First Degree Cohomology of Specht Modules and Extensions of Symmetric Powers

Let $\Sigma_d$ denote the symmetric group of degree $d$ and let $K$ be a field of positive characteristic $p$. For $p>2$ we give an explicit description of the first cohomology group $H^1(\Sigma_d,{\rm{Sp}}(\lambda))$, of the Specht module ${\rm{Sp}}(\lambda)$ over $K$, labelled by a partition $\lambda$ of $d$. We also give a sufficient condition for the cohomology to be non-zero for $p=2$ and we find a lower bound for the dimension. Our method is to proceed by comparison with the cohomology for the general linear group $G(n)$ over $K$ and then to reduce to the calculation of ${\rm{Ext}}^1_{B(n)}(S^d E,K_\lambda)$, where $B(n)$ is a Borel subgroup of $G(n)$, $S^dE$ denotes the $d$th symmetric power of the natural module $E$ for $G(n)$ and $K_\lambda$ denotes the one dimensional $B(n)$-module with weight $\lambda$. 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 $G(n)$.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 $2$. 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 $q=-1$, 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