Congruences on “character” values of permutation summands

Abstract

A class of congruences on values JL 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 QA(G) of the category of all permutation summands for G over A has been studied in [3], via a sort of numerical character 4?L of a permutation summand L. The construction of 4?L is reviewed in the first paragraph of the proof below. In this note we exhibit a class of congruences on the values of 4?L which are strong enough to imply the connectedness of the prime ideal spectrum of QA(G). The corresponding result for the character ring RK (G) was established in [2], where Lemma 7 gives analogous congruences on character values. For the Burnside ring Q(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 4?L takes values in the ring A' of integers of some sufficiently large number field, for instance K((IGI), 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/Op(H) where Op(H) is the largest normal p-subgroup of H. Congruences. For any prime number q, we have 4DL(H, b, p') --L(0Q(H), bq/, p') mod q' where Oq(H) is the smallest normal subgroup of H with H/Oq(H) a q-group, bqi 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 ip,: A' -A', be the inclusion of A' in its completion at p', and let p = p' n A. Denote the ApG-module Ap ?A L by M for simplicity. Since H is p-hypoelementary, Op(H) is the normal pSylow subgroup of H. Decompose the restriction MH of M to H as MH M'GM, where the vertices of the indecomposable ApH-summands of M' are Op(H), and Received by the editors September 7, 1995. 1991 Mathematics Subject Classification. Primary 20C10; Secondary 19A22, 20C15. ?1997 American Mathematical Society