Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

Let $\Phi$ be a classical root system and $k$ be a field of sufficiently large characteristic. Let $G$ be the classical group over $k$ with the root system $\Phi$, $U$ be its maximal unipotent subgroup and $\mathfrak{u}$ be the Lie algebra of $U$. Let $D$ be an orthogonal subset of $\Phi$ and $\Omega$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\mathfrak{u}$ at the canonical form on $\Omega$. We also find the dimension of $\Omega$ in terms of the Weyl group of $\Phi$. As a corollary, we determine all possible dimensions of irreducible complex represenations of the group $U$ for the case of finite field $k$.Comment: 11 page

Combinatorics of BB-orbits and Bruhat--Chevalley order on involutions

Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$, $x\in\mathfrak{u}$. To each involution $\sigma$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $\Omega_{\sigma}\in\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is given in terms of rook placements and is dual to A. Melnikov's results on $B$-orbits on $\mathfrak{u}$. Using results of F. Incitti, we also prove that this partial order coincides with the restriction of the Bruhat--Chevalley order to the set of involutions.Comment: 27 page

Overview of the Situation on Highly Pathogenic Avian Influenza Virus H5 in Russia in 2016–2017

In 2016–2017, several outbreaks among wild and domestic birds caused by highly pathogenic influenza virus were recorded in the territory of the Russian Federation. After isolation of the highly pathogenic influenza A (H5N8) virus in the territory of the Republic of Tyva in May 2016, the virus was introduced into the west by wild birds, where it caused a massive epizootic. The death of agricultural poultry was noted in several regions of the European part of Russia and epizootics continued until the end of 2017. This paper presents the data on circulation of highly pathogenic influenza virus over the past two years. The current status regarding H5 influenza in the territory of the Russian Federation is also considered, where for the first time since 2005 a large-scale epizooty caused by the influenza virus subtype A (H5N8) was recorded. It is shown that the territory of Russia plays an important geographical role in the spread of influenza virus by wild birds from Southeast Asia to Europe and North America, and the research also suggests the possible continuation of circulation of highly pathogenic viruses in the territory of Russia

Leptogenesis with Heavy Majorana Neutrinos Reexamined

The mass term for Majorana neutrinos explicitly violates lepton number. Several authors have used this fact to create a lepton asymmetry in the universe by considering CP violating effects in the one loop self-energy correction for the decaying heavy Majorana neutrino. We compare and comment on the different approaches used to calculate the lepton asymmetry including those using an effective Hamiltonian and resummed propagators. We also recalculate the asymmetry in the small mass difference limit.Comment: 16 pages, LaTex, 1 figure included. 2 footnotes and 1 reference adde