The Hopf structure of symmetric group characters as symmetric functions

In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.


Introduction
In [26], the authors introduced a basis of the symmetric functions {s λ } that specialize to the characters of the irreducible modules of the symmetric group when the symmetric group S n is embedded in GL n as permutation matrices. This basis provides a new perspective on the representation theory of the symmetric group which is not well understood. In addition, several outstanding open problems in combinatorial representation theory are encoded in the linear algebra related to this basis.
One such problem is the restriction problem [9,16,19,24,29,30]. Since S n can be viewed as a subgroup of GL n via the embedding using permutation matrices, it follows that any irreducible polynomial module of GL n , with character given by the Schur function s λ , is a representation of S n via this restriction. Thus, the transition coefficients, r λ,µ , between the Schur basis {s λ } and the irreducible character basis {s µ } encode the decomposition of a polynomial, irreducible GL n -module into symmetric group irreducibles, i.e., s λ = µ r λ,µsµ .
In addition, we have shown in [26] that the structure coefficient of the {s λ } basis are the reduced (or stable) Kronecker coefficients,ḡ γ α,β [1,4,5,6,7,8,2,21,22,23]. The history of these problems indicates that they are difficult and are unlikely to be resolved in a single step. In order to make further progress, we need to develop some of the properties of character bases so that they can be treated as familiar objects in the ring of symmetric functions.
The combinatorics related to the irreducible character basis involves multiset partitions and multiset tableaux. Developing analogues of the notion of 'lattice' that exists for words and column strict tableaux would likely help find a combinatorial interpretations for (1) the transition coefficients, r λ,µ , from the Schur function s λ to the irreducible character basis; and (2) the structure coefficients of the irreducible character basis,ḡ γ α,β .
The Hopf algebra structure that we develop here lays out the combinatorics and indicates that a combinatorial interpretation for the stable Kronecker and restriction problems may exist using operations on multiset tableaux. In order to develop symmetric function expressions for irreducible characters and to develop their properties, the authors introduced in [26] an intermediate basish λ which represent the characters of the trivial module in S λ1 × S λ2 × · · · × S λ ℓ(λ) induced to S n . A third character basis,x λ , was introduced in [3] and it represents the character of the module S λ ⊗ S (n−|λ|) of S |λ| × S n−|λ| induced to S n , where S λ is the irreducible module indexed by λ and S (n−|λ|) is the trivial module. Assaf and Speyer used this basis to show that the Schur expansion of thes λ basis is sign alternating in degree. We show that the power sum expansion of all three of the character bases is given by expanding them in terms of two analogues of the power sum basis, p λ and p λ (see Equations (3)-(4)).
In [25] we gave combinatorial interpretations for some products involving thes λ basis. The purpose of this paper is to further develop algebraic properties of the character and related bases. The main results in this paper are product formulae for theh λ , p λ and p λ bases; as well as coproduct formulae for theh λ , p λ , p λ ,x λ ands λ bases. The coproduct formulae correspond to restriction of characters from S n to S r × S s with r + s = n.
In [26] we provided a combinatorial interpretation for the transition coefficients from the complete homogeneous basis to thes λ -basis in terms of multiset tableaux. In representation theory, this is equivalent to computing the multiplicities when we restrict the tensor product of symmetric tensors from GL n to S n . In this paper, we give a combinatorial interpretation for the expansion of the elementary basis in the irreducible character basis. This corresponds to finding a combinatorial interpretation of the multiplicities of the restriction of the antisymmetric tensor products from GL n to S n .

Notation and Preliminaries
The combinatorial objects that arise in our work are the classical building blocks: set, multiset, partition, set partition, multiset partition, composition, weak composition, tableau, words, etc. In this section we remind the reader about the definitions of these objects as well as establish notation and usual conventions that we will use in this paper.
A partition of a non-negative integer n is a weakly decreasing sequence λ = (λ 1 , λ 2 , . . . , λ ℓ ) such that |λ| = λ 1 + λ 2 + · · · + λ ℓ = n. We call n = |λ| the size of λ. The λ i 's are called the parts of the partition and the number of nonzero parts is called the length and it is often denoted ℓ(λ) = ℓ. We use the notation λ ⊢ n to mean that λ is a partition of n. We reserve λ and µ exclusively for partitions. We denote by m i (λ) the multiplicity of i in λ, that is, the number of times it occurs as a part in λ. Another useful notation for a partition is the exponential notation where m i = m i (λ) and λ = (1 m1 2 m2 · · · k m k ). With this notation the number of permutations with cycle structure λ ⊢ n is n! z λ where The most common operation we use is that of adding a part of size n to the beginning of a partition. This is denoted (n, λ). If n < λ 1 , this sequence will no longer be an integer partition and we will have to interpret the object appropriately. Similarly, if λ is non-empty, then let λ = (λ 2 , λ 3 , . . . , λ ℓ(λ) ).
For a partition λ the set {(i, j) : 1 ≤ i ≤ λ j , 1 ≤ j ≤ ℓ(λ)} are called the cells of λ and we represent these cells as stacks of boxes in the first quadrant with the largest part at the bottom following the 'French notation'. This graphical representation of the cells is called the Young diagram of λ. A tableau is a mapping from the set of cells to a set of labels. A tableau will be represented by filling the boxes of the diagram of a partition with the labels. In our case, we will encounter tableaux where only a subset of the cells are mapped to a label. The content of a tableau is the multiset obtained with the total number of occurrences of each number.
A multiset is a collection of elements such that repetitions are allowed, multisets are denoted by { {b 1 , b 2 , . . . , b r } }. Multisets will also be represented by exponential notation so that { {1 a1 , 2 a2 , . . . , ℓ a ℓ } } represents the multiset where the value i occurs a i times.
A set partition of a set S is a set of subsets {S 1 , S 2 , . . . , S ℓ } with S i ⊆ S for 1 ≤ i ≤ ℓ, S i ∩ S j = ∅ for 1 ≤ i < j ≤ ℓ and S 1 ∪ S 2 ∪ · · · ∪ S ℓ = S. A multiset partition π = { {S 1 , S 2 , . . . , S ℓ } } of a multiset S is a similar construction to a set partition, but now S i is a multiset, and it is possible that two multisets S i and S j have non-empty intersection (and may even be equal). The length of a multiset partition is the number of non-empty multisets in the partition and it is denoted by ℓ(π) = ℓ. We will use the notation π ⊢ ⊢S to indicate that π is a multiset partition of the multiset S.
We will usem(π) to represent the partition of ℓ(π) consisting of the multiplicities of the multisets which occur in π (e.g.m( For non-negative integers n and ℓ, a composition of size n is an ordered sequence of positive integers α = (α 1 , α 2 , . . . , α ℓ ) such that α 1 + α 2 + · · · + α ℓ = n. A weak composition is such a sequence with the condition that α i ≥ 0 (zeros are allowed). To indicate that α is a composition of n we will use the notation α |= n and to indicate that α is a weak composition of n we will use the notation α |= w n. For both compositions and weak compositions, ℓ(α) := ℓ.
2.1. The ring of symmetric functions. For some modern references on this subject see for example [20,27,33,18]. The ring of symmetric functions will be denoted Sym = Q[p 1 , p 2 , p 3 , . . .]. The p k are power sum generators and they will be thought of as functions which can be evaluated at values when appropriate by making the substitution p k → x k 1 + x k 2 + . . . + x k n but they will be used algebraically in this ring without reference to their variables. Thus the degree of each p k is k. It is well-known that the monomials in the power sums p λ := p λ1 p λ2 · · · p λ ℓ(λ) where λ ⊢ n span the subspace of the symmetric functions of degree n.
For any symmetric function f ∈ Sym, f is a linear combination of the power sum basis f = λ a λ p λ , then for any expression E(x 1 , x 2 , x 3 , . . .) in the set of variables x 1 , x 2 , x 3 , . . . (and potentially y 1 , y 2 , y 3 , . . . , z 1 , z 2 , z 3 , . . .) the notation f [E] will represent The advantage of this notation is that if X n = x 1 + x 2 + · · · + x n , then f [X n ] is a symmetric polynomial in n variables since p k [X n ] = x k 1 + x k 2 + · · · + x k n is the power sum symmetric polynomial. This notation is also practical because by defining X = x 1 + x 2 + · · · , then f [X] is a symmetric series in the set of variables x 1 , x 2 , x 3 , . . . such that if the variables x i = 0 for i > n, then the resulting expression is f [X n ]. For a given alphabet X, set Sym X to be the symmetric functions (or symmetric polynomials if X is finite) in the variables X and Sym X ≃ Sym as a graded Hopf algebra.
A Hopf algebra [35] is a vector space with both a product and a coproduct that form a bialgebra structure along with some additional relations including the existence of map known as an antipode. Hopf algebras arise in many areas of mathematics, but in algebraic combinatorics the symmetric functions as a graded bialgebra is one of the prototypes for what is known as a 'combinatorial Hopf algebra' [14].
The symmetric functions have a Hopf structure on it, [14,Chapter 2], and the coproduct operation can be encoded in operations of evaluating the symmetric function on sets of variables. The p k are primitive elements of Sym so that ∆(p k ) = p k ⊗ 1 , it follows that if we represent the coproduct of f in Sweedler notation, ∆(f ) = i f (i) ⊗f (i) , then it is possible to derive, by linearity relations calculated on the power sum basis. We will calculate formulas for the coproduct operation on the character bases in Section 4 and this symmetric function notation will be helpful in our proofs since Sym ⊗ Sym ≃ Sym X,Y as a graded Hopf algebra.
The standard bases of Sym (each indexed by the set of partitions λ) are power sum {p λ } λ , homogeneous/complete {h λ } λ , elementary {e λ } λ , monomial {m λ } λ , and Schur {s λ } λ . The Hall inner product is defined by declaring that the power sum basis is orthogonal, i.e., p λ z λ , p µ = δ λµ , where we use the notation δ λµ = 1 if λ = µ and 0 if λ = µ. Under this inner product the Schur functions are orthonormal and the monomial and homogeneous functions are dual, i.e., s λ , s µ = h λ , m µ = δ λµ . We use this scalar product to represent values of coefficients by taking scalar products with dual bases. In particular, an identity that we will repeatedly use for any pair of bases {a λ } λ and {b λ } λ such that a λ , b µ = δ λµ . The scalar product on Sym naturally extends to a scalar product on Sym X , Sym ⊗ Sym and Sym X,Y . The identities that we will use to calculate coproducts on the character symmetric functions in Section 4 can be found in [20,Section 5,.
We will also refer to the irreducible character of the symmetric group indexed by the partition λ and evaluated at a permutation of cycle structure µ as the coefficient s λ , p µ = χ λ (µ). For k > 0, define Ξ k := 1, e 2πi/k , e 4πi/k , . . . , e 2(k−1)πi/k as a symbol representing the eigenvalues of a permutation matrix of a k-cycle. Then for any partition µ, let be the multiset of eigenvalues of a permutation matrix with cycle structure µ. We will evaluate symmetric functions at these eigenvalues. The notation f [Ξ µ ] represents taking the element f ∈ Sym and replacing p k in f with x k 1 + x k 2 + · · · + x k |µ| and then replacing the variables x i with the values in Ξ µ . In particular, for r, k > 0, p r [Ξ k ] = 0 unless either r = 0 (in which case the value is 1) or k divides r (in which case p r [Ξ k ] = k).
From [26], a useful tool that we will repeatedly use to establish symmetric function identities is the following proposition.

2.2.
Symmetric group characters as symmetric functions. In this section, we will define five additional bases for the ring of symmetric functions and discuss their connections to the characters of the symmetric group. Four of these appeared in [26] and the fifth basis was introduced in [3].
The following expressions were developed from character polynomials and yield two analogues of the power sum basis. For each i, r > 0, define Then if a partition γ is expressed as γ = 1 m1(γ) 2 m2(γ) · · · ℓ m ℓ (γ) , we set Note that by Möbius inversion and Equation (3) we also have that Now define three more bases, We refer to the set of functionss λ as the irreducible character basis sinces λ [X n ] are symmetric polynomials which evaluate to the irreducible characters of the symmetric group in the same way that Schur functions evaluate to the irreducible characters of polynomial GL n modules. Assaf and Speyer [3] used the notation s † λ fors λ and called them 'stable Specht polynomials.' The foundations of these symmetric functions is likely found in earlier work than that of Specht [32] on the representation theory of the symmetric groups and go back to Frobenius [12] and Young [36].
The symmetric functionsx ν appear implicitly in [3] as [M ν ]. We refer to the set of functionsx λ as the induced irreducible character basis since, for n ≥ |λ| + λ 1 , it encodes the character of the module S λ ⊗ S (n−|λ|) ↑ Sn S |λ| ×S n−|λ| , where S λ is the irreducible module indexed by λ and S (n−|λ|) is the trivial module of S n−|λ| . As a symmetric function,x λ is a sum of irreducible character basis elements with nonnegative coefficients. We refer to the set of functionsh λ as the induced trivial character basis since theh λ [Ξ µ ] are values of trivial characters induced from Young subgroups S λ1 × · · · × S λr to the symmetric group S n . Theorem 14 and Equations (34) and (35) from [26] say that if λ is a partition and n is an integer such that n ≥ |λ| + λ 1 and µ is a partition of n, thens λ [Ξ µ ] = χ (n−|λ|,λ) (µ). Proposition 16 and Lemma 15 of [26] says thath λ [Ξ µ ] = h (n−|λ|,λ) , p µ if n ≥ |λ| and 0 otherwise. It is well-known that h µ , p γ = λ⊢|µ| K λµ s λ , p γ , where K λµ are the Kostka coefficients. Then, it follows that As Assaf and Speyer [3, Proposition 5] point out, thex λ ands λ bases are related by where the sum on the left (resp. sum on the right) is over partitions ν contained in λ such that λ/ν is a horizontal strip (resp. vertical strip). That is, if the Young diagram for ν is drawn inside the Young diagram for λ, then there is at most one cell per column (resp. row) in λ which is not in ν.
Thes λ form a basis of the symmetric functions with the property thats λ [X n ] is a symmetric polynomial (of inhomogeneous degree) that evaluates to the characters of the irreducible symmetric group modules S (n−|λ|,λ) . Then, the character symmetric functions are a means of encoding the characters of families of symmetric group modules which are representation stable (a notion formalized in [11]). They are equivalent to finitely generated F I-modules, a category introduced by Church, Ellenberg and Farb in [10] to capture the fundamental properties of representation theory stability.
Let F I be the category with objects [n] := {1, 2, . . . , n} and whose morphisms are injections [n] ֒→ [m]. An FI-module is a functor V from F I to the category of C-modules. By evaluation at a set [n], the functor V realizes a family of S n modules V ([n]). The character basis is a way of encoding the character of this family as a symmetric function.
In particular, Theorem 3.3.4 of [10] shows that that if V is a finitely generated F I-module, then there a polynomial P V (X 1 , X 2 , . . .) such that the character of a permutation σ (of a sufficiently large n) acting on the module V ([n]) is an evaluation of P V . This implies by Proposition 12 of [26] that there is a corresponding symmetric function f V such that if the permutation σ has cycle structure µ, then the character of σ acting on Summarizing Theorem 3.3.4 appearing in [10] and how they apply to the irreducible character basis, we have the following proposition.
is the character of V evaluated at a permutation σ of cycle structure µ ⊢ n acting on V ([n]).

2.3.
A scalar product on characters and the Frobenius map. In [25] we introduced a scalar product on symmetric functions (coming from the usual scalar product on irreducible characters of the symmetric group) for which the irreducible character basis was orthogonal. This scalar product was useful in the application presented in that paper because it allowed us to calculate an individual coefficient of an irreducible character in an expression. In this section we develop some of the properties of this scalar product further. For all n sufficiently large, Now the right hand side of this expression is independent of n. Thus, for a sufficiently large n and for any symmetric functions f and g, the expression ν⊢n is also independent of n by linearity since {s λ } is a basis.
Therefore we may define for n ≥ 2 max(deg(f ), deg(g)) and where cyc(σ) is a partition representing the cycle structure of σ ∈ S n . We use the @-symbol as a subscript of the right angle bracket to differentiate this scalar product from the usual scalar product where s λ , s µ = δ λµ . We can relate these scalar products by using the Frobenius map which is a linear isomorphism from the class functions of the symmetric group to the ring of symmetric functions. Since we know that characters of the symmetric group (and hence class functions) can be expressed as symmetric functions, we can define the Frobenius map or characteristic map on symmetric functions We have that φ n is a map from the ring of symmetric functions to the subspace of symmetric functions of degree n. And φ n has the property that for symmetric functions f and g, where * denotes the Kronecker (or internal) product of symmetric functions. Since if |ν| ≥ |λ| + λ 1 , then the image ofs λ ,x λ andh λ are

Remark 1.
A result of Solomon [31,Proposition 3.11], translated into the language of symmetric functions, shows that the irreducible character of the rook monoid algebra CR k (n) indexed by the partition λ (with |λ| ≤ k) at a partial permutation of cycle type µ (where µ is a partition with so then the symmetric functionsx λ are the irreducible character basis of the rook monoid in the same way that the symmetric functionss λ are the irreducible character basis of the symmetric group.
This leads to the following proposition.
Proof. For partitions λ and µ, take an n which is sufficiently large (take n ≥ 2 max(|λ|, |µ|)), then (n − |λ|, λ) and (n − |µ|, µ) are both partitions and this scalar product can easily be computed on the irreducible character basis by Since this calculation holds on a basis, Equation (10) holds for all symmetric functions f and g by linearity.
Thes λ symmetric functions are the orthonormal basis with respect to the scalar product ·, · @ . This basis is triangular with respect the Schur basis (s λ is equal to s λ plus terms of lower degree) and hence it may be calculated using Gram-Schmidt orthonormalization with respect to the @-scalar product.
Remark 2. Using the fact thats λ is an orthonormal basis and s λ = γ r λ,γsγ where r λ,γ are the multiplicities in the decomposition of a polynomial, irreducible GL n -module into symmetric group irreducibles. We have that s λ , s µ @ = γ r λ,γ r µ,γ which is a positive integer.
The analogue p λ of power sums are orthogonal with respect to the @-scalar product in the same way that the power sums are orthogonal with respect to the Hall scalar product. By the definition, Equation (4), (11) p λ = µ⊢|λ| p λ , s µ s µ .
Proposition 4. For all partitions λ and µ, Proof. By Equation (11), This expression is equal to 0 if |λ| is not equal to |µ|. If they are equal, the right hand side is equal to p λ , p µ = z λ δ λµ .
One of the main motivations for introducing these bases and developing their properties is that it will hopefully lead to a combinatorial interpretation for these coefficients.
In [25] we studied several combinatorial formulae for coefficients of repeated products of character bases in terms of multiset tableaux satisfying a lattice condition. In particular, we gave a combinatorial interpretation for the coefficient ofs ν in products of the form where γ is any partition and λ i 's are positive integers. Notice that these products contain the Pieri rule as a special case. Unfortunately, we do not have the means to extend this to a combinatorial interpretation for the coefficientsḡ γ α,β . In order to gain a better understanding we develop product formulae for other bases in this section and coproduct formulae in Section 4.
3.1. Products on the power sum bases. In contrast with the power symmetric function p λ that is multiplicative, the new bases {p γ } and {p γ } are not multiplicative unless the parts of the partition are disjoint. Therefore, in this section we begin by describing the structure of products of the form p a b p a c and p a b p a c .
Recall that the k-falling factorial, (x) k , has the following product structure Therefore we can at least easily compute the following product of one of the power sum bases from this formula.
Proposition 5. For a positive integers i, r and s, If m i (λ) = 0, then p i r p λ = p (i r )∪λ .
It also implies by making a change of basis back and forth between p i r to p i s that we have the following expansion for the product of the p i r elements.

Products of induced trivial characters.
A combinatorial interpretation for the Kronecker product of two complete symmetric functions expanded in the complete basis is listed as Exercise 23 (e) in section I.7 of [20] and Exercise 7.84 (b) in [33]. The earliest reference to this result that we are aware of is due to Garsia and Remmel [13]. The formula for this Kronecker product is h Mij summed over all matrices M of non-negative integers with ℓ(λ) rows, ℓ(µ) columns and row sums are given by the vectors λ and column sums by the vector µ. For λ, µ, ν ⊢ n, we define d λµν as the coefficient of h ν in h λ * h µ . Let λ, µ and ν be partitions, then for n ≥ |λ| + |µ| + |ν|, let d λµν := d (n−|λ|,λ)(n−|µ|,µ)(n−|ν|,ν) denote the common (stable) coefficient of h (n−|ν|,ν) in h (n−|λ|,λ) * h (n−|µ|,µ) . Proof. For each partition γ such that |γ| ≥ n (n sufficiently large), calculation in terms of characters shows thath We can conclude by Proposition 1 that the structure coefficients of the induced trivial characters are the coefficients d λµν .
Let S be a multiset and T a set. The restriction of S to T is the multiset S| T = { {v ∈ S : v ∈ T } }. Then we can define the restriction of a multiset partition to the content T by π| T = { {S| T : S ∈ π} }. If necessary in this operation we throw away empty multisets in π| T . We will use the notation π#τ to represent a set of multiset partitions that will appear in the product. Let π and τ be multiset partitions on disjoint sets S and T .
That is, θ ∈ π#τ means that θ is of the form We propose the following different, but equivalent combinatorial interpretation for this product of the induced trivial character basis. Proof. We define a bijection between matrices whose row sums are (n − |λ|, λ) and whose column sums are (n − |µ|, µ) and elements of π#τ where π is a multiset partition such thatm(π) is λ and τ is a multiset partition such thatm(τ ) = µ.
Let M be such a matrix. The first row of this matrix has sum equal to n − |λ| and the sum of row i of this matrix represents the number of times that some multiset A repeats in π (it does not matter what that multiset is, just that it repeats j M ij times). The sum of column j of this matrix (for j > 1) represents the number of times that a particular part of the multiset B repeats in τ (again, it does not matter the content of that multiset, just that it is different than the others). Therefore the entry M ij is the number of times that A ∪ B repeats in the multiset θ ∈ π#τ . The value of M i1 is equal to the number of times that A appears in θ and the value of M 1j is the number of times that B appears in θ.

Example 2.
To ensure that the bijection described in the proof is clear we show the correspondence between some specific multiset partitions and the non-negative integer matrices to which they correspond. The second and third row will represent the multiplicities of { {1} } and { {2} } respectively. The second and third column will represent the multiplicities of { {3} } and { {4} } respectively. Rather than consider all multiset partitions, we will consider only the 4 that we calculated in the last example that havem(π) = 1111. The multiset partition notation is therefore not significantly different than the integer matrices notation, but there are distinct advantages to an interpretation in terms of multiset partitions. The main one is that the notion of multisets in the context of symmetric functions leads to the combinatorial objects of multiset tableaux that can be used as a possible object to keep track of stable Kronecker coefficients.

Coproducts of the character bases
The coproduct of symmetric functions corresponds to restriction from S n to S r × S t where r + t = n. This is because the coproduct operation is isomorphic to the operation of replacing one set of variables with two in the power sum symmetric function p k [X] → p k [X] + p k [Y ] and the evaluation of the symmetric function at the eigenvalues of S r × S t in S n replaces the X variables by eigenvalues of the element of S r and the Y variables by the eigenvalues of the element of S t . Therefore, formulae involving coproducts of character basis compute multiplicities for these restrictions. The main result of this section are coproduct formulae for the analogues of the power sum bases, theh,s andx bases.
For a basis {b λ } of the symmetric functions, we will refer to the coproduct structure coefficients as the coefficients of b µ ⊗b ν in ∆(b λ ) where λ, µ, ν are all partitions. For bases that are of homogeneous degree we have the restriction that |µ| + |ν| = |λ|, but for the bases here it could be the case that |µ| + |ν| ≤ |λ|. To summarize the results in this section we state the following theorem.

Coproducts on the power sum bases.
In [26] we derived an analog of the Murnaghan-Nakayama rule, see Theorem 20. This allows us to expand the power symmetric function in the irreducible character basis. This is currently the most efficient method for computing the irreducible character basis. For a partition λ, it follows from the defining relations in Equation (4) that Now this power sum analogue is important because of the following proposition. Proof. We know thath λ [Ξ γ ] = h (|γ|−|µ|) h µ , p γ (and in particular the expression is 0 if |γ| < |µ|) by Equation (6) of [26]. By Equation (16), by an application of Equation (2) and the right hand side of this expression is equal to h (m−|µ|) p λ by a second application of Equation (2).
We wish to understand as completely as possible the coproduct structure on these power sum bases. Since the coproduct is an algebra homomorphism, it suffices to understand the result on the partitions of the form (i r ).
Proposition 11. The p λ basis has the same coproduct as the power sum basis. That is, if the coefficients a µ γτ are defined by those appearing in the equation ∆(p µ ) = γ,τ a µ γτ p γ ⊗ p τ then ∆(p µ ) = γ,τ a µ γτ p γ ⊗ p τ and specifically Proof. Define the coefficients b λ αβ appearing in the coproduct of the complete symmetric function basis by In Proposition 13, we will show by direct computation that ∆(h λ ) = α,β b λ αβh α ⊗h β . We know already that the coefficient of p γ in h λ is equal to the coefficient of p γ inh λ (and both are equal to h λ , p γ /z γ by Equation (4)). Therefore the coefficient of p γ ⊗ p τ in ∆(p λ ) is equal to Now if we compute the coefficient of p γ ⊗ p τ in ∆(p λ ) it is precisely the same as the coefficient of In particular, Equation (17) holds because Now for the coproduct on the other power basis p λ we need to work slightly harder to give the expression.
Proof. Applying Equation (3) and (17), we simplify the limits of the expression to show that r k = 0 if k > r. Then we see that where in the last equality we have just rearranged the order of the summations. The sum over a and b can be combined by setting d = a + b and changing the sum over a ≥ 0, b ≥ 0 to one of d ≥ 0, a ≥ 0. Moreover, since ℓ a = 0 if ℓ < a, then we can assume that ℓ ≥ a and hence Now by expanding the binomial coefficients we know that Note that by taking the coefficient of This reduces Equation (19) to The last equality follows by taking the coefficient of

4.2.
Coproducts on the induced trivial and induced irreducible character basis. The induced trivial character basis follows the combinatorics of multiset partitions for the product. We now give the coproduct for this basis. It turns out that the coefficient ofh Hence the coproduct formula for the induced trivial character basis can be stated in the following proposition.
Proposition 13. For a partition λ, where the sum is over all pairs of weak compositions α and β of length λ whose vector sum is equal to λ.
Proof. We will show thath λ [X + Y ] = α+β=λh α [X]h β [Y ]. This will be done by applying Proposition 1 that states if f [Ξ µ ] = g[Ξ µ ] for enough partitions µ, then f = g as symmetric functions. We note that for partitions µ and ν and a positive integer n = |µ| + |ν|, Now the evaluation of the, where in the last expression all of the terms in sum are assumed to be 0 unless k = |ν| − |β|. In this case, n − k − |λ| = |µ| + |ν| − (|ν| − |β|) − |λ| = |µ| − |α|. Now we can recognize the terms in the left entry of the scalar product as those that arise as the coproduct formula on the complete basis element h (n−|λ|,λ) . If we define f ⊗ f ′ , g ⊗ g ′ = f, g f ′ , g ′ then we know (see for instance [20] I.5 p.92 example 25) that ∆(f ), g ⊗ h = f, gh . Hence we have that Equation (21) is equivalent to Now from Proposition 1 we can conclude thath λ [X + Ξ ν ] = α+β=λh α [X]h β [Ξ ν ] as a symmetric function identity and a second application allows us to conclude thath Because the relationship between theh λ basis and thex λ basis is the same as the relationship between the h λ and s λ basis, we can conclude that the coproduct rule on the induced irreducible character basis will be the same as that for the Schur basis.
Corollary 14. For a partition λ, where the coefficients c λ µν are the Littlewood-Richardson coefficients, the same coproduct structure coefficients for the Schur basis.

4.3.
Coproducts on the irreducible character basis. The method that we used in the last section to derive coproduct formula can also be used to derive the coproduct for the irreducible character basis. In this case though we reverse the expression and expands λ [X + Y ] evaluated at X = Ξ µ and Y = Ξ ν . where c λ δβ is the Littlewood-Richardson coefficient and the inner sum is over partitions η such that the skew partition β/η is a horizontal strip.
Proof. Assume that µ ⊢ N where N is "sufficiently large". We can without loss of generality assume that N is larger than 2|λ| because we will apply Proposition 1. We need to show that the following identity holds for all partitions µ, such that |µ| > n for some n which is at least as large the degree of the symmetric function.
Remark 3. The antipode of a Hopf algebra is part of its defining structure. In the case of the symmetric functions, the antipode is an involution S : Sym → Sym where S(s λ ) = (−1) |λ| s λ ′ . The result of Assaf-Speyer [3] implies (−1) |λ| S(s λ ) will be Schur positive and hence this expression expands positively in the irreducible character basis. A better understanding of the transition coefficients between the Schur basis and the irreducible character basis could be used to give a more precise formula for this expression.

Expansion of the elementary symmetric functions in the irreducible character basis
In [26] we gave the expansion of the complete symmetric function h µ in terms of the irreducible character basiss λ . In this section we give thes λ expansion of an elementary symmetric function e µ . The proof is similar to the proof of the irreducible character expansion of a complete symmetric function, see [26] Theorem 9. We will start by proving the expansion in the irreducible character basis except that we will reserve some of the detailed combinatorial calculations for last. We assume that there is a total order on the sets that appear in set partitions and create tableaux to keep track of the terms in the symmetric function expansion. The order chosen does not matter, all the matters is that there is a total order, in the examples we have chosen lexicographic order. Let shape(T ) be a partition representing the shape of a tableau T and we again use the overline notation on a partition to represent the partition with the first part removed, λ = (λ 2 , λ 3 , . . . , λ ℓ λ ).
Theorem 16. For a partition µ, where the sum is over tableaux that are of skew shape ν/(ν 2 ) for some partition ν and that are weakly increasing in rows and columns with non-empty sets as labels of the tableaux (not multisets, that is no repeated values are allowed) such that the content of the tableau is { {1 λ1 , 2 λ2 , . . . , ℓ λ ℓ } }. A set is allowed to appear multiple times in the same column if and only if the set has an odd number of entries. A set is allowed to appear multiple times in the same row if and only if the set has an even number of entries.
Proof. In Corollary 21 we will show that wherem e (π) (resp.m o (π)) is the partition representing the multiplicities of the sets in π with an even (resp. odd) number of elements (see Example 7). For the rest of this proof, we assume that the reader is familiar with the Pieri rules, [33,Chapter 7,, which state h r s λ is the sum of terms s µ where the Young diagram for µ differs from the Young diagram of λ by adding cells that may occur in the same row, but not in the same column of the diagram. Similarly, e r s λ is the sum of terms s µ where the Young diagram for µ differs from the Young diagram of λ by adding cells that may occur in the same column, but not in the same row of the diagram. Now order the occurrences of the generators in the product h n hm e(π) em o(π) so that they are in the same order as the total order that is chosen for the sets that appear in the set partitions. Using a tableau to keep track of the terms in the resulting Schur expansion of the product we will have that h n hm e(π) em o(π) = T s shape(T ) where the sum is over tableaux that have n blank cells in the first row and sets as labels in the rest of the tableau. Because we multiply by a generator h r if a set with an even number of entries occurs r times, then those sets with an even number of elements can appear multiple times in the same row, but not in the same column of the tableau. Similarly, because we multiply by a generator e r if a set with an odd number of entries occurs r times, then those sets with an odd number of elements can appear multiple times in the same column, but not in the same row of the tableau. If the shape of the tableau is of skew shape ν/(n) with n ≥ ν 2 , there are the same number of tableaux of skew-shape (ν 2 + |ν|, ν)/(ν 2 ) because there is a bijection by deleting blank cells in the first row.
Therefore we have where the sum is over those tableau described in the statement of the proposition. Our proposition now follows from Proposition 1.  In [18, Lemma 5.10.1], Lascoux showed the following result. This result will serve as the starting point for our computations. In the following expressions, the notation r|n indicates shorthand for "r divides n." In addition, for n > 0, (28) h n [Ξ r ] = δ r|n , p n [Ξ r ] = rδ r|n , e n [Ξ r ] = (−1) r−1 δ r=n .
We will need the evaluation and a combinatorial interpretation of e λ [Ξ µ ] in order to make a connection with character symmetric functions.
Note that we are using the convention that if i γ (i) ∪ i ν (i) is not a subset of the parts of µ then the weight is equal to 0.
Proposition 19. For partitions λ, τ and µ, let F µ λ,τ be the fillings of the diagram for the partition µ with λ i labels i and τ j labels j ′ such that all cells in a row are filled with the same label such that cells in any row are either all filled or all empty. For F ∈ F µ λ,τ , the weight of the filling, wt(F ) is equal to −1 raised to the number of cells filled with primed labels plus the number of rows occupied by the primed labels. Then Proof. This is precisely the analogous statement to Proposition 28 of [26]. It follows because if we fix the sequences of partitions γ ( * ) and ν ( * ) such that γ (i) ⊢ λ i and ν (j) ⊢ τ j , then the quantity µ1 i=1 m i (µ) m i (γ (1) ), . . . , m i (γ (ℓ(λ)) ), m i (ν (1) ), . . . , m i (ν (ℓ(τ )) ) is precisely the number of F in F µ λ,τ with the rows filled according to the sequences of partitions γ ( * ) and ν ( * ) . The sign of a filling is constant on this set and is equal to sgn(ν ( * ) ). Example 6. The following are all the possible fillings of the diagram (3, 3, 2, 2, 1, 1) with two 1's and two 1 ′ 's such that the rows have the same labels.
Since the weight of the filling is equal to the (−1) raised to the number of cells plus the number of rows occupied by primed entries, the first four have weight −1 and the last two have weight 1 and hence H E (2|2),332211 = −2 .
The rest of this section develops the combinatorial constructions required to show that the evaluations of e λ at roots of unity are correct.
Example 8. Consider the set P (3,1),(3,3,2,2,1,1) that consists of the following 12 pairs of set partitions and fillings The first four of these pairs have weight +1 and the remaining eight have weight −1.
Next we define a set T λ,µ , this set is defined in a similar way as P λ,µ , with the main difference that now the tableaux will contain the sets that make up the parts of π.
Definition 3. For partitions λ and µ let T λ,µ be the fillings of some of the cells of the diagram of the partition µ with subsets of {1, 2, . . . , ℓ(λ)} such that the total content of the filling is { {1 λ1 , 2 λ2 , . . . , ℓ(λ) λ ℓ(λ) } } and such that all cells in the same row have the same subset of entries. We will define the weight of one of these fillings to be −1 to the power of the size of λ plus the number of rows whose cells are occupied by a set of odd size (this is also equal to the number of cells plus the number of rows occupied by the sets of odd size). The fillings listed above are in the same order as their isomorphism with the set of pairs P (3,1),(3,3,2,2,1,1) from Example 8. As in that case we have that, the first four of these pairs have weight +1 and the remaining eight have weight −1.
The following result should be clear from the definitions listed above and the examples we have presented.
Lemma 20. There is a bijection between the sets P λ,µ and T λ,µ and C λ,µ that preserves the weight.