A NOTE ON THE λ-STRUCTURE ON THE BURNSIDE RING

SERGE BOUC AND KARL RÖKAEUS

Abstract. Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a λ-ring, {λ^{n}}_{n∈ℕ}. However, a priori λ^{n}(S), where S is a G-set, can only be computed recursively, by first computing λ^{1}(S), …, λ^{n−1}(S). In this paper we establish an explicit formula, expressing λ^{n}(S) as a linear combination of classes of G-sets.

1. Introduction

We use B(G) to denote the Burnside ring of the finite group G. Recall that, as an abelian group, B(G) is free on {[S]}_{S∈R}, where R is a set of representatives of the isomorphism classes of transitive G-sets, and that its rank equals the number of conjugacy classes of subgroups of G. When f is a function on B(G) and S is a G-set, we write f(S) for f([S]).

There is a λ-structure on B(G), {λ^{n}}_{n∈ℕ}, defined as the opposite structure^{1} of {σ^{n}}_{n∈ℕ}, where σ^{n}(S) is the class of the nth symmetric power of S. It should be considered the natural λ-structure on B(G), one reason for this being that there is a canonical homomorphism to the ring of rational representations of G, h: B(G) → R_{ℚ}(G), defined by h(S) = [ℚ[S]], and the given λ-structure on B(G) makes h into a λ-homomorphism. (Note however that this λ-structure is non-special.)

The implicit nature of the definition of the λ-structure on B(G) makes it hard to compute with. The main result of this paper is a closed formula for λ^{n}(S), where S is any G-set. To state it, we use the following notation: let μ = (μ_{1}, …, μ_{l}) ⊢ n, i.e., μ is a partition of n. We use ℓ(μ) := l to denote the length of μ, and if μ = (1^{α_{1}}, 2^{α_{2}}, …), we define the tuple α(μ) := (α_{1}, …, α_{l′}), and write \binom{ℓ(μ)}{α(μ)} for \frac{l!}{α_{1}!···α_{l′}!}. Using this notation we can express λ^{n}(S), for any G-set S, as a linear combination of classes of G-sets:

Theorem 1.1. Let n be a positive integer and let μ = (μ_{1}, …, μ_{l}) ⊢ n. For S a G-set, let P_{μ}(S) be the G-set consisting of ℓ(μ)-tuples of pairwise disjoint subsets of S, where the first one has cardinality μ_{1}, and so on. Then

(1.2)  λ^{n}(S) = (−1)^{n} ∑_{μ⊢n}(−1)^{ℓ(μ)}\binom{ℓ(μ)}{α(μ)}[P_{μ}(S)] ∈ B(G).

In particular, λ^{n}(S) = 0 when n > |S|.

This result was first stated and proved by the second author in an earlier version of this paper (preprint [Rök07b]), showing that λ^{n}(S) lies in a subring of B(G) on which h is injective, and then that the image of (1.2) in R_{ℚ}(G) is satisfied. The proof given in the present version is more intrinsic, using only the structure of Burnside rings. It relies on the construction of a ring of formal power series with coefficients in Burnside rings, and an exponential map on this ring, developed in [Bou92]. Using this framework, the proof reduces to some explicit combinatorial computations. In Section 2 we give a survey of the relevant constructions and results from [Bou92]. Then in Section 3 we use these results to obtain a formula for λ^{n}(S), which we then show to be the requested one using some combinatorial arguments.

Date: August 31, 2009.

2000 Mathematics Subject Classification. Primary 19A22; Secondary 20B05.

1Let {σ^{n}}_{n∈ℕ} be a λ-structure on the ring R and define σ_{t}(x) := ∑_{i≥0} σ^{i}(x)t^{i} ∈ R[[t]]. The λ-structure opposite to {σ^{n}} is defined by σ_{t}(x)·λ_{−t}(x) = 1 ∈ R[[t]], where λ_{t}(x) := ∑_{i≥0} λ^{i}(x)t^{i}.

Theorem 1.1 originates in the paper [Rök07a], in which the second author computes the classes of certain tori in the Grothendieck ring of varieties, in terms of the λ-structure on that ring. This formula is suggested by the corresponding class of the cohomology of the torus, and its proof uses a map from the Burnside ring of the absolute Galois group of the base field, where formula (1.2) can be applied. Actually, it was these computations that led the second author conjecture Theorem 1.1.

Acknowledgment. The second author is grateful to Professor Torsten Ekedahl for valuable discussions and suggestions concerning his investigation of the Burnside ring.

2. Background Material

An introduction to λ-rings, representation rings and the Burnside ring is given in [Knu73]. The standard reference for λ-rings is [AT69]. We now give a quick review of some definitions and results:

2.1. Posets. A G-poset P is a G-set with a partial ordering compatible with the G-action, in the sense that if s ≤ t ∈ P then gs ≤ gt for all g ∈ G. A G-map of G-posets is a map f: P → Q of posets such that gf(s) = f(gs) for s ∈ P and g ∈ G. If also f′: P → Q, then f ≤ f′ if this holds pointwise. In connection with this, when S is a G-set and we use it as a G-poset this means that we view S as a G-poset using its discrete ordering.

Let P be a G-poset. We recall the definition of the Lefschetz invariant of P: for every i ∈ ℕ, Sd_{i} P is the G-set of chains x_{0} < ⋯ < x_{i} in P of length i + 1. The Lefschetz invariant of the G-poset P, Λ_{P}, is the alternating sum ∑_{i≥0}(−1)^{i}[Sd_{i} P] ∈ B(G). The reduced Lefschetz invariant of P is Λ̃_{P} := Λ_{P} −1.

We also need the notion of homotopic posets. We say that the G-posets P and Q are simplicially homotopic, or just homotopic^{2}, if there are G-maps f: P → Q and g: Q → P such that gf ≤ Id_{P} or gf ≥ Id_{P}, and similarly for fg. if P and Q are homotopic as G-posets then Λ̃_{P} = Λ̃_{Q} (see e.g. Proposition 4.2.5 in [Bou00]). In particular, if P has a largest or smallest element then Λ̃_{P} = 0.

2.2. Results from [Bou92]. In this subsection we give a review of the definitions and results from [Bou92] that we use to prove Theorem 1.1. We use G_{n} to denote the wreath product of G with Σ_{n}, G_{n} := G≀Σ_{n} (by definition, G_{0} = 1). One defines the ring 𝔹(G) in the folling way: as a group it is the direct product of the Burnside rings B(G ≀ Σ_{n}), indexed over all n ∈ ℕ. We represent the elements of this group as a power series, ∑_{i≥0} x_{i}t^{i} where x_{i} ∈ B(G ≀ Σ_{i}). This is a ring in a natural way, see [Bou92] for the construction of the multiplication.

Let g̃ = ((g_{1}, …, g_{n}), σ), where σ ∈ Σ_{n} and g_{i} ∈ G, be an element of G_{n}. When S is a G-set we view S^{n} as a G_{n}-set by g̃(s_{1}, …, s_{n}) = (g_{1}s_{σ^{−1}1}, …, g_{n}s_{σ^{−1}n}). Moreover, let \underline{S} be the poset defined by adding a smallest element 0 to S, and define the G_{n}-poset S^{∗n} as the set of maps \{1, …, n\} → \underline{S} which are not constant equal to zero, where the partial ordering is defined pointwise, and with the G_{n}- action defined in the same way as on S^{n}, with G acting trivially on the minimal element 0.

2Note however that two non-homotopic posets may admit homotopic realizations.

Next one defines maps u_{i}: B(G) → B(G_{i}) by x ↦ Λ_{P^{i}}, where P is a G-poset such that Λ_{P} = x. Let 𝕀(G) be the ideal of 𝔹(G) consisting of those series with zero as constant coefficient. The u_{i} are then used to define an exponential map Exp: 𝕀(G) → 𝔹(G) having the property that if f, g ∈ 𝕀(G) then Exp(f + g) = Exp(f) Exp(g). In the case we are interested in, when f = xt for x ∈ B(G), we have by definition Exp(xt) = ∑_{i≥0} u_{i}(x)t^{i}. (We omit the construction in the general case, see [Bou92].) In particular, when H is a subgroup of G we have Exp([G/H]t) = ∑_{i≥0}[G_{i}/H_{i}]t^{i}. Moreover, since, for any G-set S, S = Λ_{S} = Λ̃_{S_{+}},

where S_{+} := S ∪^{▪}{•}, we have Exp(−[S]t) = ∑_{i≥0} u_{i}(− Λ̃_{S_{+}})t^{i}. By Lemme 4 in [Bou92] it follows that

(2.1)  Exp(−[S]t) = − ∑_{i≥0} Λ̃_{(S_{+})^{∗i}} t^{i}.

For every i ∈ ℕ we have a map m_{i}: B(G_{i}) → B(G), induced by taking the G_{i}-set S to the G-set Σ_{i}\S. Together the m_{i} give a homomorphism of rings m: 𝔹(G) → B(G)[[t]].

3. Proof of Theorem 1.1

The property that allows us to use the above theory on our problem is the following:

Lemma 3.1. For any x ∈ B(G) we have m(Exp(xt)) = σ_{t}(x) and m(Exp(−xt)) = λ_{−t}(x).

Proof. Let S^{n} denote the nth symmetric power. When x = [G/H], we have to show that Σ_{n}\(G_{n}/H_{n}) ≃ S^{n}(G/H) as G-sets, for every positive integer n: firstly, the map

(g_{1}, …, g_{n}, σ) ↦ (\overline{g_{1}}, …, \overline{g_{n}}): G_{n} → (G/H)^{n}

factors through G_{n}/H_{n}, for if (g_{1}, …, g_{n}, σ) ∈ G_{n} then, for any (h_{1}, …, h_{n}, τ) ∈ H_{n}, the element

(g_{1}, …, g_{n}, σ)(h_{1}, …, h_{n}, τ) = (g_{1}h_{σ1}, …, g_{n}h_{σn}, στ) ∈ G_{n}

maps to (\overline{g_{1}h_{σ1}}, …, \overline{g_{n}h_{σn}}) = (\overline{g_{1}}, …, \overline{g_{n}}) ∈ (G/H)^{n}, which is also the image of (g_{1}, …, g_{n}, σ). Denote the resulting map ϕ: G_{n}/H_{n} → (G/H)^{n}. If we give (G/H)^{n} the G_{n}-set structure (g_{1}, …, g_{n}, σ) · (\overline{f_{1}}, …, \overline{f_{n}}) = (\overline{g_{1}f_{σ1}}, …, \overline{g_{n}f_{σn}}), then ϕ is G_{n}-equivariant. Moreover it is surjective. Since both G_{n}/H_{n} and (G/H)^{n} have |G|^{n}/|H|^{n} elements, it follows that ϕ is an isomorphism of G_{n}-sets. Consequently it induces an isomorphism of G-sets Σ_{n}\(G_{n}/H_{n}) → S^{n}(G/H).

For arbitrary x the result now follows from the properties of m and Exp. For suppose that it holds for x, y ∈ B(G). Firstly m(Exp(x+y)) = m(Exp(x)) m(Exp(y)) = σ_{t}(x)σ_{t}(y) = σ_{t}(x+y). Moreover 1 = m(Exp(x)) m(Exp(−x)) = σ_{t}(x) m(Exp(−x)), hence m(Exp(−x)) = σ_{t}(−x). Since every element of B(G) is a linear combination of elements [G/H] we are done.

The second assertion follows immediately, since σ_{t}(x)λ_{−t}(x) = 1, so λ_{−t}(x) = σ_{t}(−x). □

Using this lemma together with (2.1) shows that, when S is a G-set, λ_{−t}(S) = − m(∑_{n≥0} Λ̃_{(S_{+})^{∗i}} t^{n}), hence that

(3.2)  λ^{n}(S) = (−1)^{n−1} m_{n}(Λ̃_{(S_{+})^{∗i}}).

Thus we have in some sense achieved our goal; we have expressed λ^{n}(S) in a nonrecursive way, without using λ^{i}(S) for i < n. However, we want to be more concrete, and the major step is the following proposition, which allows us to express λ^{n}(S) without using B(G_{n}).

Proposition 3.3. For S a G-set, let Ω_{≤n}(S) be the G-poset of nonempty subsets of S of cardinality ≤ n. For any n ∈ ℕ,

m_{n}(Λ̃_{S^{∗n}}) = Λ̃_{Ω_{≤n}(S)} .

Proof. Given the G-set S and a positive integer n we define the G-poset S_{n},

S_{n} := {α: S → ℕ : 1 ≤ ∑_{s∈S} α(s) ≤ n}

with the ordering given by α ≤ α′ if α(s) ≤ α′(s) for every s ∈ S, and the G-action (gα)(s) := α(g^{−1}s). Note that S_{n} is G-homotopic to Ω_{≤n}(S), for we have maps θ: S_{n} → Ω_{≤n}(S), given by α ↦ α^{−1}(ℕ∖{0}), and θ′: Ω_{≤n}(S) → S_{n} sending A ⊆ S to its characteristic function. The composition θθ′ is the identity and θ′θ ≤ Id_{S_{n}}. Hence Λ̃_{S_{n}} = Λ̃_{Ω_{≤n}(S)}, so it suffices to show that m_{n}(Λ̃_{S^{∗n}}) = Λ̃_{S_{n}}. We will do this by proving that, for every i,

Σ_{n}\ Sd_{i}(S^{∗n}) ≃ Sd_{i}(S_{n})

as G-sets.

We proceed to constructing this isomorphism: first, we have a map ϕ: S^{∗n} → S_{n}, defined by ϕ(f)(s) = |f^{−1}(s)| for s ∈ S. One checks that this is a well-defined map of G-posets (where we view S^{∗n} as a G-poset via restriction). The map ϕ is surjective, for given α ∈ S_{n} one may construct an element f in its preimage in the following way: for s ∈ S, choose E_{s} ⊆ {1, …, n} such that |E_{s}| = α(s) (possibly, E_{s} = ∅). Since ∑_{s∈S} α(s) ≤ n we may do this such that the E_{s} are mutually disjoint. We now define f ∈ S^{∗n} by f(i) = 0 if i ∉ ∪_{s∈S}E_{s} and f(i) = s if i ∈ E_{s}. It then follows that ϕ(f)(s) = |f^{−1}(s)| = |E_{s}| = α(s) for all s ∈ S, i.e., ϕ(f) = α.

Next one shows that ϕ induces, for every i, a map of G-sets Φ: Sd_{i}(S^{∗n}) → Sd_{i}(S_{n}) defined by

Φ(f_{0} < ⋯ < f_{i}) := (ϕ(f_{0}) < ⋯ < ϕ(f_{i})).

Since we already know that ϕ is a map of G-posets it suffices to show that Φ does not map chains to shorter chains, i.e., that if f < f′ then φ(f) < φ(f′). This follows since there exists an i_{0} ∈ {1, …, n} such that f(i_{0}) < f′(i_{0}), i.e., f(i_{0}) ∉ S whereas f′(i_{0}) = s_{0} ∈ S, hence f^{−1}(s_{0}) is strictly contained in f^{′−1}(s_{0}), i.e., ϕ(f)(s_{0}) < ϕ(f′)(s_{0}).

The map Φ is surjective, for ϕ is and from the construction it follows that we may choose elements in the preimages such that the chain property is not destroyed.

Finally, for c = (f_{0} < ⋯ < f_{i}) and c = (f′_{0} < ⋯ < f′_{i}) in Sd_{i}(S^{∗n}) we have Φ(c) = Φ(c′) if and only if there exists a σ ∈ Σ_{n} such that σ(c) = c′. For suppose that Φ(c) = Φ(c′). Then, for every 0 ≤ j ≤ i, ϕ(f_{j}) = ϕ(f′_{j}), i.e., for every s ∈ S we have |f_{j}^{−1}(s)| = |f_{j}^{′−1}(s)|. Since f_{0}^{−1}(s) ⊆ ⋯ ⊆ f_{i}^{−1}(s) and f_{0}^{′−1}(s) ⊆ ⋯ ⊆ f_{i}^{′−1}(s) this means that we may chose a bijection σ_{s}: f_{i}^{−1}(s) → f_{i}^{′−1}(s) such that σ_{s}(f_{j}^{−1}(s)) = f_{j}^{′−1}(s) for every 0 ≤ j ≤ i. Since the sets f_{j}^{−1}(s), for s ∈ S, are mutually disjoint there exists a σ ∈ Σ_{n} which, viewed as an automorphism of {1, …, n}, restrict to σ_{s} on f_{i}^{−1}(s) for every s ∈ S. Then, for any 1 ≤ j ≤ i and for any m ∈ {1, …, n} and s ∈ S we have that

f_{j}(m) = s ⟺ m ∈ f_{j}^{−1}(s) ⟺ σm ∈ f^{′−1}(s) ⟺ f′_{j}(σm) = s,

and also that f_{j}(m) = 0 ⟺ f′_{j}(σm) = 0. Hence σf_{j} = f′_{j} for 0 ≤ j ≤ i, i.e., σc = c′.

It follows that Φ induces an isomorphism of G-sets Σ_{n}\ Sd_{i}(S^{∗n}) → Sd_{i}(S_{n}). □

Therefore, from (3.2),

λ^{n}(S) = (−1)^{n−1} Λ̃_{Ω_{≤n}(S)} .

Theorem 1.1 therefore follows from the following computation:

Lemma 3.4. Let S be a G-set and let S_{+} := S ∪ {•}. In B(G) we then have the equality

Λ̃_{Ω_{≤n}(S_{+})} = − ∑_{μ⊢n}(−1)^{ℓ(μ)}\binom{ℓ(μ)}{α(μ)}[P_{μ}(S)].

Proof. The inclusion S → S_{+} induces an inclusion i: Ω_{≤n}(S) → Ω_{≤n}(S_{+}). By Proposition 4.2.7 of [Bou00] we have

Λ̃_{Ω_{≤n}(S_{+})} = Λ̃_{Ω_{≤n}(S)} + ∑_{A∈[G\ Ω_{≤n}(S_{+})]} ind_{G_{A}}^{G}(Λ̃_{i^{A}} Λ̃_{]A,.[}),

where i^{A} = {B ∈ Ω_{≤n}(S) : B = i(B) ⊆ A}. However, when A ≠ {•} the set i^{A} has a largest element (namely A ∖ {•}), hence Λ̃_{i^{A}} = 0. Therefore the sum after the summation sign has only one non-zero element, namely the one with index {•}, which equals − Λ̃_{]•,.[} (where ]•, .[ is the set of elements of Ω_{≤n}(S_{+}) containing •). Since ]•, .[ is homotopic (more precisely isomorphic) to Ω_{≤n−1}(S), it follows that

Λ̃_{Ω_{≤n}(S_{+})} = Λ̃_{Ω_{≤n}(S)} − Λ̃_{Ω_{≤n−1}(S)} .

It is easy to see that this last expression is the desired one: let S be a G-set and define, for any tuple of positive integers α = (α_{0}, …, α_{i}), the G-set P_{α}(S) similarly as when α is a partition of an integer. Then the map sending the sequence (S_{0} ⊂ ⋯ ⊂ S_{i}) ↦ (S_{0}, S_{1} ∖ S_{0}, …, S_{i} ∖ S_{i−1}) is an isomorphism of G-sets

Sd_{i}(Ω_{≤n}(S)) → ∪_{\begin{matrix}α=(α_{0},...,α_{i}): \\ ∑ α_{j}≤n \\ α_{j}>0 \end{matrix}} P_{α}(S), hence

[Sd_{i}(Ω_{≤n}(S))] = ∑_{\begin{matrix}α=(α_{0},...,α_{i}): \\ ∑ α_{j}≤n \\ α_{j}>0 \end{matrix}}[P_{α}(S)].

We therefore have

[Sd_{i}(Ω_{≤n}(S))] − [Sd_{i}(Ω_{≤n−1}(S))] = ∑_{\begin{matrix}α=(α_{0},...,α_{i}): \\ ∑ α_{j}=n \\ α_{j}>0 \end{matrix}}[P_{α}(S)] = ∑_{\begin{matrix}μ⊢n \\ ℓ(μ)=i+1 \end{matrix}} \binom{ℓ(μ)}{α(μ)}[P_{μ}(S)],

and consequently

Λ̃_{Ω_{≤n}(S)} − Λ̃_{Ω_{≤n−1}(S)} = ∑_{i≥0}(−1)^{i} ∑_{\begin{matrix}μ⊢n \\ ℓ(μ)=i+1 \end{matrix}} \binom{ℓ(μ)}{α(μ)}[P_{μ}(S)]

= − ∑_{μ⊢n}(−1)^{ℓ(μ)}\binom{ℓ(μ)}{α(μ)}[P_{μ}(S)].  □

References

[AT69] M. F. Atiyah and D. O. Tall, Group representations, λ-rings and the J-homomorphism, Topology 8 (1969), 253–297.

[Bou92] Serge Bouc, Exponentielle et modules de Steinberg, J. Algebra 150 (1992), no. 1, 118– 157.

[Bou00] , Burnside rings, Handbook of algebra, Vol. 2, North-Holland, Amsterdam, 2000, pp. 739–804.

[Knu73] Donald Knutson, λ-rings and the representation theory of the symmetric group, Springer-Verlag, Berlin, 1973, Lecture Notes in Mathematics, Vol. 308.

[Rök07a] Karl Rökaeus, The computation of the classes of some tori in the Grothendieck ring of

varieties, arXiv:0708.4396v2 [math.AG] (2007).

[Rök07b] , A note on the λ-structure on the Burnside ring, arXiv:0708.1470v1 [math.GR] (2007).

Serge Bouc - CNRS - LAMFA - Université de Picardie Jules Verne - 33 rue St Leu - 80039 - Amiens - France

   E-mail address: serge.bouc@u-picardie.fr

Karl Rökaeus, Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

   E-mail address: karlr@math.su.se