Related papers: ${\rm Spin}(7)$ is unacceptable

Let $L/K$ be a finite Galois extension of fields with group $\Gamma$. When $\Gamma$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the…

Let F be a finite field of odd cardinality, and let G= GL2(F). The group G \times G \times G acts on F^2 \otimes F^2 \otimes F^2 via symplectic similitudes, and has a natural Weil representation. Answering a question rasised by V. Drinfeld,…

6D spinors with $Spin(3,3)$ symmetry are utilized to efficiently encode three generations of matter. $E_{8(-24)}$ is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations…

Given a $p$-adic group $G$ equipped with an action of a finite group $\Gamma\subset\mathrm{Aut}_F(\mathbf{G})$, and a reductive fixed-point subgroup $G^\Gamma$, we establish a relationship between constructions of types for these two groups…

We summarize recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces $\Gamma \backslash G/H$, beyond the classical setting where $H$ is compact (e.g. theory of automorphic forms for arithmetic $\Gamma$) or…

For a right-angled Artin group $A_\Gamma$, the untwisted outer automorphism group $U(A_\Gamma)$ is the subgroup of $Out(A_\Gamma)$ generated by all of the Laurence-Servatius generators except twists (where a {\em twist} is an automorphisms…

Let $F$ be a local non-Archimedean field and $E$ a finite Galois extension of $F$, with Galois group $G$. If $\rho$ is a representation of $G$ on a complex vector space $V$, we may compose it with any tensor operation $R$ on $V$, and get…

The possibility of using spin connection components as basic quantization variables of a conformal version of General Relativity is studied. The considered model contains gravitational degrees of freedom and a scalar dilaton field. The…

Let $F$ be a non-Archimedean local field with odd characteristic $p$. Let $N$ be a positive integer and $G=Sp_{2N}(F)$. By work of Lomel\'i on $\gamma$-factors of pairs and converse theorems, a generic supercuspidal representation $\pi$ of…

Any complex-analytic vector bundle $\mathbb E$ admits naturally defined homotheties $\phi_{\alpha}$, $\alpha\in \mathbb C^*$, i.e. $\phi_{\alpha}$ is the multiplication of a local section by a complex number $\alpha$. We investigate the…

Let $p$ be a prime and $K$ be a $p$-adic local field. We study the stack of quasi-deRham $(\varphi,\Gamma_K)$-modules, i.e. $(\varphi,\Gamma_K)$-modules that are deRham up to twist by characters. These objects are used to construct and then…

Let $\Gamma=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(\Gamma)$. The Cayley index of $\Gamma$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a…

We determine the eigenvalues with multiplicity of each element of an alternating group in any irreducible representation. This is equivalent to determining the decomposition of cyclic representations of alternating groups into irreducibles.…

The action of the subgroup $\operatorname{G}_2$ of $\operatorname{SO}(7)$ (resp.\ $\operatorname{Spin}(7)$ of $\operatorname{SO}(8)$) on the Grassmannian space $M=\frac{\operatorname{SO}(7)}{\operatorname{SO}(5)\times\operatorname{SO}(2)}$…

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. This paper is about reversibles in the group $G$ of formally-invertible pairs of formal power series in two variables, with complex…

We prove that the plane Cremona group over a perfect field with at least one Galois extension of degree 8 is a non-trivial amalgam, and that it admits a surjective morphism to a free product of groups of order two.

Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and let $W$ be a finite-dimensional $K$-space of dimension greater than or equal to $5.$ In this paper, we give the structure of certain Weyl modules for…

A finite spin system invariant under a symmetry group G is a very illustrative example of the finite group action on a set of mappings f:X->Y. In the case of spin systems X is a set of spin carriers and Y contains 2s+1 z-components -s<=m<=s…

We describe the different classes of $\mathrm{Spin(7)}$ structures in terms of spinorial equations. We relate them to the spinorial description of $\mathrm{G}_2$ structures in some geometrical situations. Our approach enables us to analyze…

It is well-known that perturbative quantum gravity is non-renormalizable. The metric or vierbein has generally been used as the variable to quantize in perturbative quantum gravity. In this essay, we show that one can use the spin…