Lattice polytopes from Schur and symmetric Grothendieck polynomials

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the $h^\ast$-vector in the case of Schur polynomials.Comment: 37 pages, 3 tables, 4 figures; Comments Welcome; Version 2: updated references to acknowledge one result was previously known, corrected values in Table 1 and reference correct OEIS sequence; Version 3: Final Version. To appear in Electronic Journal of Combinatoric

Independence complexes of finite groups

2021 Fall.Includes bibliographical references.Understanding generating sets for finite groups has been explored previously via the generating graph of a group, where vertices are group elements and edges are given by pairs of group elements that generate the group. We generalize this idea by considering minimal generating sets (with respect to inclusion) for subgroups of finite groups. These form a simplicial complex, which we call the independence complex. The vertices of the independence complex are nonidentity group elements and the faces of size k correspond to minimal generating sets of size k. We give a complete characterization via constructive algorithms, together with enumeration results, for the independence complexes of cyclic groups whose order is a squarefree product of primes, finite abelian groups whose order is a product of powers of distinct primes, and the nonabelian class of semidirect products Cp1p3…p2n-1 rtimes Cp2p4…p2n where p1,p2,…,p2n are distinct primes with p2i-1 > p2i for all 1 ≤ i ≤ n. In the latter case, we introduce a tool called a combinatorial diagram, which is a multipartite simplicial complex under certain numerical and minimal covering conditions. Combinatorial diagrams seem to be an interesting area of study on their own. We also include GAP and Polymake code which generates the facets of any (small enough) finite group, as well as visualize the independence complexes in small dimensions

Megaregions Among the Large Marine Ecosystems of the Americas

We organized environmental observations (Sea Surface Temperature, chlorophyll concentration, and primary productivity) and biological diversity indices based on reconstructed fisheries landings obtained from the Sea Around Us project to address two objectives: 1) to understand whether adjacent Large Marine Ecosystems (LMEs) of the Americas form megaregions for assemblages of commercially-valuable fish; and 2) to assess changes in the diversity of fisheries landings in LMEs of the Americas over time (1982 to 2010). To test for similarities between LMEs, we used the seascape approach of unsupervised clustering of annual mean environmental observations and fisheries-derived diversity indices. Beta-diversity estimates based on fisheries landings were used to evaluate the degree to which species spanned LMEs. Temporal trends were computed for each dataset by linear least-squares. Three megaregions emerged when considering similarities in species composition of fisheries landings, fisheries-derived diversity indices, and characteristic environmental conditions among LMEs. These include (A) the South Brazil Shelf, East Brazil Shelf, and North Brazil Shelf LMEs, (B) the Gulf of Mexico and Southeast U.S. Continental Shelf LMEs, and (C) the Northeast U.S. Continental Shelf, Scotian Shelf, and Newfoundland-Labrador Shelf LMEs. No megaregions emerged for the Pacific Ocean. While there were some shared species assemblages between the California Current and the Gulf of Alaska, the Gulf of California, and the Pacific Central-American Coastal LMEs, these showed different average environmental conditions and fishery-derived diversity indices, so they did not cluster as a megaregion. In the Pacific Ocean, the high dissimilarity in the fisheries is in part related to different top-down pressures and strong regional differences in oceanographic properties, including upwelling and impacts of El-Niño Southern Oscillation events. Overall, between 1982 and 2010, seven LMEs diversified their fisheries (Pacific Central-America Coastal, Patagonian Shelf, South Brazil Shelf, East Brazil Shelf, North Brazil Shelf, Southeast U.S. Continental Shelf, and Newfoundland-Labrador Shelf). This may be due to a number of reasons including decreasing fishing pressure but expansion of target stocks due to management quotas, changes in regional markets, competition, effort, or a decrease in particular target stocks. Three LMEs showed increasingly less diversified fisheries, namely the California Current, the Northeast U.S. Continental Shelf, and the Caribbean Sea LMEs. While in some cases this may be related to historical overfishing, such as in the Northeast U.S. Continental Shelf LME, the California Current LME has been subjected to strong and conservative management practices. The Caribbean Sea LME was likely subjected to heavy fishing at a time of rapid environmental change

Diseases and Mortalities of Fishes and Other Animals In the Gulf of Mexico

Most mortality results from natural causes including red tide which is primarily restricted to West Florida and cold-kills that have greater influence in the warmer regions of South Texas and South Florida, but also kill a significant amount of fish and other animals in the northern Gulf. With the exception of red tide and other harmful algal blooms, the health of the Gulf has not been systematically studied. Mexico has only recently started to evaluate the health of its coastlines. Mortalities of marine animals, particularly fishes, in the Gulf caused by natural and anthropogenic events seem to interact with infectious disease agents and noninfectious diseases, but the mortalities are often attributed to the disease agents alone. “Events” that cause mortalities include eutrophication; hypoxia; algal blooms; temperature, salinity, and weather extremes; and chemical and sediment pollution. “Diseases” include those caused by infectious agents, parasites, neoplasms, and developmental abnormalities. Interactions of the effects of diseases and stressful events are considered important but little investigated