Faces of highest weight modules and the universal Weyl polyhedron

Let $V$ be a highest weight module over a Kac-Moody algebra $\mathfrak{g}$, and let conv $V$ denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv $V$, i.e. we completely classify the faces and their inclusions. In the special case where $\mathfrak{g}$ is semisimple, this brings closure to a question studied by Cellini-Marietti [IMRN 2015] for the adjoint representation, and by Khare [J. Algebra 2016; Trans. Amer. Math. Soc. 2017] for most modules. The determination of faces of finite-dimensional modules up to the Weyl group action and some of their inclusions also appears in previous work of Satake [Ann. of Math. 1960], Borel-Tits [IHES Publ. Math. 1965], Vinberg [Izv. Akad. Nauk 1990], and Casselman [Austral. Math. Soc. 1997]. For any subset of the simple roots, we introduce a remarkable convex cone which we call the universal Weyl polyhedron, which controls the convex hulls of all modules parabolically induced from the corresponding Levi factor. Namely, the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules, as well as how faces degenerate as the highest weight gets increasingly singular. To our knowledge, this cone is new in finite and infinite type. We further answer a question of Michel Brion, by showing that the localization of conv $V$ along a face is always the convex hull of the weights of a parabolically induced module. Finally, as we determine the inclusion relations between faces representation-theoretically from the set of weights, without recourse to convexity, we answer a similar question for highest weight modules over symmetrizable quantum groups.Comment: Final version, to appear in Advances in Mathematics (42 pages, with similar margins; essentially no change in content from v2). We recall preliminaries and results from the companion paper arXiv:1606.0964

Sequestration and Characterization of Soil Organic Carbon for Shelterbelt Agroforestry Systems in Saskatchewan

The increase in atmospheric concentration of carbon dioxide (CO₂) is contributing to global climate change. Agroforestry systems, such as shelterbelts, can contribute to the mitigation of increasing CO₂ levels, through carbon (C) sequestration in plant biomass and soils. However, little information is available on the storage and dynamics of soil organic carbon (SOC) for shelterbelt systems. The objective of this research was to examine the effect of shelterbelt plantings on the storage, physical stabilization and chemical composition of SOC for major shelterbelt species across Saskatchewan compared to adjacent agricultural fields. Soil and litter samples were collected for six major shelterbelt species including green ash (Fraxinus pennsylvanica), hybrid poplar (Populus spp.), Manitoba maple (Acer negundo), white spruce (Picea glauca), Scots pine (Pinus sylvestris) and caragana (Caragana arborescens) and the adjacent agricultural fields at 59 sampling sites across the agricultural region of Saskatchewan. Measurement of SOC concentration for soil samples was preceded by fumigation with concentrated HCl (12N), which was determined to be the efficient method for SOC determination in carbonate-rich soils. Physical stabilization of SOC was characterized by using the density fraction technique to separate SOC into uncomplexed, plant-derived debris (i.e. light fraction) and mineral-associated organic matter (i.e. heavy fraction). Changes in SOC composition due to shelterbelt plantation were studied using attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy and synchrotron based carbon K-edge X-ray absorption near edge structure (XANES) spectroscopy. Concentration of SOC for shelterbelts was significantly higher compared to agricultural fields throughout the soil profile (0-50 cm). Sequestration of SOC for shelterbelts varied from 6-38 Mg C ha⁻¹ under different shelterbelt species, along with 3-8 Mg C ha⁻¹ stored in the litter layer. Shelterbelts led to an increase in SOC content for both the labile light fraction and the mineral-associated heavy fraction. The increase in the heavy fraction was higher in coniferous shelterbelt species including white spruce and Scots pine, while the increase in the light fraction C was higher in hybrid poplar, Manitoba maple, green ash and caragana. These trends were attributed to differences in quality and decomposition rate of litter among shelterbelt species. Maximum amount of SOC was sequestered at the 10-30 cm soil depth, and the majority (70%) of it was in the stable mineral-associated form. Light fraction C was predominant in the surface layer (0-10 cm), where it accounted for 92% of the total sequestered C. Younger shelterbelts tended to lose SOC in the early years of shelterbelt establishment, but eventually resulted in net addition of C after about 20 years of age. SOC sequestration potential of shelterbelts was positively related to shelterbelt characteristics including stand age, tree height, diameter and crown width and density of litter layer. These variables together explained 56-67% of the inter-site variability in the amount of SOC sequestered. Analysis of molecular composition of SOC revealed shelterbelts had higher abundance of processed forms of C such as aromatic and conjugated carboxyl groups for hybrid poplar and white spruce shelterbelts and aromatic and aliphatic C moieties for Manitoba maple shelterbelts. In contrast, agricultural field soils were enriched in easily degradable C forms such as polysaccharides. These results revealed a strong effect of initial litter quality and extent of decomposition on SOC composition. Together, these findings indicate that shelterbelt planting leads to sequestration of SOC, resulting in the decrease of atmospheric CO₂ concentration. Additionally, shelterbelts also influence organo-mineral association and molecular composition of SOC, which may affect stabilization and dynamics of sequestered SOC

Recovering from Hail

Non-Peer ReviewedHailstorms can be responsible for significant economic loss to the agricultural sector in Alberta. Foliar applications of certain fungicides and nutrient blends have been advocated to promote recovery and yield of hail-damaged crops. Proper understanding of different crop- and hail-related factors is required for an accurate assessment of hail damage to crops, and for evaluations of hail-recovery product claims. This study was undertaken at three locations in Alberta during three growing seasons (2016-18) to determine the effect(s) of two levels of simulated hail severity at three different crop developmental stages including early vegetative, middle vegetative and reproductive stages. Plant growth, yield and grain quality parameters of wheat, field pea and dry bean crops were measured. Simulated hail damage led to reductions in crop height, biomass, canopy cover, grain yield and kernel weight of all three crops. Timing of simulated hail was a critical factor influencing the extent of crop damage with early damage to vegetative stages having less effect on yield compared to damage at later reproductive stages. This trend was especially evident in wheat which did not show significant reduction in yield from the damage at early tillering stages but had significant yield loss with damage at flowering. Foliar applications of fungicides and nutrient blends did not significantly improve crop recovery, grain yield or kernel weight for any of the crops in this study, and thus, their use for the recovery of hail-affected wheat, field pea and dry bean was not supported by the results of this study. Link to Video Presentation: https://youtu.be/vgUDIufNwP

Effect of precision planting and seeding rates on canola plant density and seed yield in south Alberta

Precision planters are recently being adopted for seeding canola to improve crop establishment and seed yield. This study determined the effect of seeding canola using precision planters (30.5 and 50.8 cm seeding row width) and conventional air drill seeders at different rates (20, 40, 60, 80 and 160 seeds m-2) on plant density and seed yield. The study was conducted for four years (2016 to 2019) at three locations in southern Alberta. Plant density increased with higher seeding rates following the negative exponential function distribution. The yield-density relationship was non-linear asymptotic in nature and weak-to-moderate in strength at most site-years. The parameters of yield-density relationship did not show statistically significant differences among the air drill and precision planters. When averaged among seeding rates, canola yield was higher for the narrow row precision planter at five site-years and for the air drill at two site-years out of a total of 12 site-years. Under irrigated and high-precipitation conditions, seed yield in narrow-row precision planted canola was higher than air drill seeded canola. There was an average increase of 463 kg ha-1 (10%) in the seed yield in narrow-row precision planted canola compared to the air drill seeded canola among irrigated systems. However, under water-limited conditions, seed yield in air drill seeded canola was comparable or higher than the precision planted canola. Wide-row planter led to poor crop establishment and seed yield under both irrigated and dryland conditions, attributed to higher in-row plant density due to wider row spacing.The accepted manuscript in pdf format is listed with the files at the bottom of this page. The presentation of the authors' names and (or) special characters in the title of the manuscript may differ slightly between what is listed on this page and what is listed in the pdf file of the accepted manuscript; that in the pdf file of the accepted manuscript is what was submitted by the author