Contemporary Mathematics
Volume 9, 1982
HOMOLOGY REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE*
C.W. Curtis and G.I. Lehrer
ABSTRACT. The main purpose of these notes is to present a part of
the foundations of a theory of finite group actions on simplicial
complexes with coefficient systems, and the associated group
representations in homology. We give some applications of the
theory to homology representations of Coxeter groups and finite
groups of Lie
typ~and
concentrate on the problem of giving a
homological interpretation of the endomorphism algebra of a
homology representation.
The main purpose of these notes is to present a part of the foundations of
a theory of finite group actionson simplicial complexes with coefficient systems,
and the associated group representations in homology. We give some
applications of the theory to homology representations of Coxeter groups and
finite groups of Lie type, and concentrate on the problem of giving a
homological interpretation of the endomorphism algebra of a homology
representation. For the case of representations on rational homology, this
idea was discussed in Curtis [7], while the general theory can be viewed as
a combinatorial version, in the category of G-equivariant coefficient systems,
of results of Grothendieck ([12], §§4, 5) in the category of sheaves on
topological spaces, with group actions. Some general results on coefficient
systems and their homology together with applications to the representation
theory of GLn over a finite field, were obtained by Lusztig [14]. Homotopy
properties of finite group actions on partially ordered sets, with applications
to group theory, were investigated by Quillen [16].
In §1, we give a survey of facts we shall require on partially ordered sets
with group actions (G-posets), and their associated simplical complexes and
topological spaces. This point of view follows Quillen [16], while our
discussion of orbit posets has much in common with Breden [4]. Some of the
results were presented in Curtis [7], and appear here in an improved form.
* These notes are an expanded version of lectures given by the first author at
the SRI of the Australian Mathematical Society in Hobart, 2- 4 February, 1981.
1980 14athematics Subject Classification: 20G40, 20G05, 20Hl5, 55Jl5.
Copyright© 1982, American Mathematical Society
0271-4132/81/0000-0688/$08.00
http://dx.doi.org/10.1090/conm/009/655971
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