Summary: A Construction of Endo­permutation Modules
J.L. Alperin \Lambda
1 Introduction
Endo­permutation modules for finite p­groups appear in the representation theory of finite
groups in several ways, for example as sources of simple modules and in connection with
equivalences between blocks [6, 7, 11]. The main problem is their complete description and
classification. We shall give a simple new construction of such modules, study their properties
and apply the results to the group of endo­trivial modules, determining its rank.
Let P be a fixed finite p­group and k an algebraically closed field of characteristic p. All
modules for kP will be assumed finite­dimensional and all P ­sets will be finite. If X is a
P ­set let \Delta(X ) be the kernel of the map from the kP­module kX to the trivial modules
k sending each element of X to 1 (the augmentation map). Recall that a kP­module U
is an endo­permutation module if the kP­module Hom k (U; U) = End k (U) is a permutation
module and U is an endo­trivial module if Hom k (U; U) is the direct sum of k and a free
kP­module.
Theorem 1. The module \Delta(X ) is an endo­permutation module.
Theorem 2. The module \Delta(X ) is indecomposable if, and only if, no orbit of P on X is a
homomorphic image, as P ­set, of another orbit of P on X.
Theorem 3. If \Delta(X ) is indecomposable and Q is a subgroup of P then the class [Br P
Q (\Delta(X ))]

Source: Alperin, Jon L. - Department of Mathematics, University of Chicago

Collections: Mathematics