132 research outputs found
First Degree Cohomology of Specht Modules and Extensions of Symmetric Powers
Let denote the symmetric group of degree and let be a
field of positive characteristic . For we give an explicit description
of the first cohomology group , of the Specht
module over , labelled by a partition of .
We also give a sufficient condition for the cohomology to be non-zero for
and we find a lower bound for the dimension. Our method is to proceed by
comparison with the cohomology for the general linear group over and
then to reduce to the calculation of ,
where is a Borel subgroup of , denotes the th symmetric
power of the natural module for and denotes the one
dimensional -module with weight . The main new input is the
description of module extensions by: extensions sequences, coherent triples of
extension sequences and coherent multi-sequences of extension sequences, and
the detailed calculation of the possibilities for such sequences. These
sequences arise from the action of divided powers elements in the negative part
of the hyperalgebra of .Comment: 94 page
Decompositions of some Specht modules I
We give a decomposition as a direct sum of indecomposable modules of several
types of Specht modules in characteristic . These include the Specht modules
labelled by hooks, whose decomposability was considered by Murphy. Since the
main arguments are essentially no more difficult for Hecke algebras at
parameter , we proceed in this level of generality.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1704.02413,
arXiv:1704.0241
Obtaining the diffusion coefficient for cosmic ray propagation in the Galactic Centre Ridge through time-dependent simulations of their gamma-ray emission
Recent observations by the H.E.S.S. collaboration of the Galactic Centre
region have revealed what appears to be gamma-ray emission from the decay of
pions produced by interactions of recently accelerated cosmic rays with local
molecular hydrogen clouds. Synthesizing a 3-D hydrogen cloud map from the
available data and assuming a diffusion coefficient of the form kappa(E) =
kappa_0(E/E0)^delta, we performed Monte Carlo simulations of cosmic ray
diffusion for various propagation times and values of kappa_0 and delta. By
fitting the model gamma-ray spectra to the observed one we were able to infer
the value of the diffusion coefficient in that environment (kappa = 3.0 +/- 0.2
kpc^2 Myr^-1 for E = 10^12.5 eV and for total propagation time 10^4 yr) as well
as the source spectrum (2.1 < gamma < 2.3). Also, we found that proton losses
can be substantial, which justifies our approach to the problem.Comment: 7 pages, 7 figures, accepted for publication in Astroparticle Physic
On invariant ideals associated to classical groups
AbstractWe study the G1×G2-invariant ideals of the coordinate ring of the variety of nullforms, associated to a pair of classical groups over a field of characteristic zero
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