TITLE: Ghosts in modular representation theory
AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac
Department of Mathematics
University of Western Ontario
London, ON N6A 5B7, Canada
Email addresses: schebolu@uwo.ca, jdc@uwo.ca, and minac@uwo.ca
AMS Subject classsification: Primary 20C20, 20J06; Secondary 55P42
ABSTRACT:
A \emph{ghost} over a finite group $G$ is a map between modular
representations of $G$ which is invisible in Tate cohomology. Motivated by the
failure of the \emph{generating hypothesis}---the statement that ghosts
between finite-dimensional $G$-representations factor through a
projective---we define the \emph{compact ghost number} of $kG$ to be the smallest integer $l$
such that the composition of any $l$ ghosts between finite-dimensional
$G$-representations factors through a projective. In this paper we study ghosts
and the compact ghost numbers of $p$-groups. We begin by showing that a weaker version
of the generating hypothesis, where the target of the ghost is fixed to be the
trivial representation $k$, holds for all $p$-groups.
We do this by proving that a map between finite-dimensional
$G$-representations is a ghost if and only if it is a \emph{dual ghost}.
We then compute the compact ghost
numbers of all cyclic $p$-groups and all abelian $2$-groups with $C_2$ as a
summand. We obtain bounds on the compact ghost numbers for abelian $p$-groups and for
all $2$-groups which have a cyclic subgroup of index $2$. Using these bounds
we determine the finite abelian groups which have compact ghost number at most $2$.
%Finally, using universal ghosts, we establish various sets of conditions which
%guarantee the existence of a non-trivial ghost out of a $G$-representation.
Our methods involve techniques from group theory, representation theory,
triangulated category theory, and constructions motivated from homotopy
theory.
COMMENTS: This version replaces an earlier one with file name ghost.tex. This is
a substantial improvement with many new results and major reorganisation of the
paper.