First cohomology for finite groups of Lie type: simple modules with small dominant weights

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(\lambda))$ where $L(\lambda)$ is the simple $G$-module of highest weight $\lambda$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $\lambda$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $\lambda$ is a minimal nonzero dominant weight.Comment: 24 pages, 5 figures, 6 tables. Typos corrected and some proofs streamlined over previous versio

Second cohomology for finite groups of Lie type

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(\lambda)$ be the simple rational $G$-module of highest weight $\lambda$. In this paper we establish sufficient criteria for the restriction map in second cohomology $H^2(G,L(\lambda)) \rightarrow H^2(G(\mathbb{F}_q),L(\lambda))$ to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on $p$ and $q$ provided $\lambda$ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(\lambda))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G(\mathbb{F}_q),L(\lambda))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.Comment: 29 pages, GAP code included as an ancillary file. Rewritten to include the adjoint representation in types An, B2, and Cn. Corrections made to Theorem 3.1.3 and subsequent dependent results in Sections 3-4. Additional minor corrections and improvements also implemente

Assessing the cost of global biodiversity and conservation knowledge

Knowledge products comprise assessments of authoritative information supported by stan-dards, governance, quality control, data, tools, and capacity building mechanisms. Considerable resources are dedicated to developing and maintaining knowledge productsfor biodiversity conservation, and they are widely used to inform policy and advise decisionmakers and practitioners. However, the financial cost of delivering this information is largelyundocumented. We evaluated the costs and funding sources for developing and maintain-ing four global biodiversity and conservation knowledge products: The IUCN Red List ofThreatened Species, the IUCN Red List of Ecosystems, Protected Planet, and the WorldDatabase of Key Biodiversity Areas. These are secondary data sets, built on primary datacollected by extensive networks of expert contributors worldwide. We estimate that US$160million (range: US$116–204 million), plus 293 person-years of volunteer time (range: 278–308 person-years) valued at US$14 million (range US$12–16 million), were invested inthese four knowledge products between 1979 and 2013. More than half of this financingwas provided through philanthropy, and nearly three-quarters was spent on personnelcosts. The estimated annual cost of maintaining data and platforms for three of these knowl-edge products (excluding the IUCN Red List of Ecosystems for which annual costs were notpossible to estimate for 2013) is US$6.5 million in total (range: US$6.2–6.7 million). We esti-mated that an additional US$114 million will be needed to reach pre-defined baselines ofdata coverage for all the four knowledge products, and that once achieved, annual mainte-nance costs will be approximately US$12 million. These costs are much lower than those tomaintain many other, similarly important, global knowledge products. Ensuring that biodi-versity and conservation knowledge products are sufficiently up to date, comprehensiveand accurate is fundamental to inform decision-making for biodiversity conservation andsustainable development. Thus, the development and implementation of plans for sustain-able long-term financing for them is critical

Assessing the Cost of Global Biodiversity and Conservation Knowledge

<div><p>Knowledge products comprise assessments of authoritative information supported by standards, governance, quality control, data, tools, and capacity building mechanisms. Considerable resources are dedicated to developing and maintaining knowledge products for biodiversity conservation, and they are widely used to inform policy and advise decision makers and practitioners. However, the financial cost of delivering this information is largely undocumented. We evaluated the costs and funding sources for developing and maintaining four global biodiversity and conservation knowledge products: The IUCN Red List of Threatened Species, the IUCN Red List of Ecosystems, Protected Planet, and the World Database of Key Biodiversity Areas. These are secondary data sets, built on primary data collected by extensive networks of expert contributors worldwide. We estimate that US$160 million (range: US$116–204 million), plus 293 person-years of volunteer time (range: 278–308 person-years) valued at US$14 million (range US$12–16 million), were invested in these four knowledge products between 1979 and 2013. More than half of this financing was provided through philanthropy, and nearly three-quarters was spent on personnel costs. The estimated annual cost of maintaining data and platforms for three of these knowledge products (excluding the IUCN Red List of Ecosystems for which annual costs were not possible to estimate for 2013) is US$6.5 million in total (range: US$6.2–6.7 million). We estimated that an additional US$114 million will be needed to reach pre-defined baselines of data coverage for all the four knowledge products, and that once achieved, annual maintenance costs will be approximately US$12 million. These costs are much lower than those to maintain many other, similarly important, global knowledge products. Ensuring that biodiversity and conservation knowledge products are sufficiently up to date, comprehensive and accurate is fundamental to inform decision-making for biodiversity conservation and sustainable development. Thus, the development and implementation of plans for sustainable long-term financing for them is critical.</p></div

Categories, subcategories and funding sources classification used to categorise costs.

<p>Categories, subcategories and funding sources classification used to categorise costs.</p

Sources of funding (midpoints of estimates) invested until 2013 for each knowledge product.

<p>Sources of funding (midpoints of estimates) invested until 2013 for each knowledge product.</p

Overall funds (midpoints) and volunteer days invested in the four knowledge products between 1979 and 2013, and annual cost in 2013.

<p>The mid-point is the equidistant point between the maximum and minimum values. Full details are available in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160640#pone.0160640.s004" target="_blank">S4 Table</a>.</p