Cohomology and Support Varieties for Lie Superalgebras II

In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.Comment: 28 pages, the proof of Proposition 4.5.1 was corrected, several other small errors were fixe

Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a classical Lie superalgebra and $\mathcal{F}$ be the category of finite dimensional $\mathfrak{g}$-supermodules which are completely reducible over the reductive Lie algebra $\mathfrak{g}_{\bar{0}}$. In an earlier paper the authors demonstrated that for any module $M$ in $\mathcal{F}$ the rate of growth of the minimal projective resolution (i.e., the complexity of $M$) is bounded by the dimension of $\mathfrak{g}_{\bar{1}}$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.Comment: 32 page

Choosing the Elite \u3ci\u3eRecmitment, Assessment, and Selection in Law Enforcement Tactical Teams and Military Special Forces\u3c/i\u3e

In the early morning hours of July 31, 2007, Gazi Abu Mezer and Lafi Khalil, two PcUestinians who had entered the United States after exploiting loopholes in the immigration system. were in the operational phase of a planned suicide bombing of the New York City fransit system. Their weapons: simple pipe bombs with electrical detonating switches packed into knapsacks. Two things would stand in the way of their attempt at martyrdom-their Egyptian roommate and the New York City Fblice Department\u27s Emergency Service Unit (ESU). Mossabah, the roommate, had been taken to the New York Fblice Department\u27s 88th Precinct. When the ESU tour commandex:, a veteran lieutenant arrived at the precinct, he interviewed Mossabah, having him start at the beginning of his recollections, interrupting him for clarifications and details. After digesting what he had heard, the commander began apprising his leadership of the situation and stressing that he believed Mossabah. He then looked at the roster of ESU officers working his shift, from midnight to 8 a.m. The urgency of the situation prohibited him from calling in people from home and handpicking a team to enter the apartment Mossabah shared with the two suspects; he would have to go with those already available. The commander called a sergeant and four police officers and told them to meet him at the precinct. After the five arrived, Mossabah was again asked to tell his stoty from the beginning. They interrupted him with even more questions and requests for clarification and had him draw an extensive diagram of his apartment building\u27s exterior and his apartment\u27s interior, highlighting the locations of the backpacks as well as the spaces Mezer and Khalil normally ocrupied. The commander took his team into another room and devised his tactical plan, giving out assignments as they progressed. The team would exerute a standard dynamic search warrant entry. Mossabah would lead them to the building. and they would enter using his key. The building the team saw upon exiting their vehicle looked nothing like the one Mossabah had sketched at the precinct. Thoughts of an ambush ran through the officers\u27minds. Mossabah then led the team through a narrow alley, at the end of which the team saw a building resembling the one he had drawn. The lead officer, protected by a hand-held body bunker, put the key in the lock, turned it, and opened the door. The team flooded the apartment, yelling. Police! Get down on the floor! The first officer through the door was met by a man who attempted to physically disarm him. The officer fired one shot from his 9-mm pistol, hitting his target. The wounded man stumbled backwards, toward a black canvas bag in the comer of the room. As he flipped one of four toggles on the front of the bag. another officer fired two 5.56-mm rounds into him, and he crumpled into a comer of the room. As the team pressed on, a second man in another room lunged toward a backpack in a comer. Two shots from an officer\u27s pistol dropped him before he could reach it. The entry was over in less than ninety seconds. Both men were taken into rustody. The FBI reconstructed both backpack improvised explosive devises, determining them to be functional with a blast mdius of more than 100 yards. Their analysis also revealed that the toggle thrown on the first backpack should have detonated the device. It is not known why it failed

Alfalfa Cultivar Yield Test for South Dakota: 1993 Report

Public and commercial breeding programs have produced many alfalfa cultivars in the last 15 years. That makes selecting the proper cultivar for your needs no easy task, for you should have yield information from several South Dakota locations before choosing. The Alfalfa Cultivar Yield Test reports relative forage production characteristics for available cultivars and experimental lines at several locations in South Dakota

Grass Seed Production in South Dakota Guidelines

Finding ways to supplement a farm income can be difficult, yet many producers have found alternative enterprises for their farming operation. One of which many South Dakota farmers are considering is growing grass seed. Because of the increased demand for grass seed it is fast becoming a profitable alternative to traditional or row-crop production. The economic viability of producing grass seed can vary widely. However, with an understanding of the risks involved--production costs, the weather, government programs, and the presence of weeds and damaging insects—growing grass seed can be a worthwhile investment. Producers need to be aware that decisions to produce grass seed should be based not only on the perceived profitability of such a crop but also their ability to handle risk. Government programs which influence grass seed prices, such as the Conservation Reserve Program (CRP), are subject to budget cuts and provisions. It was the CRP itself that increased the demand for grass seed by developing provisions to reduce surplus grain stocks by decreasing crop acreages. Also, given the apparent profitability of grass seed production, many producers will probably start up. This will increase grass seed supplies in the years ahead, resulting in downward pressure on prices. This bulletin is intended to aid South Dakota producers in the establishment and management of grass seed production acreages. Economic and marketing concerns are also addressed