506 research outputs found
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
Testing the Cactus code on exact solutions of the Einstein field equations
The article presents a series of numerical simulations of exact solutions of
the Einstein equations performed using the Cactus code, a complete
3-dimensional machinery for numerical relativity. We describe an application
(``thorn'') for the Cactus code that can be used for evolving a variety of
exact solutions, with and without matter, including solutions used in modern
cosmology for modeling the early stages of the universe. Our main purpose has
been to test the Cactus code on these well-known examples, focusing mainly on
the stability and convergence of the code.Comment: 18 pages, 18 figures, Late
Three Dimensional Numerical General Relativistic Hydrodynamics I: Formulations, Methods, and Code Tests
This is the first in a series of papers on the construction and validation of
a three-dimensional code for general relativistic hydrodynamics, and its
application to general relativistic astrophysics. This paper studies the
consistency and convergence of our general relativistic hydrodynamic treatment
and its coupling to the spacetime evolutions described by the full set of
Einstein equations with a perfect fluid source. The numerical treatment of the
general relativistic hydrodynamic equations is based on high resolution shock
capturing schemes. These schemes rely on the characteristic information of the
system. A spectral decomposition for general relativistic hydrodynamics
suitable for a general spacetime metric is presented. Evolutions based on three
different approximate Riemann solvers coupled to four different discretizations
of the Einstein equations are studied and compared. The coupling between the
hydrodynamics and the spacetime (the right and left hand side of the Einstein
equations) is carried out in a treatment which is second order accurate in {\it
both} space and time. Convergence tests for all twelve combinations with a
variety of test beds are studied, showing consistency with the differential
equations and correct convergence properties. The test-beds examined include
shocktubes, Friedmann-Robertson-Walker cosmology tests, evolutions of
self-gravitating compact (TOV) stars, and evolutions of relativistically
boosted TOV stars. Special attention is paid to the numerical evolution of
strongly gravitating objects, e.g., neutron stars, in the full theory of
general relativity, including a simple, yet effective treatment for the surface
region of the star (where the rest mass density is abruptly dropping to zero).Comment: 45 pages RevTeX, 34 figure
Implementation of standard testbeds for numerical relativity
We discuss results that have been obtained from the implementation of the
initial round of testbeds for numerical relativity which was proposed in the
first paper of the Apples with Apples Alliance. We present benchmark results
for various codes which provide templates for analyzing the testbeds and to
draw conclusions about various features of the codes. This allows us to sharpen
the initial test specifications, design a new test and add theoretical insight.Comment: Corrected versio
Scaling behaviour of three-dimensional group field theory
Group field theory is a generalization of matrix models, with triangulated
pseudomanifolds as Feynman diagrams and state sum invariants as Feynman
amplitudes. In this paper, we consider Boulatov's three-dimensional model and
its Freidel-Louapre positive regularization (hereafter the BFL model) with a
?ultraviolet' cutoff, and study rigorously their scaling behavior in the large
cutoff limit. We prove an optimal bound on large order Feynman amplitudes,
which shows that the BFL model is perturbatively more divergent than the
former. We then upgrade this result to the constructive level, using, in a
self-contained way, the modern tools of constructive field theory: we construct
the Borel sum of the BFL perturbative series via a convergent ?cactus'
expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our
method shows how the ?sum over trian- gulations' in quantum gravity can be
tamed rigorously, and paves the way for the renormalization program in group
field theory
- …