Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.


Introduction
Let λ/µ be a skew partition. The Schur function s λ/µ is a multivariate generating function for the semistandard tableaux of shape λ/µ. In the same vein, the dual stable Grothendieck polynomial g λ/µ is a generating function for the reverse plane partitions of shape λ/µ; these, unlike semistandard tableaux, are only required to have their entries increase weakly down columns (and along rows). More precisely, g λ/µ is a formal power series in countably many commuting indeterminates x 1 , x 2 , x 3 , . . . defined by g λ/µ = ∑ T is a reverse plane partition of shape λ/µ where x ircont(T) is the monomial x a 1 1 x a 2 2 x a 3 3 · · · whose i-th exponent a i is the number of columns (rather than cells) of T containing the entry i. As proven in [LamPyl07,§9.1], this power series g λ/µ is a symmetric function (albeit, unlike s λ/µ , an inhomogeneous one in general). Lam and Pylyavskyy connect the g λ/µ to the (more familiar) stable Grothendieck polynomials G λ/µ (via a duality between the symmetric functions and their completion, which explains the name of the g λ/µ ; see [LamPyl07,§9.4]) and to the K-theory of Grassmannians ([LamPyl07, §9.5]). We devise a common generalization of the dual stable Grothendieck polynomial g λ/µ and the classical skew Schur function s λ/µ . Namely, if t 1 , t 2 , t 3 , . . . are countably many indeterminates, then we set where t ceq(T) is the product t b 1 1 t b 2 2 t b 3 3 · · · whose i-th exponent b i is the number of cells in the i-th row of T whose entry equals the entry of their neighbor cell directly below them. This g λ/µ becomes g λ/µ when all the t i are set to 1, and becomes s λ/µ when all the t i are set to 0.
We prove this result (thus obtaining a new proof of [LamPyl07,Theorem 9.1]) first using an elaborate generalization of the classical Bender-Knuth involutions to reverse plane partitions, and then for a second time by analyzing the structure of reverse plane partitions whose entries lie in {1, 2}. The second proof reflects back on the first, in particular providing an alternative definition of the generalized Bender-Knuth involutions constructed in the first proof, and showing that these involutions are (in a sense) "the only reasonable choice".
The present paper is organized as follows: In Section 2, we recall classical definitions and introduce notations pertaining to combinatorics and symmetric functions. In Section 3, we define the refined dual stable Grothendieck polynomials g λ/µ , state our main result (that they are symmetric functions), and do the first steps of its proof (by reducing it to a purely combinatorial statement about the existence of an involution with certain properties). In Section 4, we describe the idea of constructing this involution in an elementary way without proofs. In Section 5, we prove various properties of this involution advertised in Section 4, thus finishing the proof of our main result. In Section 6, we recapitulate the definition of the classical Bender-Knuth involution, and show that our involution is a generalization of the latter. Finally, in Section 7 we study the structure of reverse plane partitions with entries belonging to {1, 2}, which (in particular) gives us an explicit formula for the t-coefficients of g λ/µ (x 1 , x 2 , 0, 0, . . . ; t), and shines a new light on the involution constructed in Sections 4 and 5 (also showing that it is the unique involution that shares certain natural properties with the classical Bender-Knuth involutions).

Acknowledgments
We owe our familiarity with dual stable Grothendieck polynomials to Richard Stanley. We thank Alexander Postnikov for providing context and motivation, and Thomas Lam and Pavlo Pylyavskyy for interesting conversations.

Notations and definitions
Let us begin by defining our notations (including some standard conventions from algebraic combinatorics).
A sequence α = (α 1 , α 2 , α 3 , . . .) of nonnegative integers is called a weak composition if the sum of its entries (denoted |α|) is finite. We shall always write α i for the i-th entry of a weak composition α.
We identify each partition λ with the subset (i, j) ∈ N 2 + | j ≤ λ i of N 2 + (called the Young diagram of λ). We draw this subset as a Young diagram (which is a left-aligned table of empty boxes, where the box (1, 1) is in the top-left corner while the box (2, 1) is directly below it; this is the English notation, also known as the matrix notation); see [Fulton97] for the detailed definition.
A skew partition λ/µ is a pair (λ, µ) of partitions satisfying µ ⊆ λ (as subsets of the plane). In this case, we shall also often use the notation λ/µ for the set-theoretic difference of λ and µ.
If λ/µ is a skew partition, then a filling of λ/µ means a map T : λ/µ → N + . It is visually represented by drawing λ/µ and filling each box c with the entry T(c). Three examples of a filling can be found on Figure 1.
A filling T : λ/µ → N + of λ/µ is called a reverse plane partition of shape λ/µ if its values increase weakly in each row of λ/µ from left to right and in each column of λ/µ from top to bottom. If, in addition, the values of T increase strictly down each column, then T is called a semistandard tableau of shape λ/µ. (See Fulton's [Fulton97] for an exposition of properties and applications of semistandard tableaux 1 .) We denote the set of all reverse plane partitions of shape λ/µ by RPP (λ/µ). We abbreviate reverse plane partitions as rpps.
Examples of an rpp, of a non-rpp and of a semistandard tableau can be found on Figure 1. is not an rpp as it has a 4 below a 6, (b) is an rpp but not a semistandard tableau as it has a 3 below a 3, (c) is a semistandard tableau (and hence also an rpp).

Symmetric functions
A symmetric function is defined to be a bounded-degree 2 power series in countably many indeterminates x 1 , x 2 , x 3 , . . . over Z that is invariant under (finite) permutations 3 of x 1 , x 2 , x 3 , . . . . The symmetric functions form a ring, which is called the ring of symmetric functions and denoted by Λ. (In [LamPyl07] this ring is denoted by Sym, while the notation Λ is reserved for the set of all partitions.) Much research has been done on symmetric functions and their relations to Young diagrams and tableaux; see [Stan99,Chapter 7], [Macdon95] and [GriRei15, Chapter 2] for expositions.
Replacing "semistandard tableau" by "rpp" in the definition of a Schur function in general gives a non-symmetric function. Nevertheless, Lam and Pylyavskyy [LamPyl07,§9] have been able to define symmetric functions from rpps, albeit using a subtler construction instead of the content cont (T).
For the rest of this section, we fix a skew partition λ/µ. Now, the dual stable Grothendieck polynomial g λ/µ is defined to be the formal power series ∑ T is an rpp of shape λ/µ Unlike the Schur function s λ/µ , it is (in general) not homogeneous, because whenever a column of an rpp T contains an entry several times, the corresponding monomial x ircont(T) "counts" this entry only once. It is fairly clear that the highest-degree homogeneous component of g λ/µ is s λ/µ (the component of degree |λ| − |µ|). Therefore, g λ/µ can be regarded as an inhomogeneous deformation of the Schur function s λ/µ .
They prove this proposition using generalized plactic algebras [FomGre06, Lemma 3.1] (and also give a second, combinatorial proof for the case µ = ∅ by explicitly expanding g λ/∅ as a sum of Schur functions).
In the next section, we shall introduce a refinement of these g λ/µ , and later we will reprove Proposition 2.2 in a bijective and elementary way.

Definition
Let t = (t 1 , t 2 , t 3 , . . .) be a sequence of further indeterminates. For any weak composition α, we define t α to be the monomial t α 1 1 t α 2 2 t α 3 3 · · · . If T is a filling of a skew partition λ/µ, then a redundant cell of T is a cell of λ/µ whose entry is equal to the entry directly below it. That is, a cell (i, j) of λ/µ is redundant if (i + 1, j) is also a cell of λ/µ and T (i, j) = T (i + 1, j). Notice that a semistandard tableau is the same thing as an rpp which has no redundant cells.
If T is a filling of λ/µ, then we define the column equalities vector (or, by way of abbreviation, the ceq statistic) of T to be the weak composition ceq (T) = (c 1 , c 2 , c 3 , . . . ) where c i is the number of j ∈ N + such that (i, j) is a redundant cell of T. Visually speaking, (ceq (T)) i is the number of columns of T whose entry in the i-th row equals their entry in the (i + 1)-th row. For instance, for fillings T a , T b , T c from Figure 1 we have ceq(T a ) = (0, 1), ceq(T b ) = (1), and ceq(T c ) = (), where we again drop trailing zeroes.
Notice that |ceq(T)| is the number of redundant cells in T, so we have for all rpps T of shape λ/µ. Let now λ/µ be a skew partition. We set Let us give some examples of g λ/µ .
Here h n (x) is the n-th complete homogeneous symmetric function.
The power series g λ/µ generalize the power series g λ/µ and s λ/µ studied before. The following proposition is clear:

The symmetry statement
Our main result is now the following: Theorem 3.3. Let λ/µ be a skew partition. Then g λ/µ (x; t) is symmetric in x.
Clearly, Theorem 3.3 implies the symmetry of g λ/µ and s λ/µ due to Proposition 3.2.
We shall prove Theorem 3.3 bijectively. The core of our proof will be the following restatement of Theorem 3.3: Theorem 3.4. Let λ/µ be a skew partition and let i ∈ N + . Then, there exists an involution B i : RPP (λ/µ) → RPP (λ/µ) which preserves the ceq statistics and acts on the ircont statistic by the transposition of its i-th and i + 1-th entries.
This involution B i is a generalization of the i-th Bender-Knuth involution defined for semistandard tableaux (see, e.g., [GriRei15, proof of Proposition 2.11]), but its definition is more complicated than that of the latter. 4 Defining it and proving its properties will take a significant part of this paper.
Proof of Theorem 3.3 using Theorem 3.4. We need to prove that g λ/µ (x; t) is invariant under all finite permutations of the indeterminates x 1 , x 2 , x 3 , . . .. The group of such permutations is generated by s 1 , s 2 , s 3 , . . ., where for each i ∈ N + , we define s i as the permutation of N + which transposes i with i + 1 and leaves all other positive integers unchanged. Hence, it suffices to show that g λ/µ (x; t) is invariant under each of the permutations s 1 , s 2 , s 3 , . . .. In other words, it suffices to show that s i · g λ/µ (x; t) = g λ/µ (x; t) for each i ∈ N + . So fix i ∈ N + . In order to prove s i · g λ/µ (x; t) = g λ/µ (x; t), it suffices to find a bijection B i : RPP (λ/µ) → RPP (λ/µ) with the property that every T ∈ RPP (λ/µ) satisfies ceq (B i (T)) = ceq (T) and ircont (B i (T)) = s i · ircont (T). Theorem 3.4 yields precisely such a bijection (even an involution).

Reduction to 12-rpps
Fix a skew partition λ/µ. We shall make one further simplification before we step to the actual proof of Theorem 3.4. We define a 12-rpp to be an rpp whose entries all belong to the set {1, 2}. We let RPP 12 (λ/µ) be the set of all 12-rpps of shape λ/µ. Lemma 3.5. There exists an involution B : RPP 12 (λ/µ) → RPP 12 (λ/µ) which preserves the ceq statistic and switches the number of columns containing a 1 with the number of columns containing a 2 (that is, switches the first two entries of the ircont statistic). This Lemma implies Theorem 3.4: for any i ∈ N + and for T an rpp of shape λ/µ, we construct B i (T) as follows: • Ignore all entries of T not equal to i or i + 1.
• Replace all occurrences of i by 1 and all occurrences of i + 1 by 2. We get a 12-rpp T ′ of some smaller shape (which is still a skew partition 5 ).
• In B(T ′ ), replace back all occurrences of 1 by i and all occurrences of 2 by i + 1.
• Finally, restore the remaining entries of T that were ignored on the first step.
It is clear that this operation acts on ircont(T) by a transposition of the i-th and i + 1-th entries. The fact that it does not change ceq(T) is also not hard to show: the set of redundant cells remains the same.

Construction of B
In this section we are going to sketch the definition of B and state some of its properties. We postpone the proofs until the next section.
For the whole Sections 4 and 5, we shall be working in the situation of Lemma 3.5. In particular, we fix a skew partition λ/µ.
A 12-table means a filling T : λ/µ → {1, 2} of λ/µ such that the entries of T are weakly increasing down columns. (We do not require them to be weakly increasing along rows.) Every column of a 12-table is a sequence of the form (1, 1, . . . , 1, 2, 2, . . . , 2). We say that such a sequence is • 1-pure if it is nonempty and consists purely of 1's, • 2-pure if it is nonempty and consists purely of 2's, • mixed if it contains both 1's and 2's. 5 Fine print: It has the form λ/µ for some skew partition λ/µ, but this skew partition λ/µ is not always uniquely determined (e.g., (3, 1, 1) / (2, 1) and (3, 2, 1) / (2, 2) have the same Young diagram). But the involution B constructed in the proof of Lemma 3.5 depends only on the Young diagram of λ/µ, and thus the choice of λ/µ does not matter. T, we define flip(T) to be the 12-table obtained from T by changing each column of T as follows:

Definition 4.1. For a 12-table
• If this column is 1-pure, we replace all its entries by 2's (so that it becomes 2-pure).
Otherwise, if this column is 2-pure, we replace all its entries by 1's (so that it becomes 1-pure).
Otherwise (i.e., if this column is mixed or empty), we do not change it.
If T is a 12-rpp then flip(T) need not be a 12-rpp, because it can contain a 2 to the left of a 1 in some row. We say that a positive integer k is a descent of a 12-table P if there is a 2 in the column k and there is a 1 to the right of it in the column k + 1. We will encounter three possible kinds of descents depending on the types of columns k and k + 1: (M1) The k-th column of P is mixed and the (k + 1)-th column of P is 1-pure.
(2M) The k-th column of P is 2-pure and the (k + 1)-th column of P is mixed.
(21) The k-th column of P is 2-pure and the (k + 1)-th column of P is 1-pure.
For an arbitrary 12-table it can happen also that two mixed columns form a descent, but such a descent will never arise in our process.
For each of the three types of descents, we will define what it means to resolve this descent. This is an operation which transforms the 12-table P by changing the entries in its k-th and (k + 1)-th columns. These changes can be informally explained by Figure 2: For example, if k is a descent of type (M1) in a 12-table P, then we define the 12-table res k P as follows: the k-th column of res k P is 1-pure; the (k + 1)th column of res k P is mixed and the highest 2 in it is in the same row as the highest 2 in the k-th column of P; all other columns of res k P are copied over from P unchanged. The definitions of res k P for the other two types of descents are similar (and will be elaborated upon in Subsection 5.3). We say that res k P is obtained from P by resolving the descent k, and we say that passing from P to res k P constitutes a descent-resolution step. (Of course, a 12-table P can have several descents and thus offer several ways to proceed by descent-resolution steps.) Now the map B is defined as follows: take any 12-rpp T and apply flip to it to get a 12-table flip(T). Next, apply descent-resolution steps to flip(T) in arbitrary order until we get a 12-table with no descents left. Put B(T) := P. (A rigorous statement of this is Definition 5.11.) In the next section we will see that B(T) is well-defined (that is, the process terminates after a finite number of descent-resolution steps, and the result does not depend on the order of steps). We will also see that B is an involution RPP 12 (λ/µ) → RPP 12 (λ/µ) that satisfies the claims of Lemma 3.5. An alternative proof of all these facts can be found in Section 7.

Proof of Lemma 3.5
We shall now prove Lemma 3.5 in detail.

Descents, separators, and benign 12-tables
In Subsection 4, we have defined a "descent" of a 12-table. Let us reword this definition in more formal terms: If T is a 12-table, then we define a descent of T to be a positive integer i such that there exists an r ∈ N + satisfying (r, i) ∈ λ/µ, (r, i + 1) ∈ λ/µ, T (r, i) = 2 and T (r, i + 1) = 1. For instance, the descents of the 12-table shown in (2) are 1 and 4. Clearly, a 12-rpp of shape λ/µ is the same as a 12-table which has no descents.
If T is a 12-table, and if k ∈ N + is such that the k-th column of T is mixed, then we define sep k T to be the smallest r ∈ N + such that (r, k) ∈ λ/µ and T (r, k) = 2. Thus, every 12-table T, every r ∈ N + and every k ∈ N + such that the k-th column of T is mixed and such that (r, k) ∈ λ/µ satisfy If T is a 12-table, then we let seplist T denote the list of all values sep k T (in the order of increasing k), where k ranges over all positive integers for which the k-th column of T is mixed. For instance, if T is 1 1 1 2 1 1 2 1 2 1 2 2 2 then sep 1 T = 4, sep 3 T = 4, and sep 5 T = 2 (and there are no other k for which sep k T is defined), so that seplist T = (4, 4, 2).
We say that a 12-table T is benign if the list seplist T is weakly decreasing. 6 Notice that 12-rpps are benign 12-tables, but the converse is not true. If T is a benign 12-table, then there exists no descent k of T such that both the k-th column of T and the (k + 1) -th column of T are mixed. (4) Let BT 12 (λ/µ) denote the set of all benign 12-tables; we have RPP 12 (λ/µ) ⊆ BT 12 (λ/µ). Recall the map flip defined for 12-tables in Definition 4.1. If T ∈ BT 12 (λ/µ) then flip(T) ∈ BT 12 (λ/µ) as well because T and flip(T) have the same mixed columns. Thus, the map flip restricts to a map BT 12 (λ/µ) → BT 12 (λ/µ) which we will also denote flip.

Plan of the proof
Let us now briefly sketch the ideas behind the rest of the proof before we go into them in detail. The map flip : BT 12 (λ/µ) → BT 12 (λ/µ) does not generally send 12-rpps to 12-rpps (i.e., it does not restrict to a map RPP 12 (λ/µ) → RPP 12 (λ/µ)). However, we shall amend this by defining a way to transform any benign 12-table into a 12-rpp by what we call "resolving descents". The process of "resolving descents" will be a stepwise process, and will be formalized in terms of a binary relation ⇛ on the set BT 12 (λ/µ) which we will soon introduce. The intuition behind saying "P ⇛ Q" is that the benign 12-table P has a descent, resolving which yields the benign 12-table Q. Starting with a benign 12table P, we can repeatedly resolve descents until this is no longer possible. We have some freedom in performing this process, because at any step there can be a choice of several descents to resolve; but we will see that the final result does not depend on the process. Hence, the final result can be regarded as a function of P. We will denote it by norm P, and we will see that it is a 12-rpp. We will then define a map B : RPP 12 (λ/µ) → RPP 12 (λ/µ) by B (T) = norm (flip T), and show that it is an involution satisfying the properties that we want it to satisfy.

Resolving descents
Now we come to the details.
Let k ∈ N + . Let P ∈ BT 12 (λ/µ). Assume (for the whole Subsection 5.3) that k is a descent of P. Thus, the k-th column of P must contain at least one 2. Hence, the k-th column of P is either mixed or 2-pure. Similarly, the (k + 1)-th column of P is either mixed or 1-pure. But the k-th and the (k + 1)-th columns of P cannot both be mixed (by (4), because P is benign). Thus, exactly one of the following three statements holds: (M1) The k-th column of P is mixed and the (k + 1)-th column of P is 1-pure.
(2M) The k-th column of P is 2-pure and the (k + 1)-th column of P is mixed.
(21) The k-th column of P is 2-pure and the (k + 1)-th column of P is 1-pure. Now, we define a new 12-table res k P as follows (see Figure 2 for illustration): • If we have (M1), then res k P is the 12-table defined as follows: The k-th column of res k P is 1-pure; the (k + 1)-th column of res k P is mixed and satisfies sep k+1 (res k P) = sep k P; all other columns of res k P are copied over from P unchanged. 7 • If we have (2M), then res k P is the 12-table defined as follows: The k-th column of res k P is mixed and satisfies sep k (res k P) = sep k+1 P; the (k + 1)th column of res k P is 2-pure; all other columns of res k P are copied over from P unchanged.
• If we have (21), then res k P is the 12-table defined as follows: The k-th column of res k P is 1-pure; the (k + 1)-th column of res k P is 2-pure; all other columns of res k P are copied over from P unchanged.
In either case, res k P is a well-defined 12-table. It is furthermore clear that seplist (res k P) = seplist P. Thus, res k P is benign (since P is benign); that is, res k P ∈ BT 12 (λ/µ). We say that res k P is the 12-table obtained by resolving the descent k in P. Then P is a benign 12-table, and its descents are 1, 2 and 4. We have sep 2 P = 4. If we set k = 1 then we have (2M), if we set k = 2 then we have (M1), and if we set k = 4 then we have (21). We can resolve each of these three descents; the results are the three 12-tables on the right.
We notice that each of the three 12-tables res 1 P, res 2 P and res 4 P still has descents. In order to get a 12-rpp from P, we will have to keep resolving these descents until none remain.
We now observe some further properties of res k P: Proposition 5.4. Let P ∈ BT 12 (λ/µ) and k ∈ N + be such that k is a descent of P.
(a) The 12-table res k P differs from P only in columns k and k + 1.

(b)
The k-th and the (k + 1)-th columns of res k P depend only on the k-th and the (k + 1)-th columns of P.

(e)
The integer k is a descent of flip (res k P), and we have res k (flip (res k P)) = flip (P) .
(f) Recall that we defined a nonnegative integer ℓ (T) for every 12-table T in Definition 5.1. We have ℓ (P) > ℓ (res k P) .
Proof of Proposition 5.4. All parts of Proposition 5.4 follow from straightforward arguments using the definitions of res k and flip and (3).

The descent-resolution relation ⇛
Definition 5.5. Let us now define a binary relation ⇛ on the set BT 12 (λ/µ) as follows: Let P ∈ BT 12 (λ/µ) and Q ∈ BT 12 (λ/µ). If k ∈ N + , then we write P ⇛ k Q if k is a descent of P and we have Q = res k P. We write P ⇛ Q if there exists a k ∈ N + such that P ⇛ k Q.
Without loss of generality, assume that u < v. We are in one of the following two cases: Case 2: We have u < v − 1. Let us deal with Case 2 first. In this case, {u, u + 1} ∩ {v, v + 1} = ∅. It follows that res v (res u A) and res u (res v A) are well-defined and res u (res v A) = res v (res u A). Setting D = res u (res v A) = res v (res u A) completes the proof in this case. Now, let us consider Case 1. The v-th column of A must contain a 1 (since v − 1 = u is a descent of A) and a 2 (since v is a descent of A). Hence, the v-th column of A is mixed. Since A is benign but has v − 1 and v as descents, it thus follows that the (v − 1)-th column of A is 2-pure and the (v + 1)-th column of A is 1-pure. We can represent the relevant portion (that is, the (v − 1)-th, v-th and (v + 1)-th columns) of the 12-table A as follows: Notice that the separating line which separates the 1's from the 2's in column v is lower than the upper border of the (v − 1)-th column (since v − 1 is a descent of A), and higher than the lower border of the (v + 1)-th column (since v is a descent of A). Let s = sep v A. Then, the cells (s, v − 1), (s, v), (s, v + 1), (s + 1, v − 1), (s + 1, v), (s + 1, v + 1) all belong to λ/µ (due to what we just said about separating lines). We shall refer to this observation as the "six-cells property". Now, B = res u A = res v−1 A, so B is represented as follows: where sep v−1 B = s (that is, the separating line in the (v − 1)-th column of B is between the cells (s, v − 1) and (s + 1, v − 1)). Now, v is a descent of B. Resolving this descent yields a 12-table res v B which is represented as follows: This, in turn, shows that v − 1 is a descent of res v B (by the six-cells property). Resolving this descent yields a 12-table res v−1 (res v B) which is represented as follows: where sep v (res v−1 (res v B)) = s. On the other hand, C = res v A. We can apply a similar argument as above to show that the 12-table res v (res v−1 C) is well-defined, and is exactly equal to the 12-table in (6). Hence, res v−1 (res v B) = res v (res v−1 C), and setting D equal to this 12-table completes the proof in Case 1.

The normalization map
The following proposition is the most important piece in our puzzle: Proposition 5.9. For every T ∈ BT 12 (λ/µ), there exists a unique N ∈ Proof of Proposition 5.9. For every T ∈ BT 12 (λ/µ), let Norm (T) denote the set Thus, in order to prove Proposition 5.9, we need to show that for every T ∈ BT 12 (λ/µ) this set Norm (T) is a one-element set.
(7) We then need to prove that Norm (T) is a one-element set.
Let Z = S ∈ BT 12 (λ/µ) | T ⇛ S . In other words, Z is the set of all benign 12-tables S which can be obtained from T by resolving one descent. If Z is empty, then T ∈ RPP 12 (λ/µ), so that Norm (T) = {T} and we are done. Hence, we can assume that Z is nonempty. Therefore T / ∈ RPP 12 (λ/µ).
Thus, every N ∈ RPP 12 (λ/µ) satisfying T * ⇛ N must satisfy Z * ⇛ N for some Z ∈ Z. In other words, every N ∈ Norm (T) must belong to Norm (Z) for some Z ∈ Z. The converse of this clearly holds as well. Hence, Let us now notice that: • By Lemma 5.6 (d) and (7), for every Z ∈ Z, the set Norm (Z) is a oneelement set.

Refined dual stable Grothendieck polynomials
October 15, 2015 Hence, (8) shows that Norm (T) is a union of one-element sets, any two of which have a nonempty intersection (and thus are identical). Moreover, this union is nonempty (since Z is nonempty). Hence, Norm (T) itself is a oneelement set. This completes our induction.
In order to complete the proof of Lemma 3.5, we need to show that B is an involution, preserves the ceq statistic, and switches the number of columns containing a 1 with the number of columns containing a 2. At this point, all of this is easy: B is an involution. Let T ∈ RPP 12 (λ/µ). We have flip (T) *   Let SST (λ/µ) denote the set of all semistandard tableaux of shape λ/µ. We define a map BK i : SST (λ/µ) → SST (λ/µ) as follows:

The classical Bender-Knuth involutions
Let T ∈ SST (λ/µ). Then every column of T contains at most one i and at most one i + 1. If a column contains both an i and an i + 1, we will mark its entries as "ignored". Now, let k ∈ N + . The k-th row of T is a weakly increasing sequence of positive integers; thus, it contains a (possibly empty) string of i's followed by a (possibly empty) string of (i + 1)'s. These two strings together form a substring of the k-th row which looks as follows: , . . . , i, i + 1, i + 1, . . . , i + 1) .
Some of the entries of this substring are "ignored"; it is easy to see that the "ignored" i's are gathered at the left end of the substring whereas the "ignored" (i + 1)'s are gathered at the right end of the substring. So the substring looks as follows:  for some a, r, s, b ∈ N. Now, we change this substring into  We do this for every k ∈ N + . At the end, we have obtained a new semistandard tableau of shape λ/µ. We define BK i (T) to be this new tableau.
Now, every semistandard tableau of shape λ/µ is also an rpp of shape λ/µ. Hence, B i (T) is defined for every T ∈ SST (λ/µ). Our claim is the following: Proposition 6.2. For every T ∈ SST (λ/µ), we have BK i (T) = B i (T).
Proof of Proposition 6.2. Recall that the map B i comes from the map B we defined on 12-rpps in Section 5. We could have constructed the map BK i from the map BK 1 in an analogous way. We define a 12-sst to be a semistandard tableau whose entries all belong to the set {1, 2}. Clearly, to prove Proposition 6.2, it suffices to prove that BK 1 (T) = B(T) for all 12-ssts T.
Let T be a 12-sst, and let k ∈ N + . The k-th row of T has the form  We can now repeatedly apply descent-resolution steps to obtain a tableau whose k-th row is  Repeating this process for every row of flip (T) (we can do this because each pure column contains only one entry, and thus each descent-resolution described above affects only one row), we obtain a 12-rpp. By the definition of B, this rpp must equal B(T). By the above description, it is also equal to BK 1 (T) (because the ignored entries in the construction of BK 1 (T) are precisely the entries lying in mixed columns), which completes the proof.

The structure of 12-rpps
In this section, we restrict ourselves to the two-variable dual stable Grothendieck polynomial g λ/µ (x 1 , x 2 , 0, 0, . . . ; t) defined as the result of substituting 0, 0, 0, . . . for x 3 , x 4 , x 5 , . . . in g λ/µ . We can represent it as a polynomial in t with coefficients in Z[x 1 , x 2 ]: where the sum ranges over all weak compositions α, and all but finitely many Q α (x 1 , x 2 ) are 0. We shall show that each Q α (x 1 , x 2 ) is either zero or has the form where M, r and n 0 , n 1 , . . . , n r are nonnegative integers naturally associated to α and λ/µ and We fix the skew partition λ/µ throughout the whole section. We will have a running example with λ = (7, 7, 7, 4, 4) and µ = (5, 3, 2).

Irreducible components
We recall that a 12-rpp means an rpp whose entries all belong to the set {1, 2}. Given a 12-rpp T, consider the set NR(T) of all cells (i, j) ∈ λ/µ such that T(i, j) = 1 but (i + 1, j) ∈ λ/µ and T(i + 1, j) = 2. (In other words, NR(T) is the set of all non-redundant cells in T which are filled with a 1 and which are not the lowest cells in their columns.) Clearly, NR(T) contains at most one cell from each column; thus, let us write NR(T) = {(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i s , j s )} with j 1 < j 2 < · · · < j s . Because T is a 12-rpp, it follows that the numbers i 1 , i 2 , . . . , i s decrease weakly, therefore they form a partition which we denoted seplist(T) := (i 1 , i 2 , . . . , i s ) in Section 5.1. This partition will be called the seplist-partition of T. An example of calculation of seplist(T) and NR(T) is illustrated on Figure 3.
We would like to answer the following question: for which partitions ν = (i 1 ≥ · · · ≥ i s > 0) does there exist a 12-rpp T of shape λ/µ such that seplist(T) = ν?
Until the end of Section 7, we make an assumption: namely, that the skew partition λ/µ is connected as a subgraph of Z 2 (where two nodes are connected if and only if their cells have an edge in common), and that it has no empty columns. This is a harmless assumption, since every skew partition λ/µ can be written as a disjoint union of such connected skew partitions and the corresponding seplist-partition splits into several independent parts, the polynomials g λ/µ get multiplied and the right hand side of (9) changes accordingly.
Note that these notions depend on the skew partition; thus, when we want to use a skew partition λ/µ rather than λ/µ, we will write that ν is nonrepresentable/irreducible/etc. with respect to λ/µ, and we denote the corre- These definitions can be motivated as follows. Suppose that a partition ν is non-representable, so there exist integers Recall that ν ⊆[a,b) =: (i r , i r+1 , . . . , i r+q ) contains all entries of ν whose support is a subset of [a, b). Thus in order for condition (10) to be true there must exist some integers j r < j r+1 < · · · < j r+q such that (i r , j r ), (i r + 1, j r ), . . . , (i r+q , j r+q ), (i r+q + 1, j r+q ) ∈ λ/µ. 10 Here and in the following, #κ denotes the length of a partition κ.
On the other hand, by the definition of the support, we must have j k ∈ supp(i k ) ⊆ [a, b) for all r ≤ k ≤ r + q. Therefore we get q It means that a non-representable partition ν is never a seplist-partition of a 12-rpp T.
Suppose now that a partition ν is reducible, so for some a < b we get an equality #ν ⊆[a,b) = b − a. Then these integers j r < · · · < j r+q should still all belong to [a, b) and there are exactly b − a of them, hence j r = a, j r+1 = a + 1, · · · , j r+q = a + q = b − 1. (11) Because supp(i r ) ⊆ [a, b) but supp(i r ) = ∅ (since ν is admissible), we have (i r , a − 1) / ∈ λ/µ. Thus, placing a 1 into (i r , a) and 2's into (i r + 1, a), (i r + 2, a), . . . does not put any restrictions on entries in columns 1, . . . , a − 1. And the same is true for columns b, b + 1, . . . when we place a 2 into (i r+q + 1, b − 1) and 1's into all cells above. Thus, if a partition ν is reducible, then the filling of columns a, a + 1, . . . , b − 1 is uniquely determined (by (11)), and the filling of the rest can be arbitrary -the problem of existence of a 12-rpp T such that seplist(T) = ν reduces to two smaller independent problems of the same kind (one for the columns 1, 2, . . . , a − 1, the other for the columns 11 b, b + 1, . . . , λ 1 ). One can continue this reduction process and end up with several independent irreducible components separated from each other by mixed columns. An illustration of this phenomenon can be seen on Figure 3: the columns 3 and 4 must be mixed for any 12-rpps T with seplist(T) = (4, 3, 3, 2).

(b)
The partition ν is the concatenation (where we regard a partition as a sequence of positive integers, with no trailing zeroes).
Proof. Part (a) has already been proven.
then i r appears in exactly one of the concatenated partitions, namely, ν ⊆[a k ,b k ) . Otherwise there is an integer k such that supp(i r ) ∩ [b k , a k+1 ) = ∅. It remains to show that such k is unique, that is, that supp(i r ) ∩ [b k+1 , a k+2 ) = ∅. Assume the contrary. The interval [a k+1 , b k+1 ) is nonempty, therefore there is an entry i of ν with supp(i) ⊆ [a k+1 , b k+1 ). It remains to note that we get a contradiction: we get two numbers i, i r with supp(i r ) being both to the left and to the right of supp(i).
(c) Fix k. Let J denote the restricted skew partition λ/µ [b k ,a k+1 ) , and let ν ′ = We are in one of the following four cases: • Case 1: We have c > b k (or k = 0) and d < a k+1 (or k = r). In this case, and we are done.
• Case 2: We have c = b k and k > 0 (but not d = a k+1 and k < r). Assume (for the sake of contradiction) that #ν ′ J ⊆[c,d) ≥ d − c. Then, the i p satisfying supp J (i p ) ⊆ [c, d) must satisfy supp(i p ) ⊆ [a k , d) (since otherwise, supp(i p ) would intersect both [b k−1 , a k ) and [b k , a k+1 ), something we have ruled out in the proof of (b)).
• Case 3: We have d = a k+1 and k < r (but not c = b k and k > 0). The argument here is analogous to Case 2.
• Case 4: Neither of the above. Exercise.
Definition 7.3. In the context of Lemma 7.2, for 0 ≤ k ≤ r the subpartitions ν ∩[b k ,a k+1 ) are called the irreducible components of ν and the nonnegative integers

The structural theorem and its applications
It is easy to see that for a 12-rpp T, the number #seplist(T) is equal to the number of mixed columns in T.
Proof of Corollary 7.7. The restriction map is injective (since, as we know, the entries of a T ∈ RPP 12 (λ/µ; ν) in any column outside of the irreducible components are uniquely determined) and surjective (as one can "glue" rpps together). Now use Theorem 7.5.
For a 12-rpp T, the vectors seplist(T) and ceq(T) uniquely determine each other: if (ceq(T)) i = h then seplist(T) contains exactly λ i+1 − µ i − h entries equal to i, and this correspondence is one-to-one. Therefore, the polynomials on both sides of (13) are equal to Q α (x 1 , x 2 ) where the vector α is the one that corresponds to ν.
Note that the polynomials P n (x 1 , x 2 ) are symmetric for all n. Since the question about the symmetry of g λ/µ can be reduced to the two-variable case, Corollary 7.7 gives an alternative proof of the symmetry of g λ/µ : Of course, our standing assumption that λ/µ is connected can be lifted here, because in general, g λ/µ is the product of the analogous power series corresponding to the connected components of λ/µ. So we have obtained a new proof of Theorem 3.3.
Another application of Theorem 7.5 is a complete description of Bender-Knuth involutions on rpps.
Take any 12-rpp T ∈ RPP 12 (λ/µ; ν) and recall that a 12-table flip(T) is obtained from T by simultaneously replacing all entries in 1-pure columns by 2 and all entries in 2-pure columns by 1.
Corollary 7.10. If ν is an irreducible partition, then, no matter in which order one resolves descents in flip(T), the resulting 12-rpp T ′ will be the same. The map T → T ′ is the unique Bender-Knuth involution on RPP 12 (λ/µ; ν).
Finally, notice that, for a general representable partition ν, descents in a 12table T with seplist(T) = ν may only occur inside each irreducible component independently. Thus, we conclude the chain of corollaries by stating that our constructed involutions are canonical in the following sense: Corollary 7.11. For a representable partition ν, the map B : RPP 12 (λ/µ; ν) → RPP 12 (λ/µ; ν) is the unique involution that interchanges the number of 1pure columns with the number of 2-pure columns inside each irreducible component.

The proof
Let ν = (i 1 , . . . , i s ) be an irreducible partition. We start with the following simple observation: Lemma 7.12. Let T ∈ RPP 12 (λ/µ; ν) for an irreducible partition ν. Then any 1-pure column of T is to the left of any 2-pure column of T.
Proof of Lemma 7.12. Suppose it is false and we have a 1-pure column to the right of a 2-pure column. Among all pairs (a, b) such that column a is 2-pure and column b is 1-pure, and b > a, consider the one with smallest b − a. Then, the columns a + 1, . . . , b − 1 must all be mixed. Therefore the set NR(T) contains {(i p+1 , a + 1), (i p+2 , a + 2), . . . , (i p+b−1−a , b − 1)} for some p ∈ N. And because a is 2-pure and b is 1-pure, each i p+k (for k = 1, . . . , b − 1 − a) must be ≤ to the y-coordinate of the highest cell in column a and > than the y-coordinate of the lowest cell in column b. Thus, the support of any i p+k for k = 1, . . . , b − 1 − a is a subset of [a + 1, b), which contradicts the irreducibility of ν.
Proof of Theorem 7.5. We proceed by strong induction on the number of columns in λ/µ. If the number of columns is 1, then the statement of Theorem 7.5 is obvious. Suppose that we have proven that for all skew partitions λ/µ with less than λ 1 columns and for all partitions ν irreducible with respect to λ/µ and for all 0 ≤ m ≤ λ 1 − # ν, there is exactly one 12-rpp T of shape λ/µ with exactly m 1-pure columns, exactly # ν mixed columns and exactly ( λ 1 − # ν − m) 2-pure columns. Now we want to prove the same for λ/µ. Take any 12-rpp T ∈ RPP 12 (λ/µ; ν) with seplist(T) = ν and with m 1-pure columns for 0 ≤ m ≤ λ 1 − #ν. Suppose first that m > 0. Then there is at least one 1-pure column in T. Let q ≥ 0 be such that the leftmost 1-pure column is column q + 1. Then by Lemma 7.12 the columns 1, 2, . . . , q are mixed. If q > 0 then the supports of i 1 , i 2 , . . . , i q are all contained inside [1, q + 1) and we get a contradiction with the irreducibility of ν. The only remaining case is that q = 0 and the first column of T is 1-pure. Let λ/µ denote λ/µ with the first column removed. Then ν is obviously admissible but may not be irreducible with respect to λ/µ, because it may happen that #ν λ/µ ⊆[2,b+1) = b − 1 for some b > 1. In this case we can remove these b − 1 nonempty columns from λ/µ and remove the first b − 1 entries from ν to get an irreducible partition again 12 , to which we can apply the induction hypothesis. We are done with the case m > 0. If m < λ 1 − #ν then we can apply a mirrored argument to the last column, and it remains to note that the cases m > 0 and m < λ 1 − #ν cover everything (since the irreducibility of ν shows that λ 1 − #ν > 0). This inductive proof shows the uniqueness of the 12-rpp with desired properties. Its existence follows from a parallel argument, using the observation that the first b − 1 columns of λ/µ can actually be filled in. This amounts to showing that for a representable ν, the set RPP 12 (λ/µ; ν) is non-empty in the case when λ 1 = #ν (so all columns of T ∈ RPP 12 (λ/µ; ν) must be mixed). This is clear when there is just one column, and the general case easily follows by induction on the number of columns 13 .