Antipode formulas for some combinatorial Hopf algebras

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.


Introduction
A Hopf algebra is a structure that is both an associative algebra with unit and a coassociative coalgebra with counit. The algebra and coalgebra structures are compatible, which makes it a bialgebra. To be a Hopf algebra, a bialgebra must have a special anti-endomorphism called the antipode, which must satisfy certain properties.
Hopf algebras arise naturally in combinatorics. Notably, the symmetric functions (Sym), quasisymmetric functions (QSym), noncommutative symmetric functions (NSym), and the Malvenuto-Reutenauer algebra of permutations (MR) are Hopf algebras, which can be arranged as in the following diagram. where we take the isomorphisms sending a ⊗ k to ak and k ⊗ a to ka.
The first diagram tells us that m is an associative product and the second that η(1 k ) = 1 A .
Definition 2.2. A co-associative coalgebra C is a k-vector space with k-linear map ∆ : C → C⊗C (the coproduct) and a counit : C → k such that the following diagrams commute: The diagram on the left indicates that ∆ is co-associative. Note that these are the same diagrams as in the Definition 2.1 with all of the arrows reversed.
It is often useful to think of the product as a way to combine two elements of an algebra and to think of the coproduct as a sum over ways to split a coalgebra element into two pieces. When discussing formulas involving ∆, we will use Sweedler notation as shown below: This is a common convention that will greatly simplify our notation.
Example 2.3. To illustrate the concepts just defined, we give the example of the shuffle algebra, which is both an algebra and coalgebra. Let I be an alphabet andĪ be the set of words on I. We declare that words on I form a k-basis for the shuffle algebra.
Given two words a = a 1 a 2 · · · a t and b = b 1 b 2 · · · b n inĪ, define their product, m(a ⊗ b), to be the shuffle product of a and b. That is, m(a ⊗ b) is the sum of all t+n n ways to interlace the two words while maintaining the relative order of the letters in each word. For example, m(a 1 a 2 ⊗ b 1 ) = a 1 a 2 b 1 + a 1 b 1 a 2 + b 1 a 1 a 2 .
We may then extend by linearity. It is not hard to see that this multiplication is associative.
The unit map for the shuffle algebra is defined by η(1 k ) = ∅, where ∅ is the empty word. Note that m(a ⊗ ∅) = m(∅ ⊗ a) = a for any word a.
For a word a = a 1 a 2 · · · a t inĪ, we define ∆(a) = t i=0 a 1 a 2 · · · a i ⊗ a i+1 a i+2 · · · a t and call this the cut coproduct of a. For example, given a word a = a 1 a 2 , ∆(a) = ∅ ⊗ a 1 a 2 + a 1 ⊗ a 2 + a 1 a 2 ⊗ ∅.
The counit map is defined by letting take the coefficient of the empty word. Hence for any nonempty a ∈Ī, (a) = 0.

2.2.
Morphisms and bialgebras. The next step in defining a Hopf algebra is to define a bialgebra. For this, we need a notion of compatibility of maps of an algebra (m, η) and maps of a coalgebra (∆, ). With this as our motivation, we introduce the following definitions.
Definition 2.4. If A and B are k-algebras with multiplication m A and m B and unit maps η A and η B , respectively, then a k-linear map f : Definition 2.5. Given k-coalgebras C and D with comultiplication and counit ∆ C , C , ∆ D , and d , k-linear map g : C → D is a coalgebra morphism if ∆ D • g = (g ⊗ g) • ∆ C and D • g = C .
Given two k-algebras A and B, their tensor product A ⊗ B is also a k-algebra with m A⊗B defined to be the composite of The unit map in A ⊗ B, η A⊗B , is given by the composite Similarly, given two coalgebras C and D, their tensor product C ⊗ D is a coalgebra with ∆ C⊗D the composite of and the counit A⊗B is the composite Definition 2.6. Given A that is both a k-algebra and a k-coalgebra, we call A a k-bialgebra if (∆, ) are morphisms for the algebra structure (m, η) or equivalently, if (m, η) are morphisms for the coalgebra structure (∆, ).
Example 2.7. The shuffle algebra is a bialgebra. We can see, for example, that This is evidence that the coproduct, ∆, is an algebra morphism.
2.3. The antipode map. A Hopf algebra is a bialgebra equipped with an additional map called the antipode map. On our way to defining the antipode map, we must first introduce an algebra structure on k-linear algebra maps that take coalgebras to algebras. Definition 2.8. Given coalgebra C and algebra A, we form an associative algebra structure on the set of k-linear maps from C to A, Hom k (C, A), called the convolution algebra as follows: for f and g in Hom k (C, A), define the product, f * g, by Note that η • is the two-sided identity element for * . We can easily see this in the shuffle algebra from Example 2.7 if we remember that (η • )(a) = η(0) = 0 for all words a = ∅. Let c be a word in the shuffle algebra, then because c 1 = c when c 2 = ∅ and c 2 = c when c 1 = ∅.
If we have a bialgebra A, then we can consider this convolution structure to be on End k (A) := Hom k (A, A).
In other words, the endomorphism S is the two-sided inverse for the identity map id A under the convolution product. Equivalently, if ∆(a) = a 1 ⊗ a 2 , (S * id A )(a) = S(a 1 )a 2 = η( (a)) = a 1 S(a 2 ) = (id A * S).
Because we have an associative algebra, this means that if an antipode exists, then it is unique.
Example 2.10. In the shuffle algebra, we define the antipode of a word by S(a 1 a 2 · · · a t ) = (−1) t a t a t−1 · · · a 2 a 1 and extend by linearity. We can see an example of the defining property by computing a 1 a 2 )).
We end this section with two useful properties that we use in later sections. The first is a well-known property of the antipode map for any Hopf algebra.
Proposition 2.11. Let S be the antipode map for Hopf algebra A. Then S is an algebra antiendomorphism: S(1) = 1, and S(ab) = S(b)S(a) for all a, b in A.
The second property allows us to translate antipode formulas between certain Hopf algebras. Lemma 2.12. Suppose we have two bialgebra bases, {A λ } and {B µ }, that are dual under a pairing and such that the structure constants for the product of the first basis are the structure constants for the coproduct of the second basis and vice versa. In other words, A λ , B µ = δ λ,µ , by assumption.

The Hopf algebra of multi-quasisymmetric functions
The multi-quasisymmetric functions (mQSym) are a K-theoretic analogue of the Hopf algebra of quasisymmetric functions (QSym), which was introduced by Gessel [3] and stemmed from work of Stanely [11]. An understanding of QSym is useful for understanding mQSym. We recommend [5], [9], and [10] for exposition on QSym and its Hopf algebra structure.
In what follows, we say that a set {A λ } continuously spans space A if everything in A can be written as a (possibly infinite) linear combination of A λ 's. Here, we assume that {A λ } comes with a natural filtration and that each filtered component is finite. Then we may talk about continuous span with respect to the topology induced by the filtration. A continuous basis for A allows elements to be written as arbitrary linear combinations of the basis elements. We say that a linear function f : A → A is continuous if it respects arbitrary linear combinations of elements in A.
3.1. (P, θ)-set-valued partitions. Following [5], we define mQSym, the Hopf algebra of multiquasisymmetric functions, by defining the continuous basis of multi-fundamental quasisymmetric functions,L α . We start with a finite poset P with n elements and a bijective labeling θ : P → [n]. LetP denote the set of nonempty, finite subsets of the positive integers. If a ∈P and b ∈P are two such subsets, we say that We next define the (P, θ)-set-valued partitions. The definition is almost identical to that of the more well-known (P, θ)-set partitions except that σ now assigns a nonempty, finite subset of positive integers to each element of the poset instead of assigning a single positive integer.
Definition 3.1. Let (P, θ) be a poset with a bijective labeling. A (P, θ)-set-valued partition is a map σ : P →P such that for each covering relation s t in P , Example 3.2. The diagram on the left shows an example of a poset P with a bijective labeling θ. We identify elements of P with their labeling. The diagram on the right shows a valid (P, θ)set-valued partition σ. Note that since 3 < 2 in the poset, we must have the strict inequality max(σ(3)) = 6 < min(σ(2)) = 30.
Given a composition α of n, we write w α to denote any permutation in S n with C(w α ) = α.
We may now define the multi-fundamental quasisymmetric functionL α indexed by composition α. Definition 3.3. Let P be a finite chain p 1 < p 2 < . . . < p k , w ∈ S k a permutation, and C(w) = α the composition of n associated to the descent set of w. We label P using w with θ(p i ) = w i . ThenL It is easy to see thatK (P,w) depends only on α. Note that this is an infinite sum of unbounded degree. The sum of the lowest degree terms inL α gives L α , the fundamental quasisymmetric function in QSym.
x 100 x 101 + . . . , an infinite sum of unbounded degree. Definition 3.5. Given a poset P with n elements, a linear multi-extension of P by [N ] is a map e : P → 2 [N ] for N ≥ n such that is in e(x) for exactly one x ∈ P , and (3) no set e(x) contains both i and i + 1 for any i.
Note that elements inJ N (P, θ) are m-permutations of [n] with N letters, where we define an m-permutation of [n] to be a word in the alphabet 1, 2, . . . , n such that no two consecutive letters are equal.  The following result is proven in [5] by giving an explicit weight-preserving bijection betweeñ A(P, θ) and the set of pairs (w, σ ) where w ∈J N (P, θ) and σ ∈Ã(C, w), where C = (c 1 < c 2 < . . . < c r ) is a chain with r elements. One can easily recover this bijection from the bijection given in the proof of Theorem 5.4 by restricting toÃ(P, θ).
Theorem 3.7 ([5], Theorem 5.6). We can writẽ We now describe how to expressL α as an infinite linear combination of L α 's, where L α is the fundamental quasisymmetric function in QSym. Let L

3.3.
Hopf structure. Next we describe the bialgebra structure of mQSym using the continuous basis of multi-fundamental quasisymmetric functions. The first step is to define the multishuffle of two words in a fixed alphabet. To that end, we give the following definition.
Definition 3.9. Let a = a 1 a 2 · · · a k be a word. We call w = w 1 w 2 · · · w r a multiword of a if there exists non-decreasing, surjective map t : [k] → [r] such that w j = a t(j) .
As an example, consider the permutation 1342 as a word in N. Then 11333422 and 1342 are both multiwords of 1342, while 34442 and 1133244 are not multiwords of 1342. Definition 3.10. Let a = a 1 a 2 · · · a k and b = b 1 b 2 · · · b n be words with distinct letters. We say that w = w 1 w 2 · · · w m is a multishuffle of a and b if the following conditions are satisfied: Eventually, we would like to multishuffle two permutations, which will not have distinct letters. To remedy this, given a permutation w = w 1 w 2 · · · w k , define w[n] = (w 1 + n)(w 2 + n) · · · (w k + n) to be the word obtained by adding n to each digit entry of w. For example, for w = 21, w[4] = 65.
Starting with permutations u = 1342 and w = 21, we see that v = 161613346252 is a multishuffle of u = 1342 and w [4] = 65, where we shift w by 4 since 4 is the largest letter in u.
If we restrict to the letters in u, v| u = 11133422 is a multiword of u, and similarly v| w [4] = 6665 is a multiword of w [4]. Note that this is an infinite sum whose lowest degree terms are exactly those of L α L β , the product of the two corresponding fundamental quasisymmetric functions.
To define the coproduct, we need the following definition.
Example 3.14. Let α = (1) and β = (2, 1) with w α = 1 and w β = 231. Theñ where the terms listed correspond to the multishuffles 1342, 3142, 3412, 3421, 13421, 131421, and 3414212 of w α and w β [1]. We also compute We give a combinatorial formula for the antipode map in mQSym in Theorem 4.9. In Section 5, we give an antipode map in terms of a new basis introduced within the section.

The Hopf algebra of Multi-noncommutative symmetric functions
The Hopf algebra of noncommutative symmetric functions (NSym) is dual to that of quasisymmetric functions. We next describe a K-theotric analogue called the Multi-noncommutative symmetric functions or MNSym. We recall its bialgebra structure as given in [5] and develop a combinatorial formula for its antipode map. 4.1. Multi-noncommutative ribbon functions and bialgebra strucure. MNSym has a basis {R α } of Multi-noncommutative ribbon functions indexed by compositions, which is an analogue to the basis of noncommutative ribbon functions {R α } for NSym. There is a bijection between compositions and ribbon diagrams sending α = (α 1 , . . . , α k ) to the skew diagram λ/µ with k rows where row k − i has α i+1 squares and there is exactly one column of overlap between adjacent rows. Thinking of {R α } as being indexed by ribbon diagrams will be useful.
We first introduce a product structure on {R α } as given in [5].
It is helpful to think of the product using ribbon diagrams. From the statement above, we haveR (2,3,2) . In pictures, this is In contrast to the product in mQSym, the product in MNSym is a finite sum whose highest degree terms are those of the corresponding product R α R β in NSym.
Proposition 4.4. The coproduct of a basis element is Note that since multishuffles of w β and w δ [i] may not have adjacent letters that are equal, we may define the descent set of a multishuffle of w β and w δ [i] in the usual way.
Example 4.5. In general, computing the coproduct in MNSym is not an easy task. However, for compositions with only one part, we have ∆(R (n) ) =R (n) ⊗ 1 +R (n−1) ⊗R (1) +R (n−2) ⊗R (2) + . . .R (1) ⊗R (n−1) + 1 ⊗R (n) because the only way a multishuffle of two permutations results in an increasing sequence is for it to be the concatenation of two increasing permutations. We use this fact in the proof of the antipode in MNSym.

4.2.
Antipode map for MNSym. Suppose we have a ribbon shape corresponding to α, a composition of n. We say that ribbon shape β is a merging of ribbon shape α if we can obtain shape β from shape α by merging pairs of boxes that share an edge. The order in which the pairs are merged does not matter, only set of boxes that were merged. Let M α,β be the number of ways to obtain shape β from shape α by merging. We will label each box in the ribbon shape to keep track of our actions.
Proof. We prove this by induction on the number of parts of the composition α.
for all k < n. Then, using Example 4.5 and Definition 2.9, we see that There are five types of terms that show up in this sum.  (4)R (1 k ) , where k ≤ n. The coefficient of this term is parts, and let β = (β 1 , β 2 , . . . , β k ) be a composition with k parts. We know that and so In the image below, let the thin rectangle represent all mergings of ω(β k ) and the square represent all mergings of ω(β 1 , . . . , β k−1 ). Then the image labeled (1) represents all mergings obtained by adding the last part of a merging of ω(β k ) to the first part of a merging of ω(β 1 , . . . , β k−1 ). The image labeled (2) represents all mergings obtained by merging the topmost box in a merging of ω(β k ) with the bottom leftmost box of a merging of ω(β 1 , . . . , β k ). These two mergings with multiplicities are exactly the shapes we want in S(R β ).
The imaged labeled (3) represents all mergings obtained by concatinating a merging of ω(β k ) with a merging of ω(β 1 , . . . , β k−1 ). We do not want these mergings to appear in S(R β ) because it is impossible for boxes that are side by side in ω(β) to be stacked one on top of the other in a merging of ω(β).
(1)  Note that while S(R α ) is a finite sum of Multi-noncommutative ribbon functions for any α, S(L α ) is an infinite sum of multi-fundamental quasisymmetric functions for any α. Since any arbitrary linear combination of multi-fundamental quasisymmetric functions is in mQSym, this is an admissible antipode formula.

(P, θ)-multiset-valued partitions.
To create a new basis for mQSym, which will be useful in finding antipode formulas, we extend the definition of a (P, θ)-set-valued partition to what we call a (P, θ)-multiset-valued partition in the natural way. In a (P, θ)-multiset-valued partition σ, we allow σ(p) to be a finite multiset ofP, keeping all other definitions the same. An example of a (P, θ)-multiset-valued partition is shown below. Now defineÂ(P, θ) to be the set of all (P, θ)-multiset-valued partitions. For each element i ∈ P, let σ −1 (i) be the multiset {x ∈ P |i = σ(x)}. In the example shown above, Using this multiset analogue of our definitions, we definê where P = p 1 < . . . < p k is a finite linear order and w ∈ S k .
Proof. Using Proposition 5.1, We have an analogue of Stanley's Fundamental Theorem of P-partitions for our new basis of L α 's. The proof of this result follows closely that of Theorem 3.7 given in [5]. Proof. We prove this result by giving an explicit weight-preserving bijection betweenÂ(P, θ) and the set of pairs (w, σ ) where w ∈J N (P, θ) and σ ∈Â(C, w) where C = (c 1 < c 2 < · · · < c l ) is a chain with l = (w) elements. Let σ ∈Â(P, θ). For each i, let σ −1 (i) denote the submultiset of [n] via θ, and let w (i) σ denote the word of length |σ −1 (i)| obtained by writing the elements of σ −1 (i) in increasing order. Note that it is possible for w j+1 . This will occur when the letter i appears more than once in some σ(s) for s ∈ P .
Let w denote the unique m-permutation such that w σ := w σ · · · is a multiword of w and t : (w σ ) → (w) be the associated function as in Definition 3.9. We know that w σ is a finite word because σ (−1) (r) = ∅ for sufficiently large r. Note that w σ uses all letters [n]. Now define σ ∈Â(C, w) by σ (c i ) = {r k | r ∈ P and w (r) σ contributes k letters to w σ | t −1 (i) } where w σ | t −1 (i) is the set of letters in w σ at the positions in the interval t −1 (i). We will show that this defines a map α : σ → (w, σ ) with the required properties.
First, w is the multi-permutation associated to the linear multi-extension e w of P by (w) defined by the condition that e w (x) contains j if and only if w j = θ(x). It follows from the definition that this e w : P → 2 [1, (w)] is a linear multi-extension. To check that σ is a multisetvalued (C, w) partition, we note that σ (c i ) ≤ σ (c i+1 ) because the function t is non-decreasing. Moreover, if w i > w i+1 , then σ (c i ) < σ (c i+1 ) because each w (r) σ is increasing. We define the inverse map β : (w, σ ) → σ by the formula The (P, θ)-multiset-valued partition σ respects θ because e w is a linear multi-extension. Thus if x < y in P and θ(x) > θ(y), then σ(x) < σ(y) since e w (x) < e w (y) and there is a descent in w between the corresponding entries of θ(x) and θ(y).
Then β • α = id follows immediately. For α • β = id, consider a subset σ (c j ) ⊂ σ(x). One checks that this subset gives rise to |σ (c j )| consecutive letters all equal to θ(x) in w σ and that this is a maximal set of consecutive repeated letters. This shows that one can recover σ . To see that w is recovered correctly, one notes that if σ (c j ) and σ (c j+1 ) contain the same letter r then w j < w j+1 so by definition w j is placed correctly before w j+1 in w To obtain the inverse map, β, read w and σ in parallel and place σ (c i ) into cell θ −1 s (w i ). For example, we put {1, 1, 2} into the cell labeled 3, and we put {2, 3} into the cell labeled 4.
The linear multi-extension, e w in this example can be represented by the filling below. Proof. Using Theorem 3.7 and the antipode in QSym, we see that =L ω(α) (−x 1 , −x 2 , . . .).

The Hopf algebra of multi-symmetric functions
We next describe the space of multi-symmetric functions, mSym. We refer the reader to [5] for details. As in previous sections, familiarity with the Hopf structure of the ring of symmetric functions, Sym, is helpful. We refer the reader to [10] for background on Sym.
6.1. Set-valued tableaux. Let P be the poset of squares in the Young diagram of a partition λ = (λ 1 , λ 2 , . . . , λ t ) and θ s be the bijective labeling of P obtained from labeling P in row reading order, i.e. from left to right the bottom row of λ is labeled 1, 2, . . . , λ t , the next row up is labeled λ t + 1, . . . , λ t + λ t−1 and so on. Note that K λ,θs is the Schur function s λ . We define mSym = λ ZK λ,θs to be the subspace of mQSym continuously spanned by theK λ,θs , where λ varies over all partitions. From this point forward, we will writeK λ in place ofK λ,θs call a (λ/µ, θ s )-set-valued partition a set-valued tableau of shape λ/µ. Example 6.1. For λ = (2, 1), we haveK λ = x 2 1 x 2 + 2x 1 x 2 x 3 + x 2 1 x 2 2 + 3x 2 1 x 2 x 3 + 8x 1 x 2 x 3 x 4 + . . ., corresponding to the following labeled poset: 2 3 1 6.2. Basis of stable Grothendieck polynomials. We next introduce another (continuous) basis for mSym, the stable Grothendieck polynomials. Stable Grothendieck polynomials originated from the Grothendieck polynomials of Lascoux and Schützenberger [6], which served as representatives of K-theory classes of structure sheaves of Schubert varieties. Through the work of Fomin and Kirillov [2] and Buch [1], the stable Grothendieck polynomials, a limit of the Grothendieck polynomials, were discovered and given the combinatorial interpretation in the theorem below. These symmetric functions play the role of Schur functions in the K-theory of Grassmannians.
Theorem 6.2 ([1], Theorem 3.1). The stable Grothendieck polynomial G λ/µ (x) is given by the formula where the sum is taken over all set-valued tableaux of shape λ/µ.
The stable Grothendieck polynomials are related to theK λ bỹ Remark 6.3. In [1], Buch studied a bialgebra Γ = ⊕ λ ZG λ spanned by the set of stable Grothendieck polynomials. Note that the bialgebra Γ is not the same as mSym. In particular, the antipode formula given in Theorem 8.2 is valid in mSym but not in Γ as only finite linear combinations of stable Grothendieck polynomials are allowed in Γ.
6.3. Weak set-valued tableaux. The following definition is needed to introduce one final basis for mSym, {J λ }.
Definition 6.4. A weak set-valued tableau T of shape λ/ν is a filling of the boxes of the skew shape λ/ν with finite, non-empty multisets of positive integers so that (1) the largest number in each box is stricty smaller than the smallest number in the box directly to the right of it, and (2) the largest number in each box is less than or equal to the smallest number in the box directly below it.
In other words, we fill the boxes with multisets so that rows are strictly increasing and columns are weakly increasing. For example, the filling of shape (3, 2, 1) shown below gives a weak set-valued tableau, T , of weight  where we sum over all reverse plane partitions of shape λ. For a skew shape λ/µ, we may define g λ /µ analogously, summing over reverse plane partitions of shape λ/µ. Example 7.2. We use the definition of g λ to compute
Using results from Section 5, the following lemma will allow us to easily prove results regarding the antipode map in mSym. Proof. We know from Theorem 3.7 that Recall that in Sym, S(s λ ) = (−1) |λ| ω(s λ ), so one may expect similar behavior fromK λ and G λ . Indeed, we obtain the theorem below.
Theorem 8.2. In mSym, the antipode map acts as follows.
(a) S( Proof. For the first assertion, we have that And for the second assertion, By Lemma 2.12, we immediately have the following results in MSym. Next, we work toward expanding S(G λ ) and S(j λ ) in terms of {G µ } and {j µ }, respectively. We introduce two theorems of Lenart as well as the notion of a hook-restricted plane partitions.
Given partitions λ and µ with µ ⊂ λ, define an elegant filling of the skew shape λ/µ to be a semistandard filling such that the numbers in row i lie in [1, i − 1]. Now let f µ λ denote the number of elegant fillings of λ/µ for µ ⊂ λ and set f µ λ = 0 otherwise. Theorem 8.4 ([7], Theorem 2.7). For a partition λ, we have where f λ µ is the number of elegant fillings of λ/µ. For the second theorem, let r λµ be the number of elegant fillings of λ/µ such that both rows and columns are strictly increasing. We will refer to such fillings as strictly elegant. Theorem 8.5 ([7], Theorem 2.2). We can expand the stable Grothendieck polynomial G λ in terms of Schur functions as follows Given two partitions, λ and µ, we now define the number P µ λ . First, P µ λ = 0 if µ λ, and P µ λ = 1 if λ = µ. If µ ⊂ λ, then P µ λ is equal to the number of hook restricted plane partitions of the skew shape λ/µ. A hook restricted plane partition is a filling of the boxes of λ/µ with positive integers such that the numbers are weakly decreasing along rows and columns with the following restrictions.
( Example 8.6. The diagram on the left shows h(b) for each box b in the shape (5, 5, 5)/(4, 2) and is also an example of a hook restricted plane partition on (5, 5, 5)/(4, 2). The diagram on the right shows another hook restricted plane partition on (5, 5, 5)/(4, 2).  Proof. We will focus on part (a), and part (b) will follow from Lemma 2.12. From Theorem 8.2, we know that so it remains to expand J λ in terms of stable Grothendieck polynomials. From Theorem 8.5, it easily follows that we can writẽ Applying ω to both sides, we haveJ λ = µ⊃λ r λµ s µ t .
Now we can use Theorem 8.4 to writeJ Thus the coefficient of G ν inJ λ is µ such that µ⊃λ and µ t ⊂ν We describe a bijection between partitions of shape ν t which contain some µ ⊃ λ such that the filling of µ/λ is stricly elegant and boxes in ν t /µ are filled such that the transpose is an elegant filling of ν/µ t and hook-restricted plane partitions of ν/λ t . Note that if we have a hook-restricted plane partition of ν t /λ, then its transpose is a hook-restricted plane partition of ν/λ t .
We first define a map φ from pairs consisting of a strictly elegant filling and the traspose of an elegant filling to a hook restricted plane partitions. Suppose we have such a filling of shape ν t and some µ with λ ⊂ µ ⊂ ν t . For any box b in ν t , let d(b) denote the southwest to northeast diagonal that contains box b. If box b is in row i and column j, then d(b) = i + j − 1. Let c(b) denote the column that contains box b and e b denote the integer in box b. To obtain a hook-restricted plane partition follow these steps: (1) if box b is in µ, fill the corresponding box in the hook-restriced plane partition with φ(b) = d(b) − e b , and (2) if box b is in ν t /µ, fill the corresponding box in the hook-restricted plane partition with φ(b) = c(b) − e b . It is easy to see that the parts of the hook-restricted plane partition correponding to shape µ and to ν t /µ are weakly decreasing in rows and columns. We now check that entries are weakly decreasing along the seams. If box b is in µ, then If box a is in ν t /µ, then 1 ≤ φ(a) = e a ≤ j a − 1, so 1 ≤ c(a) − e a = j a − e a ≤ j a − 1. If b and a are adjacent, then j b ≤ j a , so φ(b) ≥ φ(a).
Next, we check that for all boxes in ν t /λ that are adjacent to λ, φ(b) ∈ [1, h(b)], so the resulting filling is indeed a hook-restricted plane partition. Let box b be in µ in row i and column j with l boxes above b contributing to h(b) and k boxes to the left of b contributing to h(b). · · · · · · b · · Next suppose box b described above is in ν t /µ. Because the transpose of the filling of ν t /µ is an elegent filling, e b ≥ j − k. Then we have that φ(b) = c(b) − e b ≤ k ≤ k + l = h(b). Note that since rows and columns of the image of φ are weakly decreasing, we have shown that φ(b) ∈ [1, h(b)] for all boxes b.
Beginning with a hook-restricted plane partition of ν t /µ, we define a map, ψ, to recover µ and the fillings of µ/λ and ν t /µ as follows. If the integer in the box in row i and column j is greater than or equal to j, then that box is in µ and ψ(b) = d(b) − e b . Note that since e b ≥ j, ψ(b) = (i + j − 1) − e b ≤ i − 1, as is required to be strictly elegant. If the entry is less than j, that box is in ν t /µ, and ψ(b) = c(b) − e b . Note here that e b ≤ j implies that ψ(b) = c(b) − e b ≤ j − 1, which is necessary to have an elegant filling. It is easy to see that rows and columns in µ will be strictly increasing in the image of ψ and that in ν t /µ, rows will be stricly increasing and columns will be weakly increasing. Thus the image of ψ is a strictly elegant filling of µ ⊃ λ and an elegant filling of ν/µ t . Clearly the composition of φ and ψ is the identity, so they are indeed inverses.
Note that the antipode applied to G λ gives an infinite sum of stable Grothendieck polynomials (see Remark 6.3) while applying S toj λ can be written as a finite sum ofj's. This implies that while the space spanned by stable Grothendieck polynomials, Γ, is not a Hopf algebra, the space spanned byj's is a Hopf algebra. Example 8.9. To illustrate the bijection described above, consider λ = (3, 2, 1), µ = (3, 3, 2, 2), and ν t = (5,4,4,3). The figure on the left is a filling such that µ/λ is strictly elegant and the transpose of ν t /µ is elegant. The entries in µ/λ are in bold. The figure on the right is the corresponding hook-restricted plane partition of ν t /λ.