Expanding K-theoretic Schur Q-functions

We derive several identities involving Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions. Our main result is a formula conjectured by Lewis and the second author which expands each $K$-theoretic Schur $Q$-function in terms of $K$-theoretic Schur $P$-functions. This formula extends to some more general identities relating the skew and dual versions of both power series. We also prove a shifted version of Yeliussizov's skew Cauchy identity for symmetric Grothendieck polynomials. Finally, we discuss some conjectural formulas for the dual $K$-theoretic Schur $P$- and $Q$-functions of Nakagawa and Naruse. We show that one such formula would imply a basis property expected of the $K$-theoretic Schur $Q$-functions.

We refer to elements of ShYT P (λ) and ShYT Q (λ) as P -shifted and Q-shifted tableaux, respectively.We draw shifted tableaux in French notation: 4 2 3 ′ 1 2 ′ 3 ′ 3 a P -shifted tableau of shape (4, 2, 1) The weight of a shifted tableau T is the monomial x T := i≥1 x mi i where m i is the number of times that i or i ′ appears in T .For example, we have x T = x 1 x 2 2 x 3 3 x 4 for both of the shifted tableaux shown as examples above.The Schur P -and Q-functions indexed by λ are the power series x T and Q λ := T ∈ShYTQ(λ) x T . (1.1) Any way of toggling the primes in the diagonal entries of a Q-shifted tableau results in another Q-shifted tableau of the same weight, so it is clear that It is well-known that P λ and Q λ are symmetric functions of bounded degree.They were first defined in work of Schur on the projective representations of the symmetric group but have since appeared in various other contexts.We are interested in generalizations of P λ and Q λ that are similar generating functions for set-valued shifted tableaux.A set-valued shifted tableau of shape λ is a filling of SD λ by nonempty finite subsets of { 1 2 i : 0 < i ∈ Z} = {1 ′ < 1 < 2 ′ < 2 < . . .}.We consider a sequence of such subsets S 1 , S 2 , S 3 , . . . to be weakly increasing if max(S i ) ≤ min(S i+1 ) for all i.With this convention, we may define a set-valued shifted tableau to be semistandard if it satisfies the same conditions (S1)-(S3) as above.
The weight x T of a set-valued shifted tableau T is defined in the same way as in the non-set-valued case; for both tableaux in the preceding example one has x T = x 1 x 3 2 x 4 3 x 4 x 5 .Write T ij for the entry of a set-valued shifted tableau in position (i, j) and define |T | := (i,j)∈SD λ |T ij | and |λ| := |SD λ |.Then |T | − |λ| is the difference between the degree of x T and the size of SD λ .Finally, let β be a variable that commutes with each x i .The K-theoretic Schur P -and Q-functions indexed by λ are the power series in Z[β][[x 1 , x 2 , . . .]] given by (1.3) We recover P λ from GP λ and Q λ from GQ λ by setting β = 0.Both GP λ and GQ λ are symmetric in the x i variables [8, §3.4] and homogeneous of degree |λ| if we set deg(β) = −1 and deg(x i ) = 1.
Ikeda and Naruse introduced these functions in [8] for applications in Ktheory.Specializations of GP λ and GQ λ represent the structure sheaves of Schubert varieties in the K-theory of the maximal isotropic Grassmannians of orthogonal and symplectic types [8,Cor. 8.1].More precisely, the GP -and GQfunctions represent Schubert classes in connective K-theory, so can be turned into cohomology classes or elements of the Grothendieck ring of vector bundles on setting β = 0 and β = 1, respectively.The GP -and GQ-functions are also "stable limits" of connective K-theory classes of orbit closures for symplectic and orthogonal groups acting on the type A flag variety [14,15].For more results about these functions and various extensions, see [16,17,18].
Our first main result is a K-theoretic analogue of equation (1.2).The relevant identity is subtler than in the classical case, and was predicted as [11,Conj. 5.15].It expresses each K-theoretic Schur Q-function as a finite linear combination of K-theoretic Schur P -functions with integer coefficients.
The additional complexity in (1.4) compared to (1.2) is related to the fact that in a set-valued Q-shifted tableau, a diagonal entry may contain both i and i ′ .When this happens there is no simple way to remove all primes from the diagonal without changing the relevant weight.
Corollary 1.2.If µ is a strict partition then GQ µ is an N[β]-linear combination of GP -functions if and only if all distinct parts of µ differ by at least two.
Proof.It suffices by Theorem 1.1 to observe that there exists a strict partition λ ⊇ µ with ℓ(λ) = ℓ(µ) such that SD λ/µ is a vertical strip and cols(λ/µ) ≡ |λ/µ| (mod 2) if and only if We also prove a few more results.Theorem 1.1 has some enumerative consequences which we discuss in Section 4.
The GP -and GQ-functions have skew versions GP λ/µ and GQ λ/µ , which are generating functions for set-valued tableaux of shifted skew shapes.In Section 5 we derive an extension of Theorem 1.1 for these power series, along with some other related identities.
There are also dual power series gp λ and gq λ defined by Nakagawa and Naruse [16] from GP λ and GQ λ via a Cauchy identity.Section 6 contains some further results about these functions, including a dual form of Theorem 1.1 (see Corollary 6.2) and a skew Cauchy identity (see Theorem 6.9).
In Section 7, we recall a conjectural formula for gp λ and gq λ from [16].We then explain a new conjectural formula for the related functions jp λ := ω(gp λ ) and jq λ := ω(gq λ ) obtained by applying the algebra automorphism ω that sends s λ → s λ ⊤ .We show that these new conjectures would imply a conjecture of Ikeda and Naruse about the GQ-functions forming a Z[β]-basis for a ring.

Preliminaries
Fix a positive integer n and continue to let β, x 1 , x 2 , . . .be commuting variables.Theorem 2.1 (See [8]).If λ is a strict partition with r := ℓ(λ) ≤ n then where w ∈ S n acts on rational functions by permuting the x i variables while fixing β.
In fact, Ikeda and Naruse show that the sets Thm. 3.1 and Prop. 3.2].An analogous basis property is known to hold for the set of all formal power series GP λ 's and is expected to hold for the GQ λ 's; see the discussion before Corollary 7.7.
The GP -and GQ-power series are generalizations of Ivanov's factorial Pand Q-functions [9], which have another generalization studied by Okada in [19].Comparing Theorem 2.1 and [19, Lem.2.4] suggests a common generalization of these functions which might be interesting to consider in future work.
Lemma 3.1.Let λ be a strict partition with r : Then it holds that . By Theorem 2.1, the expression respectively.Then we can rewrite (3.2) as where ν is the sequence formed from ν by prepending m zeros.The subgroups S m and H n−m commute, and each h ∈ The preceding expression is therefore equal to If f (x) = 1 then the internal sums here are This proves the first identity.The other follows by taking f (x) = 2 + βx.
After expanding B δ in terms of the A λ 's, one can apply many cancellations from Lemma 3.3.We will use the next lemma to organize these cancellations.This lemma involves a certain directed graph G m for m ≥ 2 which we define inductively.In general, the vertex set of G m consists of all nonempty subsets of [m] excluding sets of the form [i] for i odd and including two copies of [i] for i even.When m ∈ {2, 3} the graph G m is given explicitly by and Our last step before proving Theorem 1.1 is to derive a simplified form of the desired identity involving the functions A λ and B λ .Lemma 3.6.Suppose µ = (q, q − 1, q − 2, . . ., p) for integers q ≥ p > 0. Then where the sum is over strict partitions λ ⊇ µ with ℓ(λ) = ℓ(µ) such that SD λ/µ is a vertical strip, and cols(λ/µ) is the number of columns occupied by SD λ/µ .
Proof.We first prove the lemma in the case when q = m and p = 1.Then For a subset S ⊆ [m], let e S := i∈S e i .It follows by expanding the definition of , which we can rewrite as the identity where χ(S) is defined to be 2 if S = [i] for any i ∈ {2, 4, 6, . . .}, 0 if S = ∅ or S = [i] for any i ∈ {1, 3, 5, . . .}, and 1 otherwise.By Lemma 3.5, the left-hand side of (3.5) is precisely where the first sum is over the (sometimes repeated) vertices of the graph G m and second sum is over the edges in G m .In view of Lemma 3.3 and property (a) in Lemma 3.5, every term in the last sum is zero so (3.4) holds.
For the general identity, observe that if λ is a strict partition with r parts then which can be rewritten as (3.3).
) are partitions then let λµ denote their concatenation; this will be another partition if λ p ≥ µ 1 .Given a strict partition µ, define Λ(µ) to be the set of strict partitions λ ⊇ µ with ℓ(λ) = ℓ(µ) such that SD λ/µ is a vertical strip.We can now prove Theorem 1.1, which states that be a nonempty strict partition.We first prove the desired identity specialized to the variables x 1 , x 2 , . . ., x n , so assume our fixed value of n has n ≥ r > 0. Let q = µ 1 and suppose m ∈ [r] is maximal with µ m = q + 1 − m.We proceed by induction on r − m.
In the base case when m = r, the result to prove is Lemma 3.6.It remains to deal with the inductive step.Assume 1 ≤ m < r and set Then γ = (q, q − 1, q − 2, . . ., p) for p := µ m and γ m ≥ ν 1 + 2. We may assume by induction that the desired identity holds when µ is replaced by ν, since this replacement transforms r → r − m and m → (some positive number) so reduces the difference r − m.This assumption and Lemma 3.6 imply that Using both parts of Lemma 3.1, we deduce that This even holds when µ = ∅, so taking the limit as n → ∞ gives the theorem.

Weight-preserving bijections
As the GQ λ 's and GP λ 's are generating functions for set-valued shifted tableaux, Theorem 1.1 has some enumerative consequences, which we describe here.
Let X ⊆ {1, 2, 3, . . .} × {1, 2, 3, . . .} be a set of positions.Given a setvalued shifted tableau T , define unprime X max (T ) to be the tableau formed from T by removing the prime from the largest element of T ij for each (i, j) ∈ X , whenever this element is not already primed.
max (T ) is also semistandard.This property may fail if X is not a subset of the main diagonal (as we see in the previous example).
If λ is a strict partition and X ⊆ {(i, i) Given strict partitions λ ⊇ µ define SetShYT P (λ : µ) to be the set of semistandard set-valued shifted tableaux The diagonal entry in row i of such a tableau can have at most one primed element if λ i = µ i and no primed elements if λ i > µ i .Finally, for a strict partition µ let Λ ± (µ) be the set of strict partitions λ ⊇ µ with ℓ(λ) = ℓ(µ) such that SD λ/µ is a vertical strip and (−1) cols(λ/µ)+|λ/µ| = ±1.Then Λ(µ) = Λ + (µ) ⊔ Λ − (µ) and this decomposition reflects the decomposition of the right side of Theorem 1.1 into positive and negative terms.
Proof.To see that the domain of the given map is indeed a disjoint union, observe that the set Λ − (µ) is empty if and only if all parts of µ differ by at least two.Since cols(µ/µ) + |µ/µ| = 0, the set Λ + (µ) is never empty and SetShYT Q (µ) and SetShYT P (λ : µ) are disjoint for all λ ∈ Λ − (µ).
It is an interesting open problem to find a bijective proof of Theorem 1.1.One way to achieve this would be to construct an explicit map realizing Corollary 4.1.This is easy to do when µ = (n) has only one nonzero part, in which case the bijection in Corollary 4.1 is a map The set SetShYT P (n : n) is contained in SetShYT Q (n) and is the union of SetShYT P (n) and the set of tableaux formed from elements of SetShYT P (n) by adding a prime to the largest number in box (1, 1).
A bijection (4.3) is given by mapping each T ∈ SetShYT P (n : n) to itself and each T ∈ SetShYT Q (n) \ SetShYT P (n : n) to the tableau in SetShYT P (n + 1) formed by adding 1  2 to the smallest primed number i ′ = i − 1 2 in box (1, 1), and then splitting this diagonal box into two adjacent boxes containing all numbers ≤ i and > i, respectively.This map is weight-preserving and would send for example.It seems difficult to generalize this idea to larger shapes.Even in the next simplest case µ = (3, 1) we do not know of a straightforward way to describe a weight-preserving bijection of the form in Corollary 4.1.
In [8, §9.2], Ikeda and Naruse derive another set of combinatorial formulas for GP λ and GQ λ as generating functions for excited Young diagrams.One could also try to find a bijective proof of Theorem 1.1 via these expressions.

Skew analogues
Let µ ⊆ λ be strict partitions.A semistandard (skew) shifted tableau of shape λ/µ is a filling of SD λ/µ := SD λ \ SD µ by positive half-integers such that rows and columns are weakly increasing, with no primed entries repeated in a row and no unprimed entries repeated in a column.
Let ShYT Q (λ/µ) be the set of all such tableaux and let ShYT P (λ/µ) be the subset in which primed entries are disallowed from diagonal positions.We define both sets to be empty if µ ⊆ λ.The skew Schur P -and Q-functions are where as usual x T := i≥1 x mi i with m i denoting the number of entries of T equal to i or i ′ .To motivate these symmetric functions, we need some addi- , then write f (x, y) for the power series f (x 1 , y 1 , x 2 , y 2 , . . . ) where x 1 , x 2 , . . .and y 1 , y 2 , . . .are separate sets of commuting variables; we also set f (x) := f (x 1 , x 2 , . . . ) = f and f (y) := f (y 1 , y 2 , . . .).If f is symmetric then specializing f (x, y) to finitely many variables gives It follows that we can write where in both sums ν ranges over all strict partitions [22, Eq. (8.2)].Define the set SetShYT Q (λ/µ) of semistandard set-valued (skew) shifted tableaux of shape λ/µ in the same way as SetShYT Q (λ), just replacing references to "fillings of SD λ " by "fillings of SD λ/µ ."Let SetShYT P (λ/µ) be the subset of tableaux in SetShYT Q (λ/µ) with no primed numbers in any diagonal boxes.The skew K-theoretic Schur P -and Q-functions are then where x T is defined in the same way as for elements of SetShYT Q (λ).When µ ⊆ λ we consider both SetShYT P (λ/µ) and SetShYT Q (λ/µ) to be empty so that GP λ/µ = GQ λ/µ = 0.These generalizations of GP λ and GQ λ were first defined in [11, §4.6] in the context of enriched set-valued P -partitions.
The K-theoretic version of (5.2) involves a variant of these power series.The removable boxes of µ are the positions (i, j) ∈ SD µ such that SD µ \ {(i, j)} is the shifted diagram of another strict partition.Let Rem(µ) be the set of removable boxes of the strict partition µ.For strict partitions µ ⊆ λ define where in both sums ν must be a strict partition.For strict partitions µ ⊆ λ we set GP λ/ /µ = GQ λ/ /µ = 0. Then the definitions of GP λ and GQ λ as set-valued shifted tableau generating functions imply that where both sums are over all strict partitions ν.
Proof.The desired identity is clear if µ = λ.Assume µ = λ so that SD λ/µ is nonempty.We start by showing that the first sum is zero.Suppose the rightmost box of SD λ/µ is in column n and SD λ/µ contains k > 0 boxes in this column.Choose a strict partition η with µ ⊆ η ⊆ λ such that SD λ/η is a vertical strip.Let L := {(i, j) ∈ SD η/µ : j < n} and R := {(i, j) ∈ SD η/µ : j = n} so that SD η/µ = L ⊔ R. Because SD λ/η is a vertical strip, there are only k + 1 possibilities for R, which must be a set of adjacent positions at the bottom of column n in SD λ/µ .Moreover, when L is fixed and η varies, each of these possibilities for R occurs exactly once.Now observe that if r := |R| then and also Since SD λ/η ⊔ R = SD λ/µ − L, we can rewrite the preceding identity as By substituting these formulas and factoring out the terms depending on L, we deduce that the sum η (−1) cols(λ/η) 2 overlap(η/µ) is a multiple of and so is zero itself.
A similar argument shows that the other sum in the lemma is zero.Suppose now that the leftmost box of SD λ/µ is in column n and SD λ/µ contains k > 0 boxes in this column.Choose a strict partition γ with µ ⊆ γ ⊆ λ such that SD γ/µ is a vertical strip.Let L := {(i, j) ∈ SD λ/γ : j = n} and R := {(i, j) ∈ SD λ/γ : j > n} so that SD λ/γ = L ⊔ R. Because SD γ/µ is a vertical strip, there are now only k + 1 possibilities for L, which must be a set of adjacent positions at the top of column n in SD λ/µ , and when R is fixed and γ varies, each of these possibilities occurs exactly once.Finally, if ℓ := |L| then we have and also Since SD γ/µ ⊔ L = SD λ/µ − R, we can rewrite the preceding identity as By substituting these formulas and factoring out the terms depending on R, we deduce that γ (−1) cols(γ/µ) 2 overlap(λ/γ) is also a multiple of (5.7), as needed.
We can now derive a skew generalization of Theorem 1.1.
There is some interest in determining when there are coincidences P λ/µ = P ν/κ and Q λ/µ = Q ν/κ among the skew Schur P -and Q-functions [1,5,21].This phenomenon is less well-understood than for skew Schur functions.
One could consider the same problem for the skew GP -and GQ-functions.In particular, any equality GP λ/µ = GP ν/κ would imply that P λ/µ = P ν/κ (and likewise for the Q-functions), but it is not clear if the converse always holds.With these questions in mind, we explain one nontrivial equality between Schur Q-functions that generalizes to the K-theoretic setting.
Define the flip of a skew shape λ/µ to be the skew shape φ(λ/µ) whose shifted diagram SD φ(λ/µ) is formed by reflecting SD λ/µ across a line perpendicular to the main diagonal, so that in French notation the bottom row becomes the rightmost column.We refer to this operation as "flipping the diagram SD λ/µ ": The following reduces to [4, Prop.IV.13] when β = 0: Proposition 5.4.Let µ ⊆ λ be strict partitions, then Proof.We first prove the GQ-identity.Suppose T ∈ SetShYT Q (λ/µ).Write min(T ) and max(T ) for the minimal and the maximal numbers appearing in any entry of T .Let n := ⌈min(T )⌉ be whichever of min(T ) or min(T ) + 1 2 is an integer, and define N := ⌈max(T )⌉ similarly.Now let φ Q (T ) be the set-valued shifted tableau of shape φ(λ/µ) formed by flipping T and then replacing each number a in each set-valued entry by n + N − 1 2 − a.The rows and columns of φ Q (T ) are weakly increasing since the rows and columns of T are weakly increasing.As exactly one of a or n + N − 1 2 − a is primed, the tableau φ Q (T ) does not have any unprimed numbers repeated in a column or primed numbers repeated in a row.We conclude that φ Q defines a map SetShYT Q (λ/µ) → SetShYT Q (φ(λ/µ)), which is clearly invertible.
Notice that the weight of φ Q (T ) is x φQ(T ) = σ(x T ), where σ is the permutation with σ(a) = n + N − a for all n ≤ a ≤ N .Since the values of n and N are determined by the monomial x T , the symmetry of GQ λ/µ implies that (5.8) The proof of the GP -identity is similar, except now for T ∈ SetShYT P (λ/µ) we define φ P (T ) from φ Q (T ) by adding 1  2 to all numbers in diagonal positions.Since all numbers in diagonal entries of φ Q (T ) are primed when T ∈ SetShYT P (λ/µ), we have x φP (T ) = x φQ(T ) and it follows that φ P is a bijection SetShYT P (λ/µ) → SetShYT P (φ(λ/µ)).We can therefore replace every letter "Q" in (5.8) by "P " to deduce that GP λ/µ = GP φ(λ/µ) .

Dual functions
Let x := −x 1+βx so that x ⊕ x = 0. Nakagawa and Naruse [16] define the dual K-theoretic Schur P -and Q-functions gp λ and gq λ to be the unique elements Both sums in this Cauchy identity are over all strict partitions λ.
The power series gp λ and gq λ are a special case of the dual universal factorial Schur P -and Q-functions in [16,Def. 3.2].One reason these specialization are interesting (compared to the more general "universal" functions) is because they have conjectural formulas as generating functions for shifted reverse plane partitions [16,Conj 5.1].We will discuss this idea in Section 7.
Both Proof.We recover the original form of i,j≥1 1−xiyj 1−xiyj after setting β = 1 by substituting x i → βx i and y j → β −1 y j for all i and j.It follows that if we define gp where the sum is over strict partitions µ ⊆ λ with ℓ(µ) = ℓ(λ) such that SD λ/µ is a vertical strip.
Proof.This follows by a standard argument similar to the proof of [22,Prop. 8.2], which is equivalent to (6.2) when β = 0. Let ∆(x; y) := i,j≥1 by equations (5.5) and (6.1).The first identity follows by substituting the formula (5.6) for GQ λ/ /µ into the first expression and extracting the coefficients of GQ µ (x)GQ ν (y).The second identity follows similarly.
Since GP λ/ /µ = GQ λ/ /µ = 0 when µ ⊆ λ, we already know from (5.6) that when µ ⊆ λ (and, as noted above, when |λ| > |µ| + |ν|).We also have for unique integers a λ µν , b λ µν ∈ Z.It is known from [3] that the GP -expansion is finite with every a λ µν ∈ N, but a priori the coefficients b λ µν could be nonzero for infinitely many strict partitions λ.We will discuss the problem of showing that the GQ-expansion is finite in Section 7. It is again clear that From this observation, to complete the proof it is enough to show that each GQ ν is a possibly infinite Z[β]-linear combination of products of the form GQ (p1) GQ (p2) • • • GQ (p k ) .This claim is a consequence of [17,Thm. 5.8], which gives a formula for a more general universal factorial Hall-Littlewood Q-function HQ L ν (x n ; t | b), essentially as a linear combination of products of other such functions indexed by one-row partitions.The notation in [17] makes it slightly nontrivial to connect this formula to our situation.However, the property needed for GQ ν follows directly from the discussion in [17, §5.2.4] (in particular, from [17,Thm. 5.10]), once one observes that the generating function in [17, Eq. (5.10)] reduces when b = 0 and n → ∞ to the expression below: Lemma 6.6.When expanded as a power series in u −1 , the expression Proof.This is essentially [7,Rem. 5.11] given [7,Def. 3.5].Here is a selfcontained proof explained to us by Joel Lewis.Let ℓ(T ) denote the number boxes in a tableau T .Then n>0 GQ (n) t n = T β |T |−ℓ(T ) x T t ℓ(T ) where the sum is over all semistandard set-valued shifted tableaux T with a nonempty one-row shape.Such a tableau T is specified uniquely by the following choices: • The finite set of positive integers {j 1 < . . .< j k } such that at least one of j i or j ′ i appears in some box of T .
• The numbers n 1 , . . ., n k > 0 such that j i or j ′ i appears in n i boxes of T .
• For each i ∈ [k], whether the first box containing j i or j ′ i contains just j i , just j ′ i , or both j i and j ′ i .
• For each i > 1, whether the first box containing j i or j ′ i contains no smaller numbers, or contains at least one of j i−1 or j ′ i−1 .
For the tableau T corresponding to this data, the value of |T | (respectively, ℓ(T )) depends on the numbers n 1 , . . ., n k and the choices in the third (respectively, fourth) bullet point.It follows that we can write n>0 GQ (n) t n as Fixing k and j 1 < • • • < j k and summing over n 1 , . . ., n k turns this into Pulling out a factor of t t+β = The desired formula follows by replacing the formal parameter t by u −1 .This completes the proof of Proposition 6.5.
For any strict partitions λ and µ we define where both sums are over all strict partitions ν.
Proof.These identities are equivalent to the first two parts of [16,Prop. 3.2].
We may now prove a dual version of Theorem 5.2.
Remark.If there were a bilinear form that made {gq λ/κ } the dual basis of {GP λ/κ } and {gp µ/ν } the dual basis of {GQ µ/ν }, then this result would follow immediately from Theorem 5.2.Since the various skew functions are not even linearly independent, no such form is available and we need another argument.

Conjectural generating functions
In this final section we discuss some conjectural formulas for gp λ/µ and gq λ/µ and two related dual functions.Let µ ⊆ λ be strict partitions.A shifted reverse plane partition of shape λ/µ is a filling of SD λ/µ by positive half-integers {1 ′ < 1 < 2 ′ < 2 < . . .} such that rows and columns are weakly increasing.Examples include which both have shape (5, 3, 2, 1)/(4, 1).The weight of a shifted reverse plane partition T is the monomial x wtRPP(T ) := i≥1 x ci+ri i where c i is the number of distinct columns of T containing i and r i is the number of distinct rows of T containing i ′ .This monomial has degree | wt RPP (T )| := i≥1 (c i + r i ), which may be less than |T | := |λ/µ|.
Remark.The cited conjecture [16,Conj. 5.1] only states these formulas when µ = ∅ and β = −1.However, this special case implies the general result.In detail, if we knew the conjecture when β = −1 then we could derive the general statement using Proposition 6.1.In turn, if we knew that ) , and hence that the combinatorial generating function were symmetric, then we would have We could then derive gp λ/ν = T ∈ShRPPP (λ/ν) (−β) |λ/ν|−| wtRPP(T )| x wt RPP (T ) by comparing the preceding identity with (6.8) and equating the coefficients of gp ν .The reductions in gq-case are similar.
Nakagawa and Naruse give another conjectural formula for gq λ as a certain Pfaffian in [17, §6.2], but we will not discuss this here.
We may also describe a conjectural generating function formula for jp λ/µ and jq λ/µ .This formula appears to be new.
A partition of a set S is a set Π of disjoint nonempty subsets B ⊆ S, called blocks, with S = B∈Π B. Assume µ ⊆ λ.We define a shifted bar tableau of shape λ/µ to be a pair T = (V, Π), where V is a semistandard shifted tableau of shape λ/µ and Π is a partition of SD λ/µ such that each block B ∈ Π is a set of adjacent boxes containing the same entry in V .Because V is semistandard, each of these blocks must consist of a contiguous "bar" within a single row or single column.One might draw a shifted bar tableau as a picture like Let ShBT Q (λ/µ) denote the set of all shifted bar tableaux of shape λ/µ and let ShBT P (λ/µ) be the subset of such pairs T = (V, Π) where V has no primed diagonal entries.Given T = (V, Π) ∈ ShBT Q (λ/µ) we set  V ) .Finally, observe that if V is fixed, then the sum of x T over all Π such that T = (V, Π) is a shifted bar tableau is i≥1 x ri+ci i (x i + 1) mi−ri−ci where r i is the number of rows of V containing an entry equal to i, c i is the number of columns of V containing an entry equal to i ′ , and m i is the number of boxes of V containing i or i ′ .partitions T of size n whose first two boxes contain distinct entries, the first of which is primed.Removing the prime from the first entry defines a weightpreserving bijection from the set of such T to ShRPP Q (n) − ShRPP P (n), so it follows that gq (n) = 2gp (n) − gp (n−1) when n ≥ 2. Since we likewise have gq (1) = 2gp (1) and gq (n) = 2gp (n) − gp (n−1) when n ≥ 2 by Corollary 6.2, we deduce by induction that gp (n) = gp (n) for all n.An (unshifted) reverse plane partition of shape λ is a filling of the (unshifted) diagram D λ := {(i, j) : 1 ≤ j ≤ λ i } by positive integers, such that rows and columns are weakly increasing.The weight of such an object is defined in the same way as for shifted reverse plane partitions.As noted in [16,Prop. 5.3], it is easy to see that gp (n) = T x wtRPP(T ) where the sum is over all reverse plane partitions T of any of the hook shapes a1 n−a := (a, 1, 1, . . ., 1) for a ∈ [n]; the relevant weight-preserving bijection is given by moving all primed entries in an element of ShRPP P (n) from the first row to the first column with primes removed.This sum is precisely n a=1 g a1 n−a where g λ is the dual stable Grothendieck polynomial discussed, for example, in [10, §9.1].
An (unshifted) bar tableau of shape λ is defined in the same way as a shifted bar tableau, except the underlying tableau is a filling of D λ (rather than SD λ ) by positive integers (rather than positive half-integers); this is called a valuedset tableau in [10, §9.8].The weight x T is the same as in the shifted case.Let jp λ := T ∈ShBTP (λ) x T and jq λ := T ∈ShBTQ(λ) x T .Then jp (n) = T x T where the sum is over all bar tableaux T of any of the hook shapes a1 n−a for a ∈ [n]; the relevant weight-preserving bijection is again given by moving all primed entries in an element of ShBT P (n) from the first row to the first column with primes removed.(Each primed entry comprised its own block in the first row and is assigned to its own block in the first column.)This sum is precisely n a=1 j a1 n−a where j λ is the generating function discussed in [10, §9.8], which has j λ = ω(g λ ) for all partitions λ [10, Prop.9.25].
Combining the last two paragraphs shows that jp (n) = ω(gp (n) ) so jp (n) = ω(gp (n) ) = ω(gp (n) ) = jp (n) for all n.Finally, we have jq (1) = 2jp (1) = 2(x 1 + x 2 + x 3 + . . .), and if n ≥ 2 then jp (n) − jp (n−1) = T x T where the sum is over one-row shifted bar tableaux T of size n whose first two entries are unprimed but not in the same block.Adding a prime to the diagonal entry gives a weight-preserving bijection from the set of such T to ShBT Q (n) − ShBT P (n), so jq (n) = 2jp (n) − jp (n−1) when n ≥ 2. Since the same formulas relate jq (n) to jp (n) by the linearity of ω, we must have jq (n) = jq (n) .Conjecture 7.3 has some consequences regarding the numbers a λ µν and b λ µν .

Corollary 4 . 1 .
For each strict partition µ there is a weight-preserving bijection SetShYT