Complete Reducibility in Good Characteristic

Let $G$ be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic $p \ge 0$. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi subgroup of $P$. In this paper we determine the $G$-conjugacy classes of non-$G$-cr simple connected subgroups of $G$ when $p$ is good for $G$. For each such subgroup $X$, we determine the action of $X$ on the adjoint module $L(G)$ and the connected centraliser of $X$ in $G$. As a consequence we classify all non-$G$-cr connected reductive subgroups of $G$, and determine their connected centralisers. We also classify the subgroups of $G$ which are maximal among connected reductive subgroups, but not maximal among all connected subgroups.Comment: 66 pages. To appear in Trans. Amer. Math. So

Finite subgroups of simple algebraic groups with irreducible centralizers

We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.Comment: 24 page

On the involution fixity of exceptional groups of Lie type

The involution fixity ${\rm ifix}(G)$ of a permutation group $G$ of degree $n$ is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if $T$ is the socle of such a group, then either ${\rm ifix}(T) > n^{1/3}$, or ${\rm ifix}(T) = 1$ and $T = {}^2B_2(q)$ is a Suzuki group in its natural $2$-transitive action of degree $n=q^2+1$. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with ${\rm ifix}(T) \leqslant n^{4/9}$. This extends recent work of Liebeck and Shalev, who established the bound ${\rm ifix}(T) > n^{1/6}$ for every almost simple primitive group of degree $n$ with socle $T$ (with a prescribed list of exceptions). Finally, by combining our results with the Lang-Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.Comment: 45 pages; to appear in Int. J. Algebra Compu

Proposed Keystone XL Pipeline: Legal Issues

[Excerpt] In 2008, TransCanada Corp. applied for a presidential permit from the State Department to construct and operate an oil pipeline across the U.S.-Canada border in a project known as Keystone XL. The Keystone XL pipeline would transport oil produced from oil sands in Alberta,Canada, to Gulf Coast refineries. The permit application was subjected to review by the State Department pursuant to executive branch authority over cross-border pipeline facilities as articulated in Executive Order 13337. After several phases of review, on November 10, 2011, the State Department announced that it would seek additional information about alternative pipeline routes before it could move forward with a national interest determination. In response, several pieces of legislation were introduced, including Title V of the Temporary Payroll Tax Cut Continuation Act of 2011. Title V dictated that President must grant the Keystone XL pipeline permit within 60 days of the law’s enactment, unless the President determined that the pipeline is not in the national interest. If the President did not make a national interest determination and took no action to grant the permit, then the law provided that the permit “shall be in effect by operation of law.” The Temporary Payroll Tax Cut Continuation Act of 2011 (P.L. 112-78), including Title V addressing the Keystone XL permit, was enacted on December 23, 2011. Pursuant to the requirements of Title V, on January 18, 2012, the State Department recommended that “the presidential permit for the proposed Keystone XL pipeline be denied and, that at this time, the TransCanada Keystone XL Pipeline be determined not to serve the national interest. ”The same day, the President stated his determination that the Keystone XL pipeline project“ would not serve the national interest. ”New legislative activity with respect to the permitting of border-crossing facilities, a subject previously handled exclusively by the executive branch, has triggered inquiries as to whether this raises constitutional issues related to the jurisdiction of the two branches over such facilities. Additionally, as states have begun to contemplate taking action with respect to the pipeline siting, some have questioned whether state siting of a pipeline is preempted by federal law. Others argue that states dictating the route of the pipeline violates the dormant Commerce Clause of the Constitution which, among other things, prohibits one state from acting to protect its own interests to the detriment of other states. This report reviews those legal issues. First, it suggests that legislation related to cross-border facility permitting is unlikely to raise significant constitutional questions, despite the fact that such permits have traditionally been handled by the executive branch alone pursuant to its constitutional “foreign affairs” authority. Next, it observes generally that state oversight of pipeline siting decisions does not appear to violate existing federal law or the Constitution. Finally, the report suggests that State Department’s implementation of the existing authority to issue presidential permits appears to allow for judicial review of its National Environmental Policy Act determinations

Simulations of Strong Gravitational Lensing with Substructure

Galactic sized gravitational lenses are simulated by combining a cosmological N-body simulation and models for the baryonic component of the galaxy. The lens caustics, critical curves, image locations and magnification ratios are calculated by ray-shooting on an adaptive grid. When the source is near a cusp in a smooth lens' caustic the sum of the magnifications of the three closest images should be close to zero. It is found that in the observed cases this sum is generally too large to be consistent with the simulations implying that there is not enough substructure in the simulations. This suggests that other factors play an important role. These may include limited numerical resolution, lensing by structure outside the halo, selection bias and the possibility that a randomly selected galaxy halo may be more irregular, for example due to recent mergers, than the isolated halo used in this study. It is also shown that, with the level of substructure computed from the N-body simulations, the image magnifications of the Einstein cross type lenses are very weak functions of source size up to \sim 1\kpc. This is also true for the magnification ratios of widely separated images in the fold and cusp caustic lenses. This means that selected magnification ratios for different the emission regions of a lensed quasar should agree with each other, barring microlensing by stars. The source size dependence of the magnification ratio between the closest pair of images is more sensitive to substructure.Comment: 28 pages, 2 tables and 14 figures. Accepted to MNRA

Reversible tuning of the surface state in a psuedo-binary Bi2(Te-Se)3 topological insulator

We use angle-resolved photoemission spectroscopy to study non-trivial surface state in psuedobinary Bi2Se0.6Te2.3 topological insulator. We show that unlike previously studied binaries, this is an intrinsic topological insulator with conduction bulk band residing well above the chemical potential. Our data indicates that under good vacuum condition there are no significant aging effects for more then two weeks after cleaving. We also demonstrate that shift of the Kramers point at low temperature is caused by UV assisted absorption of molecular hydrogen. Our findings pave the way for applications of these materials in devices and present an easy scheme to tune their properties.Comment: 4 pages, 4 figure

On extensions of the Jacobson-Morozov theorem to even characteristic

Let G be a simple algebraic group over an algebraically closed field k of characteristic 2. We consider analogues of the Jacobson-Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in $\mathfrak{g} := \text{Lie}(G)$ and also those with overalgebras isomorphic to the algebras $\text{Lie}(\text{SL}_2)$ and $\text{Lie}(\text{PGL}_2)$. This leads us to calculate the dimension of Lie automiser $\mathfrak{n}_\mathfrak{g}(k\cdot e)/\mathfrak{c}_\mathfrak{g}(e)$ for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.Comment: 22 page

A note on extremely primitive affine groups

Let G be a nite primitive permutation group on a set with nontrivial point stabilizer G . We say that G is extremely primitive if G acts primitively on each of its orbits in n f g. In earlier work, Mann, Praeger and Seress have proved that every extremely primitive group is either almost simple or of a ne type and they have classi ed the a ne groups up to the possibility of at most nitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall's conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of a ne groups in order to complete the classi cation of the extremely primitive groups. Mann et al. have conjectured that none of these a ne candidates are extremely primitive and our main result con rms this conjecture