25 research outputs found
Chopped and sliced cones and representations of Kac-Moody algebras
We introduce the notion of a chopped and sliced cone in combinatorial
geometry and prove two structure theorems for the number of integral points in
the individual slices of such a cone. We observe that this notion applies to
weight multiplicities of Kac-Moody algebras and Littlewood-Richardson
coefficients of semisimple Lie algebras, where we obtain the corresponding
results.Comment: 9 pages, 1 figur
On weight multiplicities of complex simple Lie algebras
The author introduces the notion of a chopped and sliced cone and shows that the weight multiplicities of semisimple complex Lie algebras are governed by this notion. From this he derives a presentation of the weight multiplicity function as a composition of a vector partition function and a linear map. By virtue of this presentation he obtains structural and asymptotic properties of weight multiplicities; for example a proof of G. Heckman's theorem on the Duistermaat-Heckman measure is obtained in this way. The author describes an algorithm for computing general vector partition functions and gives an implementation as a Maple prototype. Using this program he determines and states the weight multiplicity function of the Lie algebra so5(C) completely
Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann polytopes as marked poset polytopes
Stanley (1986) showed how a finite partially ordered set gives rise to two
polytopes, called the order polytope and chain polytope, which have the same
Ehrhart polynomial despite being quite different combinatorially. We generalize
his result to a wider family of polytopes constructed from a poset P with
integers assigned to some of its elements.
Through this construction, we explain combinatorially the relationship
between the Gelfand-Tsetlin polytopes (1950) and the Feigin-Fourier-Littelmann
polytopes (2010), which arise in the representation theory of the special
linear Lie algebra. We then use the generalized Gelfand-Tsetlin polytopes of
Berenstein and Zelevinsky (1989) to propose conjectural analogues of the
Feigin-Fourier-Littelmann polytopes corresponding to the symplectic and odd
orthogonal Lie algebras.Comment: 12 page
Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes
Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras
Lattice Point Generating Functions and Symmetric Cones
We show that a recent identity of Beck-Gessel-Lee-Savage on the generating
function of symmetrically contrained compositions of integers generalizes
naturally to a family of convex polyhedral cones that are invariant under the
action of a finite reflection group. We obtain general expressions for the
multivariate generating functions of such cones, and work out the specific
cases of a symmetry group of type A (previously known) and types B and D (new).
We obtain several applications of the special cases in type B, including
identities involving permutation statistics and lecture hall partitions.Comment: 19 page
Model-Based Analysis of SARS-CoV-2 Infections, Hospitalization and Outcome in Germany, the Federal States and Districts
The coronavirus disease 2019 (COVID-19) pandemic challenged many national health care
systems, with hospitals reaching capacity limits of intensive care units (ICU). Thus, the estimation of
acute local burden of ICUs is critical for appropriate management of health care resources. In this
work, we applied non-linear mixed effects modeling to develop an epidemiological SARS-CoV-2
infection model for Germany, with its 16 federal states and 400 districts, that describes infections as
well as COVID-19 inpatients, ICU patients with and without mechanical ventilation, recoveries, and
fatalities during the first two waves of the pandemic until April 2021. Based on model analyses, covariates influencing the relation between infections and outcomes were explored. Non-pharmaceutical
interventions imposed by governments were found to have a major impact on the spreading of
SARS-CoV-2. Patient age and sex, the spread of variant B.1.1.7, and the testing strategy (number of
tests performed weekly, rate of positive tests) affected the severity and outcome of recorded cases and
could reduce the observed unexplained variability between the states. Modeling could reasonably
link the discrepancies between fine-grained model simulations of the 400 German districts and the
reported number of available ICU beds to coarse-grained COVID-19 patient distribution patterns
within German regions