. This is a Coxeter group with simple generators There is a unique associative operation • : A reduced word for π ∈ S Z is an integer sequence a 1 a 2 · · · a n of shortest possible length such that π = s a 1 s a 2 · · · s an (equivalently with π = s a 1 • s a 2 • · · · • s an ). Let I Z := {π ∈ S Z : π = π −1 }. If z ∈ I Z and i ∈ Z, then This implies that I Z = {π −1 • π : π ∈ S Z }, so the following construction exists for any z ∈ I Z : Definition 5.1. An involution word for z ∈ I Z is an integer sequence a 1 a 2 · · · a n of shortest possible , 231, 213, 321}. Involution words have been studied before in various forms, for example, in [5,15,18,21,36].
The following generalization of Definition 5.1 is considered in [32,33]. A commutation for a 1 a 2 · · · a n ∈ R inv (z) is an index i ∈ [n] such that both a i and 1 + a i are fixed points of the involution Definition 5.2. A primed involution word for z ∈ I Z is a primed word whose unprimed form is in R inv (z) and whose primed letters occur at commutations. Write R + inv (z) for the set of such words.
The number of commutations is the same for every involution word for z ∈ I Z , and given by the absolute length ℓ abs (z) := |{i ∈ Z : i < z(i)}|. Therefore |R + inv (z)| = 2 ℓ abs (z) |R inv (z)|. Definitions 5.1 and 5.2 can be formulated for arbitrary Coxeter systems (see [33, §5]), but our applications only require the versions for permutations just given. The next two propositions recall a few special properties of primed involution words that will be useful later. . Suppose z ∈ I Z and X, Y ∈ Z. No word in R + inv (z) contains any of the following as consecutive subwords: Define≡ to be the transitive closure of the symmetric relation on primed words that has aXY b≡ aY Xb for all primed words a, b and X, Y ∈ Z ⊔ Z ′ with |⌈X⌉ − ⌈Y ⌉| > 1, as well as aXY Xb≡aY XY b and aX ′ Y Xb≡aY XY ′ b for all primed words a, b and X, Y ∈ Z with |X −Y | = 1, and finally with Xa≡ X ′ a and XY a≡ Y Xa for all primed words a and X, Y ∈ Z. For example,

Formulas for crystal operators
Fix an involution z ∈ I Z and define Incr + n (z) to be the set of tuples a = (a 1 , a 2 , · · · , a n ) where each a i is a strictly increasing (possibly empty) primed word such that the concatenation concat(a) := a 1 a 2 · · · a n is in R + inv (z). Define the weight of a ∈ Incr + n (z) to be wt(a) := (ℓ(a 1 ), ℓ(a 2 ), . . . , ℓ(a n )) ∈ N n . (5.2) We will make Incr + n (z) into a q + n -crystal with this weight map below. The sequence of numbersĉ i (z) := |{j ∈ Z : z(j) < i < j and z(i) > z(j)}| for i ∈ Z make up the involution code of z. One hasĉ i (z) = 0 for all but finitely many i ∈ Z. The transpose of the partition sorting (. . . ,ĉ 1 (z),ĉ 2 (z),ĉ 3 (z), . . . ) is a strict partition, called the involution shape of z in [16] and denoted µ(z). For example, if z = (1, 5)(2, 3) ∈ I Z , then µ(z) = (4, 1); this also holds if z = (k + 1, k + 5)(k + 2, k + 3) ∈ I Z for any k ∈ Z. The following is useful to note: is nonempty if and only if µ(z) has at most n nonzero parts.
Proof. It is noted in [31,Remark 3.16] that Incr + n (z) is nonempty if and only if max i∈Zĉi (z) ≤ n.
For the rest of this section we assume ℓ(µ(z)) ≤ n so that Incr + n (z) = ∅ and µ(z) ∈ N n . The crystal operators on Incr + n (z) are defined in terms of the following pairing procedure: Definition 5.6. Suppose v = v 1 v 2 · · · v p and w = w 1 w 2 · · · w q are strictly increasing words with letters in Z ⊔ Z ′ . Form a set of paired letters pair(v, w) by iterating over the letters in w from largest to smallest; at each iteration, the current letter w j is paired with the smallest unpaired letter v i with ⌈v i ⌉ > ⌈w j ⌉ (if such a letter exists) and then (v i , w j ) is added to pair(v, w).
Definition 5.9. If a 1 is empty or if the first letter of a 1 is not strictly smaller than every letter in a 2 , then f 1 (a) := 0. If a 1 has at least two letters and the first two of these are not both primed or unprimed, then reverse the primes on these letters and move the modified first letter of a 1 to the start of a 2 . Otherwise, move the first letter of a 1 to the start of a 2 . Our ultimate goal is to show that the larger set Incr + n (z) is likewise a normal q + n -crystal; this will eventually be stated as Corollary 7.18. In this section we only prove one part of this claim: Proposition 5.12. Assume z ∈ I Z has ℓ(µ(z)) ≤ n. Relative to the operators defined above and the weight map (5.2), the nonempty set Incr + n (z) is a q + n -crystal. For an example of the crystal Incr + n (z), see Figure 1. Our proof of Proposition 5.12 is at the end of this section, following several lemmas. The first of these is clear from the definitions: Lemma 5.13. If a ∈ Incr + n (z) and i ∈ [n − 1], then it holds that e i (unprime(a)) = unprime(e i (a)) and f i (unprime(a)) = unprime(f i (a)) under the convention that unprime(0) := 0. When n ≥ 2 the same identities hold for i = 1.
Since Incr n (z) is a q n -crystal, this lemma mostly reduces the proof of Proposition 5.12 to showing that e i and f i are well-defined maps Incr + n (z) → Incr + n (z) ⊔ {0}. This is nontrivial because these operators can change the locations of the primed letters in a factorization.
In Lemmas 5.14, 5.15, 5.16, and 5.17, we assume a and b are strictly increasing words with no primed letters such that the concatenation ab is a reduced word for some element of S Z . If x is a letter in a that is not the first term of a pair in pair(a, b), then we say that x is unpaired in a. If y is a letter in b that is not the second term of a pair in pair(a, b), then we say that y is unpaired in b.
Lemma 5.14. Suppose x is the last unpaired letter in a. Then x − 1 / ∈ b and there exists q ∈ N such that x + i ∈ a and x − 1 This result is essentially [4,Lem. 10.4], but since our notational conventions are quite different, we include a direct proof for completeness. pair(a, b), contradicting the assumption that x is unpaired. Let q ∈ N be maximal such that x + q ∈ a. It suffices to show that x + i ∈ b for 0 ≤ i < q but x + q / ∈ b. If x + 1 ∈ a, then (x + 1, y) ∈ pair(a, b) for some y ∈ b, and since x ∈ a is unpaired we must have y = x. If we also have x + 2 ∈ a, then (x + 1, y) ∈ pair(a, b) for some y ∈ b, and as x ∈ a is unpaired it must hold that y = x + 1. Repeating this argument shows that Finally suppose x+q ∈ b. The letters in ab that are between the subword x(x+1)(x+2) · · · (x+q) in a and the subword (x + 1)(x + 2) · · · (x + q) in b are each either greater than x + q + 1 or less than x − 1. Therefore ab belongs to the same commutation class as a word containing the consecutive subword x(x + 1)(x + 2) · · · (x + q)(x + 1)(x + 2) · · · (x + q). But it is straightforward to check that the latter word is not reduced. As ab is a reduced word, we must have x + q / ∈ b.
The proof of the following complementary result is similar. We omit the details.
Lemma 5.15. Suppose y is the first unpaired letter in b. Then y + 1 / ∈ a and there exists q ∈ N such that y + 1 − i ∈ a and y − i ∈ b for all i ∈ [q] while y − q / ∈ a and y − q − 1 / ∈ b.
We continue to let a and b be strictly increasing words with no primed letters.
Lemma 5.16. Assume that ab is a consecutive subword of an involution word w for some z ∈ I Z . Suppose there is an unpaired letter in a and let x be the last such letter. Let q ∈ N be maximal such that x + q ∈ a. Letâ be the subword of a formed by removing x, letb be the strictly increasing word formed by adding x + q to b, and letŵ be the word formed from w by replacing the subword ab byâb. Define i and j to be the respective indices of x ∈ a in w and x ∈b inŵ. 3 Then: (a) No index in {i + 1, i + 2, . . . , i + q} is a commutation in w.
(c) One hasŵ ∈ R inv (z), and if i is a commutation in w, then j + q is a commutation inŵ.
(d) If p ∈ [q] and j + p is a commutation in w, then i − 1 + p is a commutation inŵ.
As mentioned above, this fact implies parts (c) and (d). For parts (a) and (b), we note that for each k ∈ [q], the word obtained after applying k − 1 braid relations to (5.7) has the form This is a primed involution word by Proposition 5.4, so to avoid the patterns forbidden in Proposition 5.3 we must have α k+1 = 0 and α 1 + β k ∈ {0, 1}. As this applies to all k ∈ [q] and all primed involution words W with w = unprime(W ), we deduce that none of the indices i + 1, i + 2, . . . , i + q are commutations in w and that if i is a commutation in w, then none of j + 1, j + 2, . . . , j + q is also a commutation.
The last thing to check is that at most one of j + 1, j + 2, . . . , j + q is a commutation in w. Writing w k for the kth letter of w, define δ k (w) := s w k • · · · • s w 2 • s w 1 • 1 • s w 1 • s w 2 • · · · • s w k . Suppose q > 0 and j + k is a commutation in w for some minimal k ∈ [q]. Since we assume x = 1, we have w j+p = p for all p ∈ [q], so this means that k and k + 1 are both fixed points of δ j+k−1 (w). It is easy to check by induction that (k, p + 1) is then a cycle of δ j+p (w) for each p ∈ [k, q] and j + p is not a commutation for any p ∈ [k + 1, q]. This completes the proof of parts (a) and (b).
There is again a complementary result with a symmetric proof, whose details we omit.
Lemma 5.17. Assume that ab is a consecutive subword of an involution word w for some z ∈ I Z . Suppose there is an unpaired letter in b and let y be the first such letter. Let q ∈ N be maximal such that y − q ∈ b. Letb be the subword of b formed by removing y, letâ be the strictly increasing word formed by adding y − q to a, and letŵ be the word formed from w by replacing the subword ab byâb. Define i and j to be the respective indices of y ∈â inŵ and y ∈ b in w. 4 Then: (c) One hasŵ ∈ R inv (z), and if j is a commutation in w, then i − q is a commutation inŵ.
(d) If p ∈ [q] and i − p is a commutation in w, then j + 1 − p is a commutation inŵ.
We may now prove Proposition 5.12.
Proof of Proposition 5.12. First let i ∈ [n − 1]. Everything that needs to be checked to conclude that the operator f i from Definition 5.7 (respectively, e i from Definition 5.8) is a well-defined map Incr + n (z) → Incr + n (z) ⊔ {0} is immediate from Lemma 5.16 (respectively, Lemma 5.17). Now suppose b, c ∈ Incr + n (z). If e i (b) = c, then e i (unprime(b)) = unprime(c) by Lemma 5.13 so f i (unprime(c)) = unprime(b) since Incr n (z) is a q n -crystal. For this to hold, the last unpaired letter in unprime(c i ) must be the unique letter not also present in unprime(b i ), and given this observation it is clear from Definition 5.8 that f i (c) = b. If f i (c) = b, then it follows similarly that e i (b) = c. This confirms axiom (S1) in Definition 3.1. Since unprime : Incr + n (z) → Incr n (z) is a weight-preserving map, axiom (S2) holds by Lemma 5.13, so Incr + n (z) is a gl n -crystal. If n ≥ 2 and the words b 1 and b 2 are both nonempty, then unprime(min(b 1 )) = unprime(min(b 2 )) since otherwise unprime(concat(b)) ∈ R inv (z) would be equivalent under≡ to a word starting with XY X for some X, Y ∈ Z, contradicting Propositions 5. The other conditions in Definition 3.4 are clear, so Incr + n (z) is a q n -crystal. The axioms in Definition 3.12 are also mostly self-evident for Incr + n (z). The only relevant property that is not completely trivial from the definitions is the claim that e 0 and f 0 preserve the string lengths ε i and ϕ i for i ∈ {1, 1}, but this follows from Lemma 5.13. Thus Incr + n (z) is a q + n -crystal.

Coxeter-Knuth operators
Continue to fix an element z ∈ I Z . In this section we prove some additional facts about the crystal operators on Incr + n (z). These properties will be used in Section 7. A set of integers {j, k} is a cycle of z if j = z(j) = k. If i ∈ [m] is a commutation for w = w 1 w 2 · · · w m ∈ R inv (z), then the unordered pair γ i (w) := s wm · · · s w i+2 s w i+1 ({w i , 1 + w i }) is a cycle of z, and the map i → γ i (w) is a bijection from the set of commutations i ∈ [m] for w to the set of cycles of z. Each w ∈ R + inv (z) determines a corresponding set of marked cycles marked(w) := {γ i (w) : i ∈ [ℓ(w)] and w i ∈ Z ′ } where γ i (w) := γ i (unprime(w)).
We call ock i an (orthogonal) Coxeter-Knuth operator . This terminology comes from [32], where these operators are related to a map called orthogonal Edelman-Greene insertion; see Section 7.2.
Assume v i = w i and v i+2 = w i+2 . Then w i w i+1 w i+2 must have the form XY X, X ′ Y X, or Y XY ′ where X ∈ Z and Y = X ± 1, so ock i applied to w acts on this subword as the relation XY X ↔ Y XY or X ′ Y X ↔ Y XY ′ . Proposition 5.3 implies that γ i+1 (v) = γ i+1 (w) = ∅ and that i and i + 2 are not both commutations in v or w. To show that marked(v) = marked(w) it remains to check that γ i (v) = γ i+2 (w) and γ i+2 (v) = γ i (w), and this is straightforward. Lemmas 5.20 and 5.23 are key technical results that will be needed in Section 7.2.
Proof. We set up our notation as in Lemma 5.16 with a := unprime(w k ) and b := unprime(w k+1 ). Let x be the last unpaired letter in a and suppose q ∈ N is maximal such that x + q ∈ a. Let r be the number of letters greater than x + q in a, and let s be the number of letters less than x in b.
Write ∼ M N for the transitive closure of the relation on primed words that has . The lemma is equivalent to the claim that concat(w) ∼ M N concat(f k (w)). We will prove this by induction on r + s.
Applying ock 2q−1 · · · ock 5 ock 3 ock 1 transforms the left word to the right word, so (5.8) and (5.9) belong to the same equivalence class under ∼ as desired.
Suppose next that r > 0. Let h be the last letter of w k . As x is the last unpaired letter in a = unprime(w k ), there is a letter g > x + q in w k+1 with unprime(g) < unprime(h). Let g be the largest such letter. Define u k to be the word formed by removing h from w k , let u k+1 be the subword of w k+1 consisting of all letters less than g, and letũ k+1 be the subword of w k+1 consisting of all letters at least g so that w k+1 = u k+1ũk+1 . One can check that the tuple of increasing words u := (w 1 . . . , w k−1 , u k , u k+1 , h,ũ k+1 , w k+2 , . . . , w n ) has concat(u) ∼ M N concat(w) so u ∈ Incr + n+2 (z). Similarly define v k to be the word formed by removing h from the kth term of f k (w), and let v k+1 andṽ k+1 be the subwords of the (k + 1)th term of f k (w) consisting of all letters less than g and at least g, respectively. Then v := (w 1 . . . , w k−1 , v k , v k+1 , h,ṽ k+1 , w k+1 , . . . , w n ) has concat(v) ∼ M N concat(f k (w)) so v ∈ Incr + n+2 (z). Moreover, it is easy to see that f k (u) = v, so by induction on r + s we have concat(u) ∼ MÑ concat(v) forÑ := N − ℓ(ũ k+1 )− 1 = N − ℓ(ṽ k+1 )− 1. This means that concat(w) ∼ M N concat(u) ∼ M N concat(v) ∼ M N concat(f k (w)).
Suppose instead that s > 0. Our argument is similar to the previous case. Let g be the first letter of w k+1 . Then there is a smallest letter h ≤ x− 1 in w k with unprime(g) < unprime(h). Define u k+1 to be the word formed by removing g from w k+1 , and let u k andũ k be the subwords of w k consisting of all letters at most h and greater than h, respectively, so that w k = u kũk . Then u := (w 1 . . . , w k−1 , u k , g,ũ k , u k+1 , w k+2 , . . . , w n ) has concat(u) ∼ M N concat(w) so u ∈ Incr + n+2 (z). Similarly define v k to be the word formed by removing g from the (k + 1)th term of f k (w), and let v k andṽ k be the subwords of the kth term of f k (w) consisting of all letters at most h and greater than h, respectively. Then v := (w 1 . . . , w k−1 , v k , g,ṽ k , v k+1 , w k+1 , . . . , w n ) likewise has concat(v) ∼ M N concat(f k (w)) so v ∈ Incr + n+2 (z). We again have f k (u) = v, so by induction concat(u) ∼M N concat(v) forM := M + ℓ(u k ) + 1 = N + ℓ(u k ) + 1, and this implies that We conclude by induction on r + s that concat(w) ∼ M N concat(f k (w)) in all cases.
The following is clear from Lemmas 5.19 and 5.20. Recall that the subset Incr n (z) ⊂ Incr + n (z) defined in (5.4) is q n -subcrystal.
This map is not usually a quasi-isomorphism of q n -crystals.
Proof. If S is any set of 2-cycles of z, then C S := {a ∈ Incr + n (z) : marked(a) = S} is a gl n -subcrystal by Corollary 5.21, and unprime : C S → Incr n (z) is a gl n -crystal isomorphism by Lemmas 5.13 and 5.18. As Incr + n (z) is a disjoint union of such gl n -subcrystals C S , the result follows.
Proof. If q = 0, then f 1 (a) = 0 since a 1 is empty. Suppose q = 1 so that w = concat(a). If wt(a) 2 = 0 and 1 ∈ Des(w), then the first and only letter in a 1 is strictly larger than the first letter of a 2 , so f 1 (a) = 0. If wt(a) 2 = 0 or 1 / ∈ Des(w), then f 1 (a) is formed from a by moving the only letter in a 1 to the beginning of a 2 , so concat(f 1 (a)) = concat(a) = w as claimed.
Now assume q ≥ 2 so that w = ock q−2 · · · ock 1 ock 0 (concat(a)). Write v i for the ith letter of a 1 . If v 1 and v 2 are both primed or both unprimed, then letṽ 1 := v 1 andṽ 2 := v 2 , and otherwise form ṽ 1 andṽ 2 by reversing the primes on v 1 and v 2 , respectively. Applying ock 0 to concat(a) replaces the first two letters v 1 v 2 byṽ 2ṽ1 . Each successive application of ock j for j = 1, 2, . . . , q − 2 then transposesṽ 1 and the letter to its right. Thus w is formed from concat(a) by changing the first q Since w is a primed involution word, it follows from Proposition 5.3 that ⌈v 1 ⌉ = ⌈ṽ 1 ⌉ cannot be equal to the first letter of unprime(a 2 a 3 · · · a n ). Thus if a 2 is nonempty, then q belongs to Des(w) precisely when the first letter of a 1 is not strictly smaller than every letter in a 2 . Therefore if wt(a) 2 = 0 and q ∈ Des(w), then f 1 (a) = 0 as claimed. If instead we have wt(a) 2 = 0 or q / ∈ Des(w), then f 1 (a) is formed from a by changing a 1 = v 1 v 2 v 3 · · · v q toṽ 2 v 3 · · · v q and then addingṽ 1 to the start of a 2 . Comparing this to the description of w above shows that concat(f 1 (a)) = w.

Crystal operators on shifted tableaux
Continue to let n be a positive integer. Recall that if λ is a strict partition, then ShTab + n (λ) is the set of semistandard shifted tableaux of shape λ with all entries at most n, and ShTab n (λ) is the subset of such tableaux with no primed diagonal entries. Results in [2,17,19] describe a q n -crystal structure on ShTab n (λ). Here, we extend this to a q + n -crystal on the larger set ShTab + n (λ).

Skew shifted tableaux
If λ and µ are strict partitions with SD µ ⊂ SD λ , then we write µ ⊂ λ and set SD λ/µ := SD λ \ SD µ . A (semistandard) skew shifted tableau of shape λ/µ is a map SD λ/µ → {1 ′ < 1 < 2 ′ < 2 < . . . } such that entries are weakly increasing along rows and columns, with no unprimed entries repeated in a column and no primed entries repeated in a row. If T is a skew shifted tableau and i ≤ j are positive integers, then the set of positions T −1 ({i ′ < i < · · · < j ′ < j}) is equal to SD λ/µ for some strict partitions µ ⊂ λ. We write T | [i,j] for the skew shifted tableau obtained by restricting T to this subdomain.
A skew shifted tableau is a rim if its domain has no positions (i 1 , j 1 ), (i 2 , j 2 ) with i 1 < i 2 and j 1 < j 2 . A rim whose domain is connected is a ribbon; in French notation, this appears as or some analogous shape. If T is a skew shifted tableau, then the subtableau T | [i,i] is always a rim, and therefore a disjoint union of ribbons, which we call the i-ribbons of T .
Let T be a skew shifted tableau whose domain fits inside [k] × [k] for some positive integer k. If C i is the sequence of primed entries in column i of T , read in order, and R i is the sequence of unprimed entries in row i of T , read in order, then the shifted reading word of T is This does not depend on k. For example, if , then shword(T ) = 845237169. The order of the letters in shword(T ) defines a total order on the boxes of T which we call the shifted reading word order . In the preceding example, this order is (1, 5) < (1, 3) < (2, 3) < (1, 2) < (2, 2) < (2, 4) < (1, 1) < (1, 4) < (1, 6).
The shifted reading word is the same as the hook reading word appearing in [2,Def. 3.4.] and in earlier literature, but different from the reading wording of a shifted tableau defined in [17, §4]. Note that toggling the primes on the diagonal of T has no effect on shword(T ).
Our descriptions of f i (T ) and e i (T ) for i > 0 are closely modeled on [2, Defs. 3.5 and 3.9]. Below, when we refer to "interchanging the primes" on two elements of Z ⊔ Z ′ = 1 2 Z, we mean the operation that adds a prime to one number while removing the prime from the other if the two are not both primed or both unprimed, and otherwise leaves the numbers unchanged. (a) If T x,y+1 = i + 1 ′ , then form f i (T ) by changing T xy to i + 1 ′ and T x,y+1 to i + 1: , then form f i (T ) by changing T xy to i + 1: Txy T x,y+1 The conjugate [17, §4] of a skew shifted tableau is given by transposing the locations of all boxes and then adding 1 2 to all entries. If T xy = i ′ then f i (T ) is formed by first conjugating T , then following the rules in cases L1(a)(b)(c) above (note that case L1(d) will not apply), and then applying the inverse of conjugation to the result. This corresponds explicitly to the following operations: Then we form f i (T ) from T as follows: (a) If T x+1,y = i, then form f i (T ) by changing T xy to i and T x+1,y to i + 1 ′ : (b) If T x+1,y = i and T x,y+1 / ∈ {i, i + 1 ′ }, then form f i (T ) by changing T xy to i + 1 ′ : (c) If T x+1,y = i and T x,y+1 ∈ {i, i + 1 ′ }, then form f i (T ) by changing T xy to i and Txỹ to i + 1 ′ , where (x,ỹ) is the first position in the i-ribbon containing (x, y) that is southeast of (x, y) with Txỹ = i and Tx ,ỹ+1 / ∈ {i, i + 1 ′ }:  Properties (i), (ii), and (iii) must also hold when T has primed diagonal entries, since any counterexample would remain so on removing all primes from the diagonal. The same reasoning applies to property (iv) since (v) in case L2(c) it always holds that x = y.
This follows since if x = y in case L2(c), then we must have T x,y+1 = i + 1 ′ for (x, y) to be the last position in unpaired i (T ) with T xy ∈ {i ′ , i}, but then removing the prime from T xy would produce a counterexample to (i). Finally, it is easy to check that (vi) in case L1(d) it must hold that Tx −1,x−1 ∈ {i ′ , i} for (x, y) to be in unpaired i (T ).
Let primes(T ) denote the total number of primed entries in T and let primes diag (T ) denote the number of primed diagonal entries in T . Remark 6.3 implies the following lemma. Next, we define raising operators e i . Recall that i > 0 and T is a skew shifted tableau. Definition 6.5. Consider the positions (x, y) in unpaired i (T ) with T xy ∈ {i + 1 ′ , i + 1}. If there are no such positions, then set e i (T ) = 0. Otherwise, let (x, y) be the first such position and construct a skew shifted tableau tableau e i (T ) with the same shape as T by the following procedure: (R1) First assume T xy = i + 1. Then we form e i (T ) from T as follows: (a) If T x,y−1 = i + 1 ′ , then form e i (T ) by changing T xy to i + 1 ′ and T x,y−1 to i: then form e i (T ) by changing T xy to i: (c) If T x,y−1 = i + 1 ′ and T x−1,y ∈ {i, i + 1 ′ }, then form e i (T ) by changing T xy to i + 1 ′ and Txỹ to i, where (x,ỹ) is the first position in the (i + 1)-ribbon containing (x, y) that is southeast of (x, y) with Txỹ = i + 1 ′ and Tx −1,ỹ / ∈ {i, i + 1 ′ }: (R2) Alternatively suppose T xy = i + 1 ′ . Then we form e i (T ) from T as follows: (a) If T x−1,y = i, then form e i (T ) by changing T xy to i and T x−1,y to i ′ : (b) If T x−1,y = i and T x,y−1 / ∈ {i ′ , i}, then form e i (T ) by changing T xy to i + 1 ′ :  These hold since it is a straightforward exercise to show that any counterexample leads to contradiction of the fact that (x, y) is the first unpaired position with T xy ∈ {i + 1 ′ , i + 1}. Next: (iv) in case R1(c) it always holds that x = y.
This follows since if x = y in case R1(c), then T x,y+1 = i as (x, y) is the last position in unpaired i (T ) with T xy ∈ {i + 1 ′ , i + 1}, but then adding a prime to T xy would give a counterexample to (i). Also: (v) in case R1(c) there always exists a position (x,ỹ) as described.
This holds since any counterexample would remain so on removing all primes from the diagonal, contradicting the fact that the raising operators in [2,Def. 3.9] are well-defined. Alternatively, (v) can be shown by mimicking the last paragraph of the proof of [2,Thm. 3.8]. Finally: (vi) in case R2(d) it must hold that Tx +1,x+1 ∈ {i + 1 ′ , i + 1} for (x, y) to be in unpaired i (T ).
An analogue of Lemma 6.4 follows from this remark. Lemma 6.7. Suppose e i (T ) = 0. Then primes diag (T ) = primes diag (e i (T )). If case R2(d) applies in Definition 6.5, then primes(e i (T )) = primes(T ) − 1. In all other cases primes(e i (T )) = primes(T ) and the sets of the primed diagonal positions in e i (T ) and T coincide.
To define the remaining operators f 1 , e 1 , f 0 , and e 0 , we require that T be a semistandard shifted tableau (rather than a skew shifted tableau). When T has no primed diagonal entries, the next two definitions reduce to the formulas in Definition 6.8. If T has a 2 ′ in its first row, or no entries equal to 1 ′ or 1, then set f 1 (T ) := 0: Otherwise form f 1 (T ) by changing the last entry equal to 1 ′ or 1 in the first row of T to 2 ′ , unless this entry is unprimed and on the diagonal, in which case it changes to 2: , and f 1 Definition 6.9. If T 11 ∈ {2 ′ , 2}, then form e 1 (T ) from T by subtracting one from this entry: Given a skew shifted tableau T , form unprime diag (T ) by removing the primes from all diagonal entries. This operation commutes with the maps e i and f i in the following sense. Lemma 6.11. Let T be a skew shifted tableau of shape λ/µ and suppose i is a positive integer. Then e i (unprime diag (T )) = unprime diag (e i (T )) and f i (unprime diag (T )) = unprime diag (f i (T )) on setting unprime diag (0) := 0. When µ = ∅ the same identities hold for i = 1.
Proof. The desired identities are easily checked when µ = ∅ and i = 1. Assume i ∈ [n − 1]. Since T and unprime diag (T ) have the same shifted reading word, if follows that f i (T ) = 0 if and only if f i (unprime diag (T )) = 0. Assume this does not occur, and let (x, y) be the unpaired position that arises in Definition 6.2 when applying f i to T . This (x, y) must also be the unpaired position that arises in Definition 6.2 when evaluating f i (unprime diag (T )).
The properties in Remark 6.3 ensure that whichever case of Definition 6.2 applies when evaluating f i (T ), the same case applies when evaluating f i (unprime diag (T )), with one exception. Outside this exception, it is evident that f i (unprime diag (T )) = unprime diag (f i (T )). The exception is that if x = y and case L2(b) of Definition 6.2 is invoked when applying f i to T , then case L1(b) applies when evaluating f i (unprime diag (T )). However, in this situation it is clear from Definition 6.2 that we again have f i (unprime diag (T )) = unprime diag (f i (T )).
Checking that e i (unprime diag (T )) = unprime diag (e i (T )) follows by a similar argument.
Choose strict partitions µ ⊂ λ and let ShTab + n (λ/µ) be the set of (semistandard) skew shifted tableaux of shape λ/µ with all entries at most n. Let ShTab n (λ/µ) be the subset of tableaux in ShTab + n (λ/µ) with no primed diagonal entries. The weight of T ∈ ShTab + n (λ/µ) is the vector wt(T ) := (a 1 , a 2 , . . . , a n ) where a i is the number of entries in T equal to i ′ or i.
Restricted to ShTab n (λ/µ), the operators e i and f i for i ∈ {1, 1, 2, . . . , n − 1} coincide with the ones in [2,17,19]. From those papers (see in particular [2, Thm 4.8]), we know that if λ ∈ N n is any strict partition, then these operators make ShTab n (λ) into a connected normal q n -crystal. Corollary 6.12. When nonempty, the set ShTab n (λ/µ) is a normal (but not necessarily connected) gl n -crystal relative to the operators e i and f i defined above.
Proof. Let k := ℓ(µ). If B is a gl k+n -crystal with weight mapwt and crystal operatorsẽ i andf i then B may be regarded as a gl n -crystal with weight map wt(b) := (wt(b) k+1 , . . . ,wt(b) k+n ) ∈ Z n and crystal operators e i :=ẽ k+i and f i :=f k+i for i ∈ [n − 1]. This gives a functor F from gl k+n -crystals to gl n -crystals. It is straightforward to see that F (B ⊗ C) ∼ = F (B) ⊗ F (C) and F (B k+n ) ∼ = ½ ⊔ · · · ⊔ ½ ⊔ B n , so this functor sends normal gl k+n -crystals to normal gl n -crystals. Now consider the subset of T ∈ ShTab k+n (λ) that have i in all boxes in row i of SD µ ⊂ SD λ and only entries greater than k in SD λ/µ . This is a union of full subcrystals of the normal gl n -crystal F (ShTab k+n (λ)), and is isomorphic to the prospective crystal structure on ShTab n (λ/µ). Theorem 7.16 and Corollary 7.17 will give q + n -analogues of the above facts for sets of shifted tableaux with primes allowed on the diagonal. Here, we only prove this easier statement: Proposition 6.13. When nonempty, the set ShTab + n (λ) (respectively, ShTab + n (λ/µ)) is a q + n -crystal (respectively, gl n -crystal) relative to the operators e i and f i defined above.
The set ShTab + n (λ) is empty if and only if ℓ(λ) > n. For an example of this crystal see Figure 2.
If primes(f i (T )) = primes(T ), then Lemma 6.4 tells us that applying f i to T must not invoke case L1(d) in Definition 6.2 while Lemma 6.7 tells us that applying e i to f i (T ) must not invoke case R2(d) in Definition 6.5. But this means that the sets of primed diagonal positions in T , f i (T ), and e i (f i (T )) all coincide, so we have e i (f i (T )) = T in view of (6.1).
If primes(f i (T )) = primes(T ), on the other hand, then Lemmas 6.4 and 6.7 imply that we must be in case L1(d) of Definition 6.2 when applying f i to T and in case L2(d) of Definition 6.5 when applying e i to f i (T ). The only way that (6.1) can hold in this situation is if the unpaired position (x, y) arising in both definitions is the same, but then we again have e i (f i (T )) = T .
When e i (T ) = 0 a symmetric argument shows that f i (e i (T )) = T . Since ShTab n (λ/µ) is a gl n -crystal and unprime diag is a weight-preserving map, this suffices by Lemma 6.11 to show that ShTab + n (λ/µ) is a gl n -crystal. Taking µ = ∅, we conclude that ShTab + n (λ) is also a gl n -crystal. Checking the remaining axioms to show that ShTab + n (λ) is a q + n -crystal is straightforward.

Highest and lowest weights
Let λ ∈ N n be a strict partition. Results in [19] identify the unique q n -highest and q n -lowest weight elements in the connected normal q n -crystal ShTab n (λ). Define T highest λ to be the shifted tableau of shape λ whose entries in row i are all i. Define T lowest λ to be the unique shifted tableau of shape λ with no primed diagonal positions whose entries along the ribbon SD (λ i+1 ,λ i+2 ,λ i+3 ,... )/(λ i+2 ,λ i+3 ,... ) are each (n − i) ′ or n − i, for i = 0, 1, 2, . . . , n − 1. The latter construction depends on n, although we suppress this in our notation. If n = 5, then . Theorem 6.14 ([19, Thm. 3.3]). The q n -crystal ShTab n (λ) is connected with unique q n -highest weight element T highest λ and unique q n -lowest weight element T lowest λ . Lemma 6.11 shows that unprime diag : ShTab + n (λ) → ShTab n (λ) is a weight-preserving map that commutes with all crystal operators, excluding e 0 and f 0 . It follows that unprime diag commutes with the involutions σ i for all i ∈ [n − 1] in (3.2), and hence also with the operators e i , f i , e i ′ , and f i ′ for all i ∈ [n − 1] used in Definition 3.9.
Since unprime diag (T ) = 0 if T = 0, the map unprime diag must send q + n -highest and q + n -lowest weight elements in ShTab + n (λ) to q n -highest and q n -lowest weight elements in ShTab n (λ). Consulting Definition 3.19, we deduce that T ∈ ShTab + n (λ) is a q   (a) If α i = 0, then σ i (U ) is formed by subtracting 1 from every entry of U equal to i + 1 ′ or i + 1.
(b) If α i+1 = 0, then σ i (U ) is formed by adding 1 to every entry of U equal to i ′ or i.
(U ). The shifted tableau U has no boxes containing i or i ′ and a positive number of boxes containing i + 1 or i + 1 ′ , which we call changeable boxes. Consider the shifted reading word order restricted to the changeable boxes of U . We claim that for each integer 0 ≤ j ≤ α i+1 the tableau e j i (U ) is formed from U by subtracting 1 from the entries in the first j changeable boxes in this order.
This certainly holds when j = 0. Now suppose the claim is true for some integer 0 ≤ j < α i+1 and let T := e j i (U ). By hypothesis all letters equal to i + 1 in shword(T ) occur after all letters equal to i so the pairing in Definition 6.1 is trivial. Therefore when applying e i to T , the unpaired position (x, y) in Definition 6.5 is the (j + 1)th changeable box and we have T xy = U xy ∈ {i + 1 ′ , i + 1}.
Consider the case of Definition 6.5 that applies to compute e i (T ) = e j+1 i (U ). If T xy = i+1 ′ , then it is impossible to be in case R2(a) or R2(c), since then an i would occur after i + 1 in shword(T ). Similarly if T xy = i + 1, then it is impossible to be in case R1(a) or R1(c), since by hypothesis (x, y) contributes the first letter equal to i + 1 or i + 1 ′ in shword(T ). Thus we must be in case R1(b) or R2(b), so e i (T ) is formed from T by subtracting one from T xy . This means that e j+1 i (U ) = e i (T ) is formed from U by subtracting one from the entries in the first j + 1 changeable boxes, which proves our claim by induction. Invoking the claim with j = α i+1 proves part (a) of the lemma.
. Then σ i (T ) is formed from T by interchanging the primes on the entries in boxes (i, i) and (i + 1, i + 1) and changing the last λ i − λ i+1 boxes containing i in row i from For example, σ 1 2 ′ 2 1 1 1 1 = 2 2 1 ′ 1 2 ′ 2 and σ 1 Proof. By hypothesis the integer k : where there are two possibilities for each of the diagonal entries i * ∈ {i ′ , i} and i+1 * ∈ {i+1 ′ , i+1}. Therefore the sequence unpaired i (T ) from Definition 6.1 consists of the last k boxes containing i in row i. Each time we apply f i to f j−1 i (T ) for j = 1, 2, . . . , k − 1, case L1(b) in Definition 6.2 changes the jth of these boxes, ordered right to left, from i to i + 1. When we finally apply f i to f k−1 i (T ), case L1(d) in Definition 6.2 results in the next box changing from i to i + 1 ′ and the primes on the entries in boxes (i, i) and (i + 1, i + 1) being interchanged. Thus σ i (T ) is as described. . To prove parts (a) and (b) it suffices to check that e 0 (respectively, f 0 ) transforms σ 1 σ 2 · · · σ i−1 (T ) to σ 1 σ 2 · · · σ i−1 (U ) when T ii is primed (respectively, unprimed). This is straightforward using Lemma 6.16. In part (c) we have (j, j) / ∈ T , so Lemma 6.15 implies that σ 1 σ 2 · · · σ j−1 (T ) is formed from T by adding 1 to all of its entries, so σ 1 σ 2 · · · σ j−1 (T ) has no boxes containing 1 ′ or 1 and therefore e Then σ i (T ) is formed from T by interchanging the primes on the entries in boxes (j, j) and (j + 1, j + 1) and changing all entries equal to i + 1 ′ or i + 1 in the first row to i. which have all entries in {i ′ , i} and {i + 1 ′ , i + 1}, respectively. The integer k := ϕ i (T ) − ε i (T ) = wt(T ) i − wt(T ) i+1 is therefore equal to λ n−i+1 − λ n−i < 0 so σ i (T ) = e −k i (T ). Compared to the proof of Lemma 6.16, it is less trivial but still straightforward to see that the sequence unpaired i (T ) from Definition 6.1 consists of all of the boxes with entries in {i + 1 ′ , i + 1} in the first row of T . There are −k such boxes, only the first of which (going left to right) has a primed entry. Applying e i to T invokes case R2(d) in Definition 6.5, causing this primed entry to change to i and the primes on the entries in boxes (j, j) and (j + 1, j + 1) to be interchanged. Successively applying e i to e i (T ), 1 − k additional times, changes the entries in the remaining unpaired boxes one by one from i + 1 to i via case R1(b) in Definition 6.5. Thus σ i (T ) is as described. Proof. The proof is similar to the one given for Lemma 6.17, just using Lemmas 6.15 and 6.18 in place of Lemma 6.16. We omit the details. Theorem 6.20. The q + n -crystal ShTab + n (λ) is connected with unique q + n -highest and q + n -lowest weight elements given by T highest λ andT lowest λ respectively.
Proof. It follows from (6.2) and Lemma 6.17 that T highest λ is the only element of ShTab + n (λ) that could be a q + n -highest weight. Since e is the unique q + n -highest weight element and ShTab + n (λ) is connected. A similar argument using (6.3) and Lemma 6.19 shows thatT lowest λ is the unique q + n -lowest weight vector in ShTab + n (λ).

Dual equivalence operators
Suppose T is standard shifted tableau with n boxes. 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.
Thus we would have The standardization of a semistandard shifted tableau T is given as follows. List the boxes of T in the order such that one box comes before another if its entry is weakly smaller and the letter it contributes to shword(T ) appears first going left to right. Then form standardize(T ) from T by changing the entry in the ith box to i ′ if primed and to i otherwise. For example, standardize      If q = 0 or if wt(T ) 2 = 0 and q ∈ Des(U ), then f 1 (T ) = 0, and otherwise standardize(f 1 (T )) = U .
Suppose q ≥ 2 so that U = d q−2 · · · d 1 d 0 (standardize(T )). The shifted tableau T contains at most one entry equal to 1 ′ , which can only appear in the (1, 1) position, and all 1's in T appear in the first row. Therefore shword(standardize(T )) contains 12345 . . . q as a consecutive subword. Applying d 0 to standardize(T ) changes the unique entry 2 to 2 ′ , so the shifted reading word of d 0 (standardize(T )) contains 21345 . . . q as a (not necessarily consecutive) subword. It follows that d 1 applied to d 0 (standardize(T )) acts as s 2 , which reverses the primes on entries 2 ′ and 3. The shifted reading word of d 1 d 0 (standardize(T )) therefore contains 31245 . . . q as a subword, so d 2 applied to d 1 d 0 (standardize(T )) acts as s 3 , which reverses the primes on entries 3 ′ and 4. Continuing in this way, we deduce that U is formed from standardize(T ) by simply adding a prime to entry q, which is contained in the off-diagonal position (1, q). Now we are ready to prove the last part of the lemma. If wt(T ) 2 = 0, then f 1 (T ) is formed from T by changing entry T 1q from 1 to 2 ′ in which case we have standardize(f 1 (T )) = U as claimed. Assume wt(T ) 2 = 0. Then q ∈ Des(U ) if and only if q + 1 appears before q in shword(U ), which occurs if and only if the first row of T contains an entry equal to 2 ′ , which would have to occur in position (1, q + 1). Thus if q ∈ Des(U ), then f 1 (T ) = 0 by Definition 6.8, while if q / ∈ Des(U ), then applying f 1 to T again changes entry T 1q from 1 to 2 ′ , in which case standardize(f 1 (T )) = U .

Crystal morphisms
Continue to fix a positive integer n. Below, we describe several morphisms between the families of q + n -crystals introduced above. Specifically, we explain how each crystal of words W + n (m) ∼ = (B + n ) ⊗m may be embedded in a crystal of factorizations Incr + n (z) and how each crystal Incr + n (z) may be embedded in a union of shifted tableau crystals ShTab + n (λ). This will allow us to prove Theorem 1.5 from the introduction and to show that Incr + n (z) and ShTab + n (λ) are always normal q + n -crystals.

From words to increasing factorizations
Let p ∈ Z. Then R + inv (s p ) = {p ′ , p} where s p = (p, p + 1) ∈ S Z . The following is an easy exercise: Proposition 7.1. The standard q + n -crystal B + n is isomorphic to Incr + n (s p ) via the map that sends i → (∅, . . . , ∅, p, ∅, . . . , ∅) and i ′ → (∅, . . . , ∅, p ′ , ∅, . . . , ∅) where in both n-tuples all but the ith terms are empty words. Proposition 7.2. The q + n -crystal Incr + n (y) ⊗ Incr + n (z) is isomorphic to Incr + n (y ⊕ z) via the map a ⊗ b → (a 1 b 1 , a 2 b 2 , · · · a n b n ) where b i is the word formed by adding M to every letter of b i .
Proof. Denote the given map Incr + n (y) ⊗ Incr + n (z) → Incr + n (y ⊕ z) by Φ. Verifying that Φ is a weightpreserving bijection is a standard exercise using the discussion in Section 5.1. One can check that Φ commutes with the operators e 1 , f 1 , e 0 , and f 0 by inspecting the relevant formulas in Theorem 3.14 and Section 5.2.
Fix i ∈ [n − 1]. It remains to show that Φ commutes with e i and f i . We will just demonstrate that Φ • f i = f i • Φ since the argument for e i is similar. Suppose a ∈ Incr + n (y) and b ∈ Incr + n (z). First consider the unpaired letters in a i and a i+1 relative to pair(a i , a i+1 ) as well as the unpaired letters in b i and b i+1 relative to pair(b i , b i+1 ) as given in Definition 5.6. Notice that the value of ϕ i (a) is the number of unpaired letters in a i while the value of ε i (a) is the number of unpaired letters in a i+1 . A similar description applies to ϕ i (b) and ε i (b).
We now turn to the unpaired letters in a i b i and a i+1 b i+1 relative to pair(a i b i , a i+1 b i+1 ). Any letters in a i+1 that were unpaired in the (a i , a i+1 )-pairing are now matched with letters in b i that arise as shifts of unpaired letters in the , then the last unpaired letter x ∈ a i b i is just M plus the last unpaired letter in b i . If there is no such unpaired letter, then Φ(f i (a ⊗ b)) = f i (Φ(a ⊗ b)) = 0. Otherwise, applying f i to Φ(a ⊗ b) removes x from a i b i and adds a new letter y ≥ x to a i+1 b i+1 , and it is easy to see that the result is the same as applying Φ to , then the last unpaired letter in a i b i is the same as the last unpaired letter in a i , and applying f i to Φ(a ⊗ b) gives the same result as applying Φ to Corollary 7.3. There is a q + n -crystal isomorphism W + n (m) ∼ = Incr + n (s 2 s 4 s 6 · · · s 2m ). Thus each connected normal q + n -crystal is isomorphic to a full subcrystal of Incr + n (z) for some z ∈ I Z .
Proof. The first claim follows by induction from Proposition 7.2, taking y = s 2 s 4 · · · s 2m−2 , z = s 1 , M = 2m − 1, and N = 2, with the m = 1 base case provided by Proposition 7.1. The second claim follows from the first claim since (B + n ) ⊗m ∼ = W + n (m) for all m ∈ N.

From increasing factorizations to shifted tableaux
Fix an involution z ∈ I Z . We now wish to relate the q + n -crystals Incr + n (z) and ShTab + n (λ). We will do this by making use of a correspondence between increasing factorizations and pairs of shifted tableaux described in [32, §3]. Recall that if i ∈ Z, then i ′ := i − 1 2 so ⌈i ′ ⌉ = ⌈i⌉ = i.
. Suppose a ∈ Incr + n (z) and w = w 1 w 2 · · · w m = concat(a). Let ∅ = T 0 , T 1 , . . . , T m be the sequence of shifted tableaux in which T i for i ∈ [m] is formed by inserting w i into T i−1 according to the following procedure: 1. Start by inserting w i into the first row. At each stage, an entry x is inserted into a row or column. Let y andỹ be the first entries in the row or column with ⌈x⌉ ≤ ⌈y⌉ and ⌈x⌉ < ⌈ỹ⌉.
2. If no such entries y andỹ exist, then x is added to the end of the row or column, with the exception that if x is added to the main diagonal, then its value is changed to ⌈x⌉. We say the process to form T i ends in column insertion if we are inserting into a column at this stage or if ⌈x⌉ = x is added to the main diagonal. Otherwise, the process ends in row insertion.
3. If y andỹ are distinct, then the primes on these entries are interchanged and x + 1 is inserted into the next row (if we were inserting into a row and y is not on the main diagonal) or the next column (if we were inserting into a column or y is on the main diagonal).
4. If y =ỹ is off the main diagonal, then x replaces y and we insert y into the next row (if we were inserting into a row) or the next column (if we were inserting into a column). If y =ỹ is on the main diagonal, then ⌈x⌉ replaces y and we insert y − (⌈x⌉ − x) into the next column.
Finally define P O EG (a) := T m and construct Q O EG (a) as the shifted tableau with the same shape as P O EG (a) that contains j (respectively, j ′ ) in the box added to T i−1 to form T i if w i is in the jth factor of a and the insertion process ends in row insertion (respectively, column insertion).
We make no distinction between w 1 w 2 · · · w n and the sequence of 1-letter words (w 1 , w 2 , . . . , w n ). This lets us view each w ∈ R + inv (z) as an element of Incr + n (z) for n = ℓ(w) so we can evaluate P O EG (w) and The map a → (P O EG (a), Q O EG (a)) is called orthogonal Edelman-Greene insertion in [32]. It is a shifted version of the Edelman-Greene correspondence from [7], as well as the "orthogonal" counterpart to a "symplectic" insertion algorithm studied in [20,30,31]. Restricted to the subset R inv (z) R + inv (z), the map is a special case of shifted Hecke insertion from [35]. A (shifted) tableau is increasing if its rows and columns are strictly increasing. The row reading word row(T ) of a (shifted) tableau T is formed by reading its rows from left to right, but starting with the top row in French notation.
One important feature of the map a → (P O EG (a), Q O EG (a)) is that it is a bijection from Incr + n (z) to the set of pairs (P, Q) of shifted tableaux of the same shape, in which Q is semistandard with all entries at most n, and P is an increasing with no primes on the main diagonal and row(P ) ∈ R + inv (z) [32,Thm. 3.15]. For our applications, we need a few other technical properties from [32]: Example 7.7. Let a = (4, 1 ′ 35, ∅, 4 ′ , ∅, 2) and w = concat(a) = 41 ′ 354 ′ 2 as in Example 7.5. Then a ∈ Incr + n (z) and w ∈ R + inv (z) for z = (1, 3)(2, 6)(4, 5) ∈ I Z and n = 6. Most of the subtlety of orthogonal Edelman-Greene insertion has to do with the distribution of primed entries in the output tableaux. When primes are disallowed, we already have some strong results that relate this algorithm to the relevant q n -crystal structures: 31,Thm. 3.32]). The map a → Q O EG (a) is a quasi-isomorphism of q n -crystals z∈I Z Incr n (z) → strict partitions λ∈N n ShTab n (λ), and the full q n -subcrystals of z∈I Z Incr n (z) are the subsets on which a → P O EG (a) is constant. We will show that this statement extends to q + n -crystals. The proof requires a lemma.

Proof. Let
Define U := d j l · · · d j 2 d j 1 (standardize(T )). We argue that standardize(f k (T )) = U . Applying Q O EG to the left side of the previous displayed equation gives ) by Lemma 7.6(e), = standardize(f k (T )) by Theorem 7.8, = standardize(unprime diag (f k (T ))) by Lemma 6.11, = unprime diag (standardize(f k (T ))) by definition, while applying Q O EG to the right side gives Q O EG (ock j l · · · ock j 2 ock j 1 (concat(a))) = d j l · · · d j 2 d j 1 (standardize(T )) = unprime diag (U ) by parts (e) and (f) of Lemma 7.6 and part (a) of Proposition 6.21. Thus unprime diag (standardize(f k (T ))) = unprime diag (U ), so to prove that standardize(f k (T )) = U it suffices to show that standardize(f k (T )) and U share the same set of primed diagonal positions. Since the same positions in f k (T ) and standardize(f k (T )) have primed entries, it is enough to check f k (T ) and U have the same primed diagonal positions. For each j ∈ [n] let j be the unique position containing j or j ′ in standardize(T ). Then the domain of the skew shifted tableau T | [i,i+1] , which is a union of two rims, consists of precisely the boxes M , M +1 , . . . , N . These boxes therefore contain at most two diagonal positions, which must occur in consecutive rows.
Suppose there are less than two diagonal positions among M , M +1 , . . . , N . Then it is clear from Definition 6.2 that f k (T ) has the same set of primed diagonal positions as T , and it follows from Proposition 6.21 that U has the same set of primed diagonal positions as standardize(T ). As standardize(T ) and T have identical sets of primed positions, the same diagonal positions in f k (T ) and U are primed as desired. This reasoning also applies if M , M +1 , . . . , N include two diagonal positions but the entries of T in these positions are both primed or both unprimed.
The case left to consider is the following: assume M , M +1 , . . . , N involve exactly two diagonal positions, say in rows r − 1 and r, and exactly one of these positions has a primed entry in T . Let D(T ) := {(i, i) ∈ T : T ii ∈ Z ′ } be the set of primed diagonal positions in T and set S := {(r − 1, r − 1), (r, r)}. Lemma 6.4 implies that where △ denotes the symmetric set difference. On the other hand, it follows from Proposition 6.21 that as we apply d j for j = j 1 , j 2 , . . . , j l successively to standardize(T ), only the entries in positions M , M +1 , . . . , N vary. In particular, the parity of the number of primed positions changes precisely when j or j ′ appears in box (r − 1, r − 1) and j + 2 or j + 2 ′ appears in box (r, r). Proposition 6.21(e) tells us that if this happens, then applying d j interchanges the primes on these diagonal boxes, and otherwise the set of primed diagonal positions is unchanged by d j . Thus D(U ) = D(standardize(T )) △ S if primes(U ) ≡ primes(standardize(T )) (mod 2), D(standardize(T )) otherwise.
Theorem 7.10. The map a → Q O EG (a) is a quasi-isomorphism of q + n -crystals z∈I Z Incr + n (z) → strict partitions λ∈N n ShTab + n (λ), and the full q + n -subcrystals of z∈I Z Incr + n (z) are the subsets on which a → P O EG (a) is constant.
This statement is a strict generalization of Theorem 7.8. Although Incr + n (z) and ShTab + n (λ), viewed as q n -crystals, contain Incr n (z) and ShTab n (λ) as subcrystals, the sets Incr + n (z) \ Incr n (z) and ShTab + n (λ) \ ShTab n (λ) are not normal as q n -crystals. Therefore even the weaker statement that Q O EG gives a q n -crystal morphism in Theorem 7.10 cannot be deduced from Theorem 7.8.
Proof. Let z ∈ I Z and a = (a 1 , a 2 , . . . , a n ) ∈ Incr + n (z). Property (b) in Lemma 7. We conclude that unprime(f i (a)) = unprime(b) since • → (P O EG (•), Q O EG (•)) is a bijection. Now to prove that f i (a) = b, it is enough by Lemma 5.18 to show that marked(f i (a)) = marked(b), i.e., that f i (a) and b have the same set of marked cycles as defined at the start of Section 5.3.
Thus w = ock j l · · · ock j 2 ock j 1 (v), so by Lemma 5.19 and Corollary 5. If λ is the shape of P , then Q O EG defines a weight-preserving bijection {b ∈ Incr + n (z) : P O EG (b) = P } → ShTab + n (λ) that commutes with all crystal operators. Since ShTab + n (λ) is a connected q + n -crystal by Theorem 6.20, all full q n -subcrystals of Incr + n (z) must be analogous subsets on which P O EG is constant, so Q O EG is a quasi-isomorphism of q + n -crystals.
Proof. Let Λ n be the set of strict partitions in N n . The diagram z∈I Z Incr + n (z) z∈I Z Incr n (z) λ∈Λn ShTab + n (λ) λ∈Λn ShTab n (λ) commutes and all four arrows are surjective maps. Since the top horizontal arrow and both vertical arrows are quasi-isomorphisms of gl n -crystals by Corollary 5.22 and Theorems 7.8 and 7.10, the bottom horizontal arrow must also be a quasi-isomorphism of gl n -crystals. This prove the result when µ = ∅. This suffices for the general case, since applying the functor F from Corollary 6.12 to a map preserves the property of being a quasi-isomorphism, and ShTab + n (λ/µ) and ShTab n (λ/µ) may be identified with gl n -subcrystals of F (ShTab + k+n (λ)) and F (ShTab k+n (λ)) for k = ℓ(µ) with the former mapped onto the latter by unprime diag .
It was easy to describe a coarse decomposition of W + n (m) and Incr + n (z) into gl n -subcrystals on which the unpriming maps W + n (m) → W n (m) and Incr + n (z) → Incr n (z) are injective. It does not seem to be straightforward to do the same for unprime diag : ShTab + n (λ) → ShTab n (λ). A harder open problem is to describe the full gl n -subcrystals of ShTab + n (λ). 5

From words to shifted tableaux
Combining Sections 7.1 and 7.2 shows how to embed each crystal of words W + n (m) in a union of shifted tableau crystals ShTab + n (λ). This will lead to a proof of Theorem 1.5. Fix m, n ∈ N and consider a word w = w 1 w 2 · · · w m ∈ W + n (m). Define w ⊤ to be the n-tuple of strictly increasing primed words a = (a 1 , a 2 , . . . , a n ) in which the unprimed letters in a i are the indices j ∈ [m] with w j = i, and the primed letters in a i are given by adding primes to each j ∈ [m] with w j = i ′ . For example, if n = 3 and w = 2 ′ 211 ′ 2 ′ , then w ⊤ = (34 ′ , 1 ′ 25 ′ , ∅).
Form double(w) by applying the map with i → 2i and i ′ → (2i) ′ for i ∈ Z to the letters of w. For a tableau T , define double(T ) by applying the same map to every entry. For an n-tuple of primed words a = (a 1 , . . . , a n ), let double(a) := (double(a 1 ), . . . , double(a n )).
The correspondence w → (P O HM (w), Q O HM (w)) extends Haiman's shifted mixed insertion algorithm from [14] and is called orthogonal mixed insertion in [32]. It can be defined in a self-contained way via a certain insertion process (see [32,Def. 5.17]) and gives a bijection from W + n (m) to the set of pairs (P, Q) of shifted tableaux of size m with the same shape, in which P is semistandard with all entries at most n and Q is standard with no primes on the diagonal [32,Prop. 5.4 and Thm. 5.21]. The following is clear from the observations above and Theorem 7.10: Corollary 7.13. The map w → P O HM (w) is a quasi-isomorphism of q + n -crystals m∈N W + n (m) → strict partitions λ∈N n ShTab + n (λ), and the full q + n -subcrystals of m∈N W + n (m) are the subsets on which w → Q O HM (w) is constant.
Corollary 7.14. A connected normal q + n -crystal has a unique q + n -highest weight element. The weight of this element is a strict partition λ ∈ N n and the crystal is isomorphic to ShTab + n (λ).
Proof. Corollary 7.13 tells us that any connected normal q + n -crystal is isomorphic to a full q + nsubcrystal of ShTab + n (λ) for some λ ∈ N n , so the result follows from Theorem 6.20.
Proposition 7.15. Suppose B is a normal q + n -crystal. Then b ∈ B is a q + n -lowest weight element if and only if σ w + 0 (b) ∈ B is a q + n -highest weight element.
The only claim in Theorem 1.5 left to prove is that for each strict partition λ ∈ N n there is a connected normal q + n -crystal with highest weight λ. Thus it suffices to check the following: Theorem 7.16. If λ ∈ N n is a strict partition, then ShTab + n (λ) is a connected normal q + n -crystal.
If z ∈ I Z is vexillary, then the normal q n -crystal Incr n (z) is also connected, but the latter may be connected without z being vexillary; see [16,Cor. 4.56], which via [31,Cor. 3.34] gives a more complicated pattern avoidance condition characterizing when this happens.