PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 2, February 1997, Pages 393–396 S 0002-9939(97)03825-2

CONGRUENCES ON “CHARACTER” VALUES OF PERMUTATION SUMMANDS

XIANGYONG WANG

(Communicated by Ronald M. Solomon)

Abstract. A class of congruences on “character” values Φ_{L} of a permutation summand L are exhibited, from which follows the connectedness of the prime ideal spectrum of the Grothendieck ring of permutation summands.

Let G be a finite group and A the ring of integers in a number field K. An AG-lattice is called a permutation lattice if it has an A-basis, necessarily finite, which is permuted by the action of G. It will be called a permutation summand (for G over A), if it is a direct summand, as AG-module, of a permutation lattice. The Grothendieck ring Ω_{A}(G) of the category of all permutation summands for G over A has been studied in [3], via a sort of numerical character Φ_{L} of a permutation summand L. The construction of Φ_{L} is reviewed in the first paragraph of the proof below.

In this note we exhibit a class of congruences on the values of Φ_{L} which are strong enough to imply the connectedness of the prime ideal spectrum of Ω_{A}(G). The corresponding result for the character ring R_{K}(G) was established in [2], where Lemma 7 gives analogous congruences on character values. For the Burnside ring Ω(G) of finite G-sets, the connectedness fails [1], because there are too few congruences on the number of fixed points of G-sets.

The function Φ_{L} takes values in the ring A′ of integers of some sufficiently large number field, for instance K(ζ_{|G|}), and is defined on triples (H, b, 𝔭′) of G over A. Here 𝔭′ is a non-zero prime ideal of A′ so that if p is the unique prime number in 𝔭′ then H is a p-hypoelementary subgroup of G and b is a generator of H/O_{p}(H) where O_{p}(H) is the largest normal p-subgroup of H.

Congruences. For any prime number q, we have

Φ_{L}(H, b, 𝔭′) ≡ Φ_{L}(O^{q}(H), b_{q′}, 𝔭′) mod 𝔮′

where O^{q}(H) is the smallest normal subgroup of H with H/O^{q}(H) a q-group, b_{q′} is the q′-part of the element b, and 𝔮′ is any prime ideal above q.

Proof. Notations are consistent with those used in [3]. Let i_{𝔭′} : A′ → A′_{𝔭′} be the inclusion of A′ in its completion at 𝔭′, and let 𝔭 = 𝔭′ ∩ A. Denote the A_{𝔭}G-module A_{𝔭} ⊗_{A} L by M for simplicity. Since H is p-hypoelementary, O_{p}(H) is the normal p-Sylow subgroup of H. Decompose the restriction M_{H} of M to H as M_{H} ≃ M′⊕M′′, where the vertices of the indecomposable A_{𝔭}H-summands of M′ are O_{p}(H), and

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors September 7, 1995.

1991 Mathematics Subject Classification. Primary 20C10; Secondary 19A22, 20C15.

the vertices of M′′ are proper subgroups of O_{p}(H). By the definition of Φ_{L} (cf. [3] (2.1)), we have

i_{𝔭′}Φ_{L}(H, b, 𝔭′) = trace of b acting on M′.

If the action of b on M′ has eigenvalues λ_{1}, ..., λ_{r} in A′_{𝔭′}, then Φ_{L}(H, b, 𝔭′) = ∑_{i} ξ_{i}, where ξ_{i} is the preimage of λ_{i} under i_{𝔭′}. We will call this the pretrace of b on M′ for convenience.

Denote O_{p}(O^{q}(H)) = O_{p}(H) ∩ O^{q}(H) by Q, and further decompose M′′ as M′′ ≃ M′′_{1} ⊕ M′′_{2}, where the vertices of the indecomposable A_{𝔭}H-summands of M′′_{1} contain Q and those of M′′_{2} do not.

Since every indecomposable A_{𝔭}H-summand of M′ ⊕ M′′_{1} has vertex P between Q and O_{p}(H) from the above decomposition of M_{H}, it is an A_{𝔭}H-summand of ind_{P}^{H}(A_{𝔭}) by [3](1.1). Its restriction to O^{q}(H) is then an A_{𝔭}O^{q}(H)-summand of ind_{Q}^{O^{q}(H)}(A_{𝔭}) by Mackey decompositon, hence has vertex Q. Every indecomposable summand of the restriction (M′′_{2})_{O^{q}(H)} has vertex properly contained in Q as the vertex can only drop after restriction. Therefore from the above decomposition of M_{H}, the restriction of M to O^{q}(H) has the decomposition M_{O^{q}(H)} ≃ (M′ ⊕M′′_{1})_{O^{q}(H)} ⊕(M′′_{2})_{O^{q}(H)}, where the vertices of the indecomposable A_{𝔭}O^{q}(H)- summands of (M′ ⊕ M′′_{1})_{O^{q}(H)} are Q, and the vertices of (M′′_{2})_{O^{q}(H)} are proper subgroups of Q. Again by the definition of Φ_{L}, applied to (O^{q}(H), b_{q′}, 𝔭′), we obtain

Φ_{L}(O^{q}(H), b_{q′}, 𝔭′) = pretrace of b_{q′} acting on M′ ⊕ M′′_{1}.

Now the congruence follows from

Claim. i) pretrace of b on M′ ≡ pretrace of b_{q′} on M′ mod 𝔮′; ii) pretrace of b_{q′} on M′′_{1} ≡ 0 mod 𝔮′.

Proof of Claim. i) If m is a sufficiently large power of q, we have b^{m} = b_{q′}^{m}, and the eigenvalues of b^{m} on M′ have preimages ξ_{1}^{m}, ..., ξ_{r}^{m} under i_{𝔭′}. Hence

(pretrace of b)^{m} = (∑_{i} ξ_{i})^{m} ≡ ∑_{i} ξ_{i}^{m} = pretrace of b^{m} mod 𝔮′

and, for the same reason,

(pretrace of b_{q′})^{m} ≡ pretrace of b_{q′}^{m} mod 𝔮′.

Combining gives

(pretrace of b)^{m} ≡ (pretrace of b_{q′})^{m} mod 𝔮′

from which i) follows.

ii) We may assume M′′_{1} ≠ 0. Then Q ⊊ O_{p}(H), hence p must be equal to q, and H/Q = (O_{p}(H)/Q) × (O^{q}(H)/Q) is nilpotent.

Since Q acts trivially on M′′_{1} by [3](1.1), M′′_{1} can be considered as an A_{𝔭}H/Qmodule. By [4]§2, this module has the structure,

M′′_{1} ≃ ∑_{j} N_{j} ⊗_{A_{𝔭}} ind_{D_{j}}^{O_{p}(H)/Q}(A_{𝔭})

for some A_{𝔭}O^{q}(H)/Q-lattices N_{j} and some p-subgroups D_{j} of O_{p}(H)/Q. These D_{j} are actually the vertices of M′′_{1}, hence are properly contained in O_{p}(H)/Q.

CONSEQUENCES ON “CHARACTER” VALUES OF PERMUTATION SUMMANDS 395

If the eigenvalues of b_{q′} on N_{j} have preimages ξ_{1}^{(j)}, ..., ξ_{r_{j}}^{(j)} under i_{𝔭′}, then the eigenvalues of b_{q′} on N_{j} ⊗_{A_{𝔭}} ind_{D_{j}}^{O_{p}(H)/Q}(A_{𝔭}) have preimages ξ_{1}^{(j)}, ..., ξ_{r_{j}}^{(j)} each repeated |O_{p}(H)/Q : D_{j}| times. Thus

pretrace of b_{q′} on M′′_{1} = ∑_{j} |O_{p}(H)/Q : D_{j}| ∑_{i} ξ_{i}^{(j)} ≡ 0 mod pA′,

as required. This completes the proof of the claim, hence of the congruence.

We want to examine the prime ideal spectrum Spec(Ω_{A}(G)) of the commutative ring Ω_{A}(G). Let T_{G}(A) be the set of triples (H, b, 𝔭′), and (A′)^{T_{G}(A)} the ring of all maps on triples with values in A′. Since the ring homomorphism Φ : Ω_{A}(G) → ℧_{A}(G) has a nilpotent kernel [3], and ℧_{A}(G) is a subring of (A′)^{T_{G}(A)} with finite Z-rank, it induces the surjection

Spec((A′)^{T_{G}(A)}) ⟶^{going-down} Spec(imΦ) ⟶^{Φ^{−1}} Spec(Ω_{A}(G)).

Spec(A′) consists of the ideal 0 and the maximal ideals of A′. The spectrum of (A′)^{T_{G}(A)} can be identified with T_{G}(A) × Spec(A′) : with each T ∈ T_{G}(A) and each 𝔮′ ∈ Spec(A′) we associate the prime ideal 𝔮′_{T} consisting of those f ∈ (A′)^{T_{G}(A)} such that f(T) ∈ 𝔮′. The image of 𝔮′_{T} in Spec(Ω_{A}(G)) is the prime ideal P_{𝔮′,T} corresponding to the prime ideal 𝔮′_{T} ∩ imΦ in imΦ by Φ^{−1}, i.e.

P_{𝔮′,T} = {x ∈ Ω_{A}(G) : Φ_{x}(T) ∈ 𝔮′}.

Lemma. With above notation, then

(1) P_{0,T} ⊂ P_{𝔮′,T};

(2) If 𝔮′ is a maximal ideal of A′ above a prime number q and T = (H, b, 𝔭′) is a triple, we denote the triple (O^{q}(H), b_{q′}, 𝔭′) by T^{q}. Then P_{𝔮′,T} = P_{𝔮′,T^{q}}.

Proof. (1) is clear.

(2) By the congruences we have Φ_{x}(T) ≡ Φ_{x}(T^{q}) mod 𝔮′ for x ∈ Ω_{A}(G). Thus x ∈ P_{𝔮′,T} ⟺ Φ_{x}(T) ∈ 𝔮′ ⟺ Φ_{x}(T^{q}) ∈ 𝔮′ ⟺ x ∈ P_{𝔮′,T^{q}}.

Corollary. Spec(Ω_{A}(G)) is connected.

Proof. Let C be the connected component of the point P_{0,(1)} in Spec(Ω_{A}(G)) where (1) is the cyclic triple (cf. [3] §3) of the trivial subgroup. By (1) of the Lemma, the closure \overline{\{P_{0,T} \}} contains P_{𝔮′,T} for all 𝔮′. So it suffices to show that C contains \overline{\{P_{0,T} \}} for every triple T. We proceed by induction on the order of the subgroup H appearing in the triple T = (H, b, 𝔭′).

If H is trivial this follows by the definition of C so we suppose H is non-trivial. Choose a prime number q so O^{q}(H) ⊊ H, and a prime ideal 𝔮′ of A′ containing q. By (2) and (1) of the Lemma we have P_{𝔮′,T} = P_{𝔮′,T^{q}}, hence \overline{\{P_{0,T} \}} ∩ \overline{\{P_{0,T^{q}}\}} is not empty, and \overline{\{P_{0,T} \}} ∪ \overline{\{P_{0,T^{q}}\}} is connected. But \overline{\{P_{0,T^{q}}\}} ⊆ C by the induction hypothesis, and therefore \overline{\{P_{0,T} \}} ⊆ C.

Acknowledgement

I thank Alfred Weiss for his encouragement and interest in this work.

References

1. A. Dress, A characterization of solvable groups, Math. Z. 110 (1969), 213–217. MR 40:1491 2. J. P. Serre, Linear Representations of Finite Groups, Springer-Verlag, New York, 1977. MR

56:8675

3. X. Wang, A. Weiss, Permutation Summands over Z, J. Number Theory 47 (1994), 413–434. MR 95d:20008

4. A. Weiss, Torsion units in integral group rings, J. Reine Angew. Math. 415 (1991), 175–187. MR 92c:20009

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Current address: Department of Computer Science, University of Toronto, Toronto, Ontario, Canada M5S 1A4

  E-mail address: xywang@cs.toronto.edu