Related papers: The Isomorphism of $H_4$ and $E_8$

An operation of associative, commutative and distributive multiplication on { Euclidean vector space} $\mathbb{E}_4$ is introduced by a skew circulant matrix. The resulting algebra $\mathbb{W}$ over $\mathbb{R}$ is isomorphic to $\mathbb{C}…

Covariant or invariant functions under a compact linear group can be expressed in terms of functions defined in the orbit space of the group. The semialgebraic relations defining the orbit spaces of all finite coregular real linear groups…

Independently trained transformers compute the same function in residual-stream bases that differ by a uniform random rotation on $\mathrm{SO}(d_{\mathrm{model}})$. We call this phenomenon polymorphism: same function, mutually…

Starting from basic identities of the group E8, we perform progressive reductions, namely decompositions with respect to the maximal and symmetric embeddings of E7xSU(2) and then of E6xU(1). This procedure provides a systematic approach to…

We provide the classification of real forms of complex D=4 Euclidean algebra $\mathcal{\epsilon}(4; \mathbb{C}) = \mathfrak{o}(4;\mathbb{C})) \ltimes \mathbf{T}_{\mathbb{C}}^4$ as well as (pseudo)real forms of complex D=4 Euclidean…

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the…

In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $\mathbb{F}_q$ of odd degree over $\mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $\pm\sqrt{q}$. This extends…

Witt (1996) has shown that for an elliptical potential, the four images of a quadruply lensed quasar lie on a rectangular hyperbola that passes through the unlensed quasar position and the center of the potential as well. Wynne and…

Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we…

Suppose $F=\mathbb{Q}(\sqrt{-p_1\dotsm p_t})$ is an imaginary quadratic number field with distinct primes $p_1,\dots,p_{t}$, where $p_i\equiv 1\pmod{4}$ ($i=1,\dots,t-1$) and $p_t\equiv 3\pmod{4}$. We express the possible values of the…

We argue that $\mathcal N=8$ supergravity in four dimensions exhibits an exceptional $E_{8(8)}$ symmetry, enhanced from the known $E_{7(7)}$ invariance. Our procedure to demonstrate this involves dimensional reduction of the $\mathcal N=8$…

Hexagonal SiGe is a promising material for combining electronic and photonic technologies. In this work, the energetic, structural, elastic and electronic properties of the hexagonal polytypes (2$H$, 4$H$ and 6$H$) of silicon and germanium…

We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description…

In this note we investigate the convex hull of those $n \times n$-permutation matrices that correspond to symmetries of a regular $n$-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein…

The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two $m$-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of…

The symmetric group on 4 letters has the reflection group $D_{3}$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_{3}$ and $D_{3}$. The isomorphism is used to construct an orthogonal basis of…

We construct interacting Sp(4,H)/Z_2 pair matrix models inside the compact E6 matrix models. Generally, models based on the compact E6 seem to always include doubly the degrees of freedom that we need physically. In this paper, we propose…

We study the spatial structure of four valence neutrons in the ground state of $^8$He and $^{18}$C nuclei using a core+4$n$ model. For this purpose, we employ a density-dependent contact interaction among the valence neutrons, and solve the…

The explicit matrix realizations of the reversion anti-automorphism and the spin group depend on the set of matrices chosen to represent a basis of 1 -vectors for a given Clifford algebra. On the other hand, there are iterative procedures…

We study the deformation complex of a canonical morphism $i$ from the properad of (degree shifted) Lie bialgebras $\mathbf{Lieb}_{c,d}$ to its polydifferential version $\mathcal{D}(\mathbf{Lieb}_{c,d})$ and show that it is quasi-isomorphic…