1,057 research outputs found
Ghost numbers of Group Algebras
Motivated by Freyd's famous unsolved problem in stable homotopy theory, the
generating hypothesis for the stable module category of a finite group is the
statement that if a map in the thick subcategory generated by the trivial
representation induces the zero map in Tate cohomology, then it is stably
trivial. It is known that the generating hypothesis fails for most groups.
Generalizing work done for -groups, we define the ghost number of a group
algebra, which is a natural number that measures the degree to which the
generating hypothesis fails. We describe a close relationship between ghost
numbers and Auslander-Reiten triangles, with many results stated for a general
projective class in a general triangulated category. We then compute ghost
numbers and bounds on ghost numbers for many families of -groups, including
abelian -groups, the quaternion group and dihedral -groups, and also give
a general lower bound in terms of the radical length, the first general lower
bound that we are aware of. We conclude with a classification of group algebras
of -groups with small ghost number and examples of gaps in the possible
ghost numbers of such group algebras.Comment: 28 pages; v2 improves introduction and has many other minor changes
throughout. appears in Algebras and Representation Theory, 201
Ghosts in modular representation theory
A ghost over a finite p-group G is a map between modular representations of G
which is invisible in Tate cohomology. Motivated by the failure of the
generating hypothesis---the statement that ghosts between finite-dimensional
G-representations factor through a projective---we define the 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 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 then compute the ghost numbers of all cyclic p-groups and all abelian
2-groups with C_2 as a summand. We obtain bounds on the 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 ghost
number at most 2. Our methods involve techniques from group theory,
representation theory, triangulated category theory, and constructions
motivated from homotopy theory.Comment: 15 pages, final version, to appear in Advances in Mathematics. v4
only makes changes to arxiv meta-data, correcting the abstract and adding a
do
Hilbert space of curved \beta\gamma systems on quadric cones
We clarify the structure of the Hilbert space of curved \beta\gamma systems
defined by a quadratic constraint. The constraint is studied using intrinsic
and BRST methods, and their partition functions are shown to agree. The quantum
BRST cohomology is non-empty only at ghost numbers 0 and 1, and there is a
one-to-one mapping between these two sectors. In the intrinsic description, the
ghost number 1 operators correspond to the ones that are not globally defined
on the constrained surface. Extension of the results to the pure spinor
superstring is discussed in a separate work.Comment: 45 page
The Spectrum of Open String Field Theory at the Stable Tachyonic Vacuum
We present a level (10,30) numerical computation of the spectrum of quadratic
fluctuations of Open String Field Theory around the tachyonic vacuum, both in
the scalar and in the vector sector. Our results are consistent with Sen's
conjecture about gauge-triviality of the small excitations. The computation is
sufficiently accurate to provide robust evidence for the absence of the photon
from the open string spectrum. We also observe that ghost string field
propagators develop double poles. We show that this requires non-empty BRST
cohomologies at non-standard ghost numbers. We comment about the relations of
our results with recent work on the same subject.Comment: 33 pages, 10 figure
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