Questions tagged [symmetry]

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Classification of maximal point groups

Have the maximal (without finite proper overgroups of the same dimensionality) finite point groups been fully classified in any dimensionality of Euclidean space greater than 4?
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66 views

A PDE involving a diffeomorphism of $\mathbb{S}^1$

This question is a special case of this one. Let $s(\theta)>0, b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$. Do there exist a diffeomorphism $\phi:\mathbb{S}^1 \to \...
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Does this geometric PDE have a solution?

Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes. Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
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1answer
189 views

Are there simplicial spheres with “non-geometric symmetries”?

Let $\Delta$ be a simplicial sphere, that is, a finite (abstract) simplicial complex whose canonical geometric realization $|\Delta|$ is homeomorphic to a sphere $\mathbf S^d\subset\Bbb R^{d+1}$. ...
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Minimal symmetry of a fibre bundle

Let $F \to E \to B$ be a topological fibre bundle with fibre $F$ and base $B$. It can be characterized by a map $B \to BAut(F)$. If it can also be characterized as a map $B \to BG$ (or say $G$ is a ...
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An easy way to recognize the edges of an orbit polytope?

Given a finite (orthogonal) matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and a point $x\in\Bbb R^d$. The corresponding orbit polytope is $$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \...
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Can a spherical simplicial complex have more than one “central” inversion?

Let $\Delta$ be a finite connected simplicial complex. Call a simplicial map $\phi:\Delta\to\Delta$ an inversion if $\phi$ is an involution, that is $\phi\circ\phi=\mathrm{id}$, and $\phi$ is not ...
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81 views

Is the projective symmetry group of a polytope more general than its linear symmetry group?

Give a (convex) polytope $P\subset\Bbb R^d$ (the convex hull of finitely many points). Consider its linear and projective symmetry groups: \begin{align} \DeclareMathOperator{\Aut}{Aut} \...
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2answers
262 views

Is there an area-preserving concentric diffeomorphism of the ellipse?

$\DeclareMathOperator\Vol{Vol}$This is a cross-post. Let $0<b<1$ be a fixed parameter, and let $(R(\theta),\theta)$ be the polar coordinates of the ellipse $$E=\{(x,y) \in \mathbb R^2 \, | \, \...
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96 views

Symmetry of points on unit sphere determined by relation between triples of points

Suppose we have $n$ points on the 3D unit sphere, $X = (\pmb{r}_{1}, \pmb{r}_{2}, ..., \pmb{r}_{n})$. I am interested in knowing to what extent the rotational symmetry of $X$ is determined by the ...
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Isomorphic Coxeter groups

After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?
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1answer
89 views

Qualitative analysis of the equation and symmetry (point on sphere)

A point moves on the surface of sphere ($R>0$ - radius) along the curve defined by the differential equation in spherical coordinate system: $R^2(|\dot \theta|^2 + w^2 \sin^2 \theta)=(at)^2$, ...
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304 views

Which reflection groups can be enlarged?

Based on this question (which focuses on the case $E_8$) I wonder the following: Question: For each finite reflection group $\Gamma\subseteq\mathrm O(\Bbb R^d)$, what is the largest finite group $\...
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1answer
410 views

Can $E_8$ be enlarged?

Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
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1answer
235 views

What is the symmetry group of this compound of two polytopes?

The geometric shape in question is a compound of two polytopes: an 11-hypercube with edge length $2$ and an 11-simplex with edge length $\sqrt6$ whose vertices are a subset of the hypercube’s. What is ...
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Are there uniform compounds of 135 $BC_8$ polytopes?

The Coxeter group $D_8$ is an index-135 subgroup of $E_8$. One of the consequences of this is that the rectified 8-orthoplex, whose coordinates can be given as all permutations and sign changes of $\{...
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244 views

Can every involution of a symmetric directed graph be written as a power of another symmetry?

Let $D=(V,A)$ be a finite directed graph, and suppose that $D$ is vertex-transitive, $D$ is edge-transitive, and between any two vertices there is at most one edge, in particular, if $(v,w)\in A$ ...
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Are there any other regular compounds?

Ever since I first read Coxeter’s definition of a regular compound (which seems to be the most commonly used), I didn’t like it on account of it being completely different than for properly connected ...
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1answer
222 views

Making use of extra symmetries; more examples?

TL; DR. In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing extra symmetries. Explicit examples come from compact groups, and I ...
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3answers
512 views

The randomness of modular squaring

For each arithmetic function $f:\mathbb{N}\rightarrow \mathbb{N}$ and each $n\in \mathbb{N}$ you can define a relation $f_{\textsf{mod } n}:[n]\times[n] \rightarrow \{0,1\}$ with $$f_{\textsf{mod } n}(...
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Vertex configuration to tile repeat unit

I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
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70 views

Projection of cocyclic Gaussian primes on the real axis

I just stumbled upon https://math.stackexchange.com/questions/2372062/4-concylic-points-of-gaussian-primes after a quick Google search about cocyclic Gaussian primes. As I've been investigating about ...
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37 views

How dense can a transitive sets of points be?

How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points? As a measure for density I use ...
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1answer
129 views

When is a $k$-distance-transitive graph already distance-transitive?

Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets $$D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\...
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1answer
73 views

A diameter 2 arc-transitive graph whose complement is not arc-transitive?

A graph $G=(V,E)$ is arc-transitive if its symmetry group acts transitively on ordered pairs of adjacent vertices. In general, the complement of an arc-transitive graph is not arc-transitive. But I ...
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Can we combine the symmetries of two polytopes to create a more symmetric polytope?

Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$. The symmetry group $\mathrm{Aut}(P_i)\subset\...
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1answer
106 views

A polytope with congruent facets and an insphere that is not facet-transitive?

Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties? All facets are congruent, it has an ...
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Balanced Gray codes for powers of 2

All of the binary 4-bit cyclic balanced Gray code sequences can be formed from simple reversals, bit-permutations, and circular shifts of the one Wikipedia example: ...
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1answer
110 views

Decomposition of the homogeneous polynomial ring $\{\mathbb R[x_{ij}]_{1\le i,j\le n}\}$ of degree 2 into Specht modules

I have tried to decompose this as following spans over the real field. $V_1=\operatorname{span} \langle x_{ij}^2\rangle$ $V_2=\operatorname{span} \langle x_{ij}x_{jk}\rangle$ $V_3=\operatorname{...
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A characterization of root systems via their intersections with halfspaces

In a recent preprint I obtained a nice characterization of root systems as a side product. I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
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Equiangular lines with symmetry requirements

Listing all possible arrangements of equiangular lines is non-trivial. Does the problem become any easier when we additionally require that the symmetry group of that line arrangement acts ...
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1answer
226 views

Examples of Calabi-Yau manifolds with $\mathbb{T}^2$ symmetry

I want to know if there exists examples of Calabi-Yau manifolds with $\mathbb{T}^2$-invariant $SU(n)$-structure. In particular these actions are both Killing and holomorphic. I am especially ...
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Definition of a moment map with physical context

This was originally posted on Math Stack Exchange, but without an answer. I thus move it here, and hope it's not because I express it unclearly. Suppose $(M,\omega)$ is a symplectic manifold "well" ...
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55 views

A matrix that commutes with all symmetries of a vertex-transitive polytope

Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope. Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\...
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1answer
3k views

Number of solutions and minimal clues in Sixy Sudoku

Sixy Sudoku is a variation on Latin squares and traditional sudoku played on a $6 \times 6$ grid with an initial clue of several cells filled in with a subset of the digits $1$–$6$. The task is to ...
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81 views

Where can I learn about the discrete symmetries of the complex projective plane (or space)?

I understand that $CP^1$ is the Riemann Sphere. I guess all its discrete symmetries were known for a long time and well-classified. (But suggestions or good references where this is worked out in a ...
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48 views

Matroids which are transitive on minimal basis exchanges

I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that. Consider a finite matroid $M$. Define a ...
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91 views

Symmetric subgraph configurations

Let $G,H$ be two simple graphs. Let's call a subgraph of $H$ that is isomorphic to $G$ a $G$-subgraph. Consider the following construction: Construction: Let $\mathcal G=\mathcal G(G,H)$ be a graph ...
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54 views

Are spherical maps with low distortion locally expanding?

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\Hom}[1]{\text{Hom}(#1)}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\S}{\mathbb{S}}$ The question in a nutshell: Are the "best" spherical maps ...
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569 views

Why does $\sqrt 5$ occur in manageable situations of these scenarios?

Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
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Can we approximate this matrix field with an invertible matrix field?

Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set $$\begin{equation*} A(x,y)=\left( \begin{array}{cc} x & -y \\ y & x \end{array} \right) \end{...
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2answers
230 views

Is a vertex- and edge-transitive polytope already a uniform polytope?

I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive). Question: Is every such ...
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Do cocycles “break” symmetry?

In an article by A. Borovik, “Is mathematics special?”, he talks about the fact that carrying is a cocycle. He then says [A student] discovered that carry is doing what cocycles frequently do: they ...
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1answer
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Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
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1answer
208 views

How large can a symmetric generating set of a finite group be?

Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...
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Cencov's “categories of figures”

In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-...
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2answers
143 views

Maximal edge length of symmetric polytopes

For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ...
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93 views

Is every “higher-order” harmonic morphism conformal?

$\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ $\newcommand{\TstarM}{...
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Counting without one-to-one correspondence? [closed]

Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of ...
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Does a map which preserve harmonic forms preserve co-closed forms (locally)?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth. Let $1 \le k \le d-1$ be fixed....