ON THE STRUCTURE OF THE HOPF REPRESENTATION

RING OF THE SYMMETRIC GROUPS

Essam Abotteen

Emporia State University

The main purpose of this paper is to prove a structure theorem of the graded Hopf representation ring of the symmetric groups R(S). We establish a Hopf ring isomorphism between R(S) and the graded polynomial Hopf ring in an infinite number of variables

C = Z[y_{1}, y_{2}, …, y_{k}, …] ,

by using the λ–operations in R(S) given in a previous paper [8] in terms of outer plethysms.

1. Introduction. In [8] λ–operations are introduced in the graded Hopf representation ring of the symmetric groups

R(S) = {R(S_{n}) : n ≥ 0}

in terms of outer plethysms and it has been shown that with respect to these operations R(S) is a special λ–ring. Zelevinsky [9] developed a complete structure theory of Hopf algebras satisfying the positivity and self adjointness, which is similar to the classical theory of Hopf algebras with commutative multiplication and comultiplication over a field of characteristic zero [7]. In this Hopf algebra approach, Zelevinsky showed R(S) is isomorphic to the polynomial Hopf ring

C = Z[y_{1}, y_{2}, …, y_{k}, …]

in an infinite number of variables over the integers. Following an elegant proof of Liulevicius [5] we are going to reproduce the Zelevinsky structure theorem in the context of the λ–ring structure in R(S).

Let S_{n} be the symmetric group of degree n. Let R(S_{n}) denote the Grothendieck representation group of S_{n}, then we have a graded group

R(S) = {R(S_{n}) : n ≥ 0}

by setting

(R(S))_{2n} = R(S_{n}) ,

and

(R(S))_{2n+1} = 0 ,

for all non–negative integers n; where

R(S_{0}) = Z .

A multiplication in R(S) is the map

m_{p,q} : R(S_{p}) ⊗ R(S_{q}) → R(S_{p+q}) ,

defined by

m_{p,q} = Ind_{S_{p}×S_{q}}^{S_{p+q}} ∘ Ψ_{p,q} ,

where

Ψ_{p,q} : R(S_{p}) ⊗ R(S_{q}) → R(S_{p} × S_{q})

is the canonical isomorphism, and

Ind_{S_{p}×S_{q}}^{S_{p+q}} : R(S_{p} × S_{q}) → R(S_{p+q}) ,

the map induced by the embedding of S_{p} × S_{q} as a subgroup of S_{p+q}. A comultiplication on R(S) is the map

△_{n} : R(S_{n}) → ∑_{p+q=n} R(S_{p}) ⊗ R(S_{q}) ,

defined by

△_{n} = ∑_{p+q=n} Ψ_{p,q}^{−1} ∘ Res_{S_{p}×S_{q}}^{S_{p+q}} .

In [6] and [9] the following result was proved:

\underline{Theorem 1.1}. R(S) is a graded Hopf ring with respect to the multiplication m_{p,q} and the comultiplication △_{n}.

2. Outer Plethysms and λ–Rings. Let S_{n} and S_{k} be symmetric groups of degree n and k respectively. The wreath product [S_{n}]S_{k} of S_{n} by S_{k} (the usual notation for [S_{n}]S_{k} is S_{n} ≀ S_{k}) is the set S_{n}^{k} × S_{k} with a multiplication defined by

(a_{1}, a_{2}, …, a_{k}; σ)(b_{1}, b_{2}, …, b_{k}; τ) = (a_{1}b_{σ^{−1}(1)}, …, a_{k}b_{σ^{−1}(k)}; στ) ,

where a_{i}, b_{i} ∈ S_{n} for k ≥ i ≥ 1 and σ, τ ∈ S_{k}. Clearly under this multiplication, [S_{n}]S_{k} is a group. Let M be an S_{n}–module and N be an S_{k}–module. Then M^{⊗_{k}} ⊗ N is an [S_{n}]S_{k}–module where the group action is given by:

(a_{1}, …, a_{k}; σ)(m_{1} ⊗ ⋯ ⊗ m_{k} ⊗ n) = (a_{1}m_{σ^{−1}(1)} ⊗ ⋯ ⊗ a_{k}m_{σ^{−1}(k)} ⊗ σn ),

where a_{i} ∈ S_{n}, σ ∈ S_{k}, m_{i} ∈ M and n ∈ N. In what follows ⊗ means ⊗_{C} and we interpret M^{⊗_{0}} as the 1–dimensional S_{n}–module C, on which S_{n} acts trivially. The map β : [S_{n}]S_{k} → S_{kn} given by

β((a_{1}, a_{2}, …, a_{k}; σ)) = \binom{(j − 1)n + i}{(σ(j) − 1)n + a_{σ(j)}(i)} ,

n ≥ i ≥ 1, k ≥ j ≥ 1, is a canonical embedding of [S_{n}]S_{k} into S_{kn}. Thus we have the induction homomorphism

β_{!} = Ind_{[S_{n}]S_{k}}^{S_{kn}} : R([S_{n}]S_{k}) → R(S_{kn}) .
\underline{Definition}. The \underline{outer plethysm} on R(S) is a map φ_{[M]} : R(S) → R(S) given by
φ_{[M]}([N]) = [Ind_{[S_{n}]S_{k}}^{S_{kn}}(M^{⊗_{k}} ⊗ N)] ,

where M is an S_{n}–module and N is an S_{k}–module.

In James–Kerber’s notation [3], the outer plethysm φ_{[M]}([N]) is denoted by [M] ⊙ [N]. In [8], this outer plethysm is used to construct λ–operations on R(S), and it is shown that with respect to these operations R(S) is a special λ–ring (see [1, 2, 4] for definitions and basic results about λ–rings). The λ–operations λ : R(S) → R(S) are defined for each non–negative integer k by

λ^{k}([M]) = [M] ⊙ [η_{k}] ,

where [M] ∈ R(S_{n}) and [η_{k}] ∈ R(S_{k}), is the trivial sign representation of S_{k}.

Zelevinsky [9] has shown that R(S) is a Hopf ring, which is isomorphic to the polynomial Hopf ring

C = Z[y_{1}, y_{2}, …, y_{k}, …]

in an infinite number of variables. Our goal in this paper is to establish a Hopf ring isomorphism between R(S) and C in terms of the λ–operations in R(S).

3. The Classical Hopf Algebra C. Let Z[t] be the graded polynomial algebra on one indeterminate t, where the grade of t is 2. Let Γ = Z[t]^{∗} be the graded dual of Z[t] with generators y_{k} defined by y_{k}(t^{k}) = 1. If △ is the comultiplication, then

△(y_{k}) = ∑_{i+j=k} y_{i} ⊗ y_{j} .

Let

C = Z[y_{1}, y_{2}, …, y_{k}, …] ,

with y_{0} = 1, and define a Hopf ring structure on C by making the inclusion i : Γ → C a homomorphism of corings.

The map i has the following universal property: if A is a graded associative algebra and φ : Γ → A is a homomorphism of graded abelian groups then there exists a unique homomorphism of graded algebras \overline{φ} : C → A such that the following diagram commutes

\begin{matrix}C \\ ↑ & i \\ Γ & ⟶_{φ} & A . \end{matrix}

Moreover if A is a Hopf algebra and φ is a map of coalgebras then \overline{φ} is a Hopf algebra homomorphism.

By taking the graded dual over Z we obtain a ring homomorphism i^{∗} : C^{∗} → Z[t], such that for any partition

π = {1^{π_{1}}, 2^{π_{2}}, …, k^{π_{k}}}

of k (in notation, π ⊢ k) we have

i^{∗}((y_{π})^{∗}) = \{ \begin{matrix}t^{k}, & if π = \{k\}; \\ 0, & otherwise, \end{matrix}

where

y_{π} = ∏_{i=1}^{k} y_{π_{i}} .

The map i^{∗} has the following universal property: if B is a graded coalgebra with each B_{n} a free abelian group of finite rank, then given a homomorphism of graded abelian groups Θ : B → Z[t] there exists a unique coalgebra homomorphism \overline{Θ} : B → C^{∗} such that the following diagram commutes

C^{∗}
↓_{i^{∗}}
Z[t] ⟵_{Θ} B .

Moreover if B is a Hopf algebra and Θ is an algebra homomorphism then \overline{Θ} is a Hopf algebra homomorphism.

Define a ring homomorphism h : Γ → Z[t] by setting h(y_{n}) = t^{n}. Then extend this to an algebra homomorphism which we also denote by h, h : C → Z[t].

Recall that to determine if a Hopf algebra homomorphism is a monomorphism it is sufficient to show that it is a monomorphism on the primitive elements, these are the elements x of a Hopf algebra with the property

△(x) = x ⊗ 1 + 1 ⊗ x ,

where △ is the comultiplication. The primitives of C have been studied by Newton (cf. [5]). The following is Theorem C of [5].

\underline{Theorem 3.1}. If R is a commutative ring with unit, then the primitives of R ⊗ C_{2n} are all of the form r ⊗ p_{n} where r ∈ R and p_{n} ∈ C_{2n} and they satisfy the following recursion relation:

p_{n} − y_{1}p_{n−1} + y_{2}p_{n−2} + ⋯ + (−1)^{n−1}y_{n−1}p_{1} + (−1)^{n}ny_{n} = 0 .

\underline{Lemma 3.2}. For each n, h(p_{n}) = (−1)^{n+1}t^{n}.

\underline{Proof}. It follows immediately from Newton’s recursion relation.

Now let h^{∗} : Γ → C^{∗} be the graded dual of the map h : C → Z[t]. Note that i^{∗} ∘ h^{∗} = h, and since h is an algebra homomorphism, h^{∗} is a coalgebra homomorphism, hence its extension h^{∗} : C → C^{∗} is a Hopf algebra homomorphism satisfying for each integer k ≥ 1 the following relation

h^{∗}(y_{k}) = ∑_{π⊢k}(y_{π})^{∗} .

\underline{Theorem 3.3}. The map h^{∗} : C → C^{∗} is a Hopf algebra isomorphism.

\underline{Proof}. First note that i^{∗} ∘h^{∗} = h, and since h(p_{n}) = (−1)^{n+1}t^{n}, then h^{∗} is a monomorphism of Hopf algebras. But since C and C^{∗} are free abelian groups of the same rank in each grading, then h^{∗} is an isomorphism.

4. A Hopf Ring Structure of R(S). In this section we are going to show how to use the λ–operations of R(S) to establish a Hopf algebra isomorphism between the graded Hopf representation ring R(S) and the Hopf ring

C = Z[y_{1}, y_{2}, …, y_{k}, …] .

First recall that the only primitive irreducible representation in R(S) is the trivial 1–dimensional representation of S_{1}, namely ρ_{1} = [1_{S_{1}}]. Define a map Λ : C → R(S), by

Λ(y_{k}) = λ^{k}(1_{S_{1}}) = [1_{S_{1}}] ⊙ [η_{k}] .

\underline{Theorem 4.1}. The map Λ : C → R(S) is a Hopf ring homomorphism.

\underline{Proof}. It is routine to verify that Λ is a ring homomorphism. Thus it remains to show Λ commutes with the comultiplication. Consider

△_{k}(Λ(y_{k})) = △_{k}([1_{S_{1}}] ⊙ [η_{k}])

= △_{k}([Ind_{[S_{1}]S_{k}}^{S_{k}}1_{S_{1}}^{⊗_{k}} ⊗ η_{k}])

= △_{k}([1_{S_{1}}^{⊗_{k}} ⊗ η_{k}])

= ∑_{p+q=k}(Ψ_{p,q}^{−1} ∘ Res_{S_{p}×S_{q}}^{S_{p+q}})([1_{S_{1}}^{⊗_{k}} ⊗ η_{k}])

= ∑_{p+q=k} Ψ_{p,q}^{−1}([Res_{S_{p}×S_{q}}^{S_{p+q}}(1_{S_{1}}^{⊗_{k}} ⊗ η_{k})])

= ∑_{p+q=k} Ψ_{p,q}^{−1}([(1_{S_{1}}^{⊗_{p}} ⊗ η_{p}) ⊗ (1_{S_{1}}^{⊗_{q}} ⊗ η_{q})])

= ∑_{p+q=k}[(1_{S_{1}}^{⊗_{p}} ⊗ η_{p})] ⊗ [(1_{S_{1}}^{⊗_{q}} ⊗ η_{q})]

= ∑_{p+q=k} Λ(y_{p}) ⊗ Λ(y_{q})

= (Λ ⊗ Λ)(△_{k}(y_{k})) .

Hence, Λ is a homomorphism of Hopf rings.

To prove that Λ is a Hopf ring isomorphism we need the following

\underline{Lemma 4.2}. The map Φ : R(S) → C^{∗} defined by

Φ([M])(y) =< [M], Λ(y) > ,

where [M] ∈ R(S), y ∈ C, and < ·, · > denotes the Schur inner product in R(S), is a ring homomorphism.

\underline{Proof}. Recall that the multiplication and comultiplication in R(S) are dual under the Schur inner product. If [M] ∈ R(S_{n}) and [N] ∈ R(S_{m}) and i : S_{n} × S_{m} → S_{n+m} is the inclusion map, then

< Ind_{S_{n}×S_{m}}^{S_{n+m}}([M] ⊗ [N]), Λ(y) >
=< [M].[N], Λ(y) >
=< [M] ⊗ [N], △(Λ(y)) >
=< [M] ⊗ [N], (Λ ⊗ Λ)(△(y)) >
= ∑ < [M], △(y′) >< [N], △(y′′) > ,

where

△(y) = ∑ y′ ⊗ y′′ .
Φ([M].[N])(y) =< [M].[N], Λ(y) >
= ∑ Φ([M])(y)Φ([N])(y′′)

= (Φ([M]) ⊗ Φ([N]))(∑ y′ ⊗ y′′)
= (Φ([M]) ⊗ Φ([N]))△(y)
= (Φ([M]).Φ([N]))(y) .

Hence Φ is a ring homomorphism.

\underline{Theorem 4.3}. The map of Hopf rings Λ : C → R(S) is an isomorphism.

\underline{Proof}. Consider the following

C
↓_{h^{∗}}
C^{∗} ⟵_{Φ} R(S) .

First we are going to show that the diagram commutes. Since Φ∘Λ is a ring homomorphism, it is sufficient to show that

(Φ ∘ Λ)(y_{k}) = h^{∗}(y_{k}) = ∑_{π⊢k} y_{π}^{∗} ,

for all k ≥ 1. For any y_{π} ∈ C, where

π = {π_{1}, π_{2}, …, π_{k}} ⊢ k

we have

(Φ ∘ Λ(y_{k}))(y_{π}) = Φ([1_{S_{1}}^{⊗_{k}} ⊗ η_{k}])(y_{π})

=< [1_{S_{1}}^{⊗_{k}} ⊗ η_{k}], Λ(y_{π}) >

= ∑_{i=1}^{k} < [1_{S_{1}}^{⊗_{i}} ⊗ η_{k}] ⊗ [1_{S_{1}^{⊗_{(k−i)}}} ⊗ η_{k−i}], Λ(y_{π}) >

= ∑_{i=1}^{k} < Λ(y_{i}) ⊗ Λ(y_{k−i}), Λ(y_{π}) >

= ∑_{i=1}^{k} < (Λ ⊗ Λ)(△(y_{k})), Λ(y_{π}) >

=< △(Λ(y_{k})), Λ(y_{π}) > =< Λ(y_{k}), m(Λ(y_{π})) >

= ∏_{i=1}^{k} < Λ(y_{k}), Λ(y_{π_{i}}) >

= δ_{kπ_{k}} = ∑_{k⊢k}(y_{π})^{∗} = h^{∗}(y_{k}) .

The third equality from the end is true because of the self–adjointness property in R(S). Hence the diagram commutes. To finish the proof note that since h^{∗} is an isomorphism, Φ is surjective. However, the rank of R(S_{k}) is the same as the rank of C_{2k}^{∗}. Hence Φ is an isomorphism. Thus Λ = Φ^{−1} ∘ h^{∗} is an isomorphism of Hopf rings. This completes the proof.

\underline{References}

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6. A. Liulevicius, “Finite G–Sets and Hopf Algebras of Representation Rings,” Proc. Sympos. on Algebraic Topology in Honor of Jose Adem, Contemporary Math., 12 (1982) Amer. Math. Soc., 183–199.

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8. H. Uehara, E. Abotteen, and M. W. Lee, “Outer Plethysms and λ–Rings,” Archive der Mathematik, 46 (1986), 216–224.

9. A. V. Zelevinsky, “Representations of Finite Classical Groups : A Hopf Algebra Approach,” Lecture Notes in Mathematics, No. 869, Berlin, Springer–Verlag, 1981.

↓ ↓
↓ ↓
\overline{φ} \overline{Θ}

Λ