36,479 research outputs found
Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups
In this paper, we make the case that Clifford algebra is the natural
framework for root systems and reflection groups, as well as related groups
such as the conformal and modular groups: The metric that exists on these
spaces can always be used to construct the corresponding Clifford algebra. Via
the Cartan-Dieudonn\'e theorem all the transformations of interest can be
written as products of reflections and thus via `sandwiching' with Clifford
algebra multivectors. These multivector groups can be used to perform concrete
calculations in different groups, e.g. the various types of polyhedral groups,
and we treat the example of the tetrahedral group in detail. As an aside,
this gives a constructive result that induces from every 3D root system a root
system in dimension four, which hinges on the facts that the group of spinors
provides a double cover of the rotations, the space of 3D spinors has a 4D
euclidean inner product, and with respect to this inner product the group of
spinors can be shown to be closed under reflections. In particular the 4D root
systems/Coxeter groups induced in this way are precisely the exceptional ones,
with the 3D spinorial point of view also explaining their unusual automorphism
groups. This construction simplifies Arnold's trinities and puts the McKay
correspondence into a wider framework. We finally discuss extending the
conformal geometric algebra approach to the 2D conformal and modular groups,
which could have interesting novel applications in conformal field theory,
string theory and modular form theory.Comment: 14 pages, 1 figure, 5 table
The Birth of out of the Spinors of the Icosahedron
is prominent in mathematics and theoretical physics, and is generally
viewed as an exceptional symmetry in an eight-dimensional space very different
from the space we inhabit; for instance the Lie group features heavily in
ten-dimensional superstring theory. Contrary to that point of view, here we
show that the root system can in fact be constructed from the icosahedron
alone and can thus be viewed purely in terms of three-dimensional geometry. The
roots of arise in the 8D Clifford algebra of 3D space as a double
cover of the elements of the icosahedral group, generated by the root
system . As a by-product, by restricting to even products of root vectors
(spinors) in the 4D even subalgebra of the Clifford algebra, one can show that
each 3D root system induces a root system in 4D, which turn out to also be
exactly the exceptional 4D root systems. The spinorial point of view explains
their existence as well as their unusual automorphism groups. This spinorial
approach thus in fact allows one to construct all exceptional root systems
within the geometry of three dimensions, which opens up a novel interpretation
of these phenomena in terms of spinorial geometry.Comment: 14 pages, 2 figures, 1 tabl
Inverse spectral positivity for surfaces
Let be a complete non-compact Riemannian surface. We consider
operators of the form , where is the non-negative
Laplacian, the Gaussian curvature, a locally integrable function, and
a positive real number. Assuming that the positive part of is
integrable, we address the question "What conclusions on and can
one draw from the fact that the operator is non-negative ?"
As a consequence of our main result, we get a new proof of Huber's theorem and
Cohn-Vossen's inequality, and we improve earlier results in the particular
cases in which is non-positive and or
The empirics of economic geography: How to draw policy implications?
Using both reduced-form and structural approaches, the spectrum of policy recommendations that can be drawn from empirical economic geography is pretty large. Reduced-form approaches allow the researchers to consider many variables that impact on regional disparities, as long as they are careful about interpretation and endogeneity issues. Structural approaches have the opposite advantages. Less issues can be simultaneously addressed, but one can be more precise in terms of which intuitions are considered and the underlying mechanisms and effects at work. Many regional policy issues remain unanswered, opening some interesting future lines of research.agglomeration economies; regional policy; structural estimation; instrumental variables
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