Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young's seminormal basis

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, $\Delta(\lambda)$ denote the Weyl module of $G$ of highest weight $\lambda$ and $\iota_{\lambda,\mu}:\Delta(\lambda+\mu)\to \Delta(\lambda)\otimes\Delta(\mu)$ be the canonical $G$-morphism. We study the split condition for $\iota_{\lambda,\mu}$ over $\mathbb{Z}_{(p)}$, and apply this as an approach to compare the Jantzen filtrations of the Weyl modules $\Delta(\lambda)$ and $\Delta(\lambda+\mu)$. In the case when $G$ is of type $A$, we show that the split condition is closely related to the product of certain Young symmetrizers and, under some mild conditions, is further characterized by the denominator of a certain Young's seminormal basis vector. We obtain explicit formulas for the split condition in some cases

Gene duplication and an accelerated evolutionary rate in 11S globulin genes are associated with higher protein synthesis in dicots as compared to monocots

Background: Seed storage proteins are a major source of dietary protein, and the content of such proteins determines both the quantity and quality of crop yield. Significantly, examination of the protein content in the seeds of crop plants shows a distinct difference between monocots and dicots. Thus, it is expected that there are different evolutionary patterns in the genes underlying protein synthesis in the seeds of these two groups of plants. Results: Gene duplication, evolutionary rate and positive selection of a major gene family of seed storage proteins (the 11S globulin genes), were compared in dicots and monocots. The results, obtained from five species in each group, show more gene duplications, a higher evolutionary rate and positive selections of this gene family in dicots, which are rich in 11S globulins, but not in the monocots. Conclusion: Our findings provide evidence to support the suggestion that gene duplication and an accelerated evolutionary rate may be associated with higher protein synthesis in dicots as compared to monocots

Anisotropic emission of thermal dielectrons from Au+Au collisions at sNN=200\sqrt{s_{NN}}=200~GeV with EPOS3

Dileptons, as an electromagnetic probe, are crucial to study the properties of a Quark-Gluon Plasma (QGP) created in heavy ion collisions. We calculated the invariant mass spectra and the anisotropic emission of thermal dielectrons from Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC) energy $\sqrt{s_{NN}}=200$~GeV based on EPOS3. This approach provides a realistic (3+1)-dimensional event-by-event viscous hydrodynamic description of the expanding hot and dense matter with a very particular initial condition, and a large set of hadron data and direct photons (besides $v_{2}$ and $v_{3}$ !) can be successfully reproduced. Thermal dilepton emission from both the QGP phase and the hadronic gas are considered, with the emission rates based on Lattice QCD and a vector meson model, respectively. We find that the computed invariant mass spectra (thermal contribution + STAR cocktail) can reproduce the measured ones from STAR at different centralities. Different compared to other model predictions, the obtained elliptic flow of thermal dileptons is larger than the STAR measurement referring to all dileptons. We observe a clear centrality dependence of thermal dilepton not only for elliptic flow $v_{2}$ but also for higher orders. At a given centrality, $v_{n}$ of thermal dileptons decreases monotonically with $n$ for $2\leq n\leq5$.Comment: 10pages, 12fig