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 $A_3$ 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 E8E_8 out of the Spinors of the Icosahedron

$E_8$ 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 $E_8$ features heavily in ten-dimensional superstring theory. Contrary to that point of view, here we show that the $E_8$ root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of three-dimensional geometry. The $240$ roots of $E_8$ arise in the 8D Clifford algebra of 3D space as a double cover of the $120$ elements of the icosahedral group, generated by the root system $H_3$. 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 $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $\Delta + aK + W$, where $\Delta$ is the non-negative Laplacian, $K$ the Gaussian curvature, $W$ a locally integrable function, and $a$ a positive real number. Assuming that the positive part of $W$ is integrable, we address the question "What conclusions on $(M,g)$ and $W$ can one draw from the fact that the operator $\Delta + aK + W$ 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 $W$ is non-positive and $a = 1/4$ or $a \in (0,1/4)$

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