# Questions tagged [symmetry]

The symmetry tag has no usage guidance.

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### Minimal Non-Symmetric Closed Tracks Under 45-Degree Rotations

Given an arbitrary number of clockwise and anti-clockwise 45-degree track turns, what is the smallest closed track, that is neither axisymmetric nor rotationally symmetrical? For example, AAAAAAAA ...

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### Internal symmetries of partial differential relation via the nonholonomic jet bundle

On a smooth n-dimensional Riemannian manifold $M$, suppose I have the kth order partial differential relation (PDR) written in the form:
$$\mathcal{R}=\mathcal{R}\left(x^{i},u^{i},u_{j}^{i},u_{jk}^{i}\...

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143
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### Question on density of certain set of matrices

Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I ...

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### Does the edge-graph of a centrally symmetric polytope determine which vertices are antipodal?

Given two origin symmetric convex polytopes $P_1$ and $P_2$ (that is $P_i=-P_i$) with the same edge-graph, but potentially of different dimensions and combinatorial types.
Let $\phi: G_{P_1}\to G_{P_2}...

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### Can a polytopal graph be "centrally symmetric" in more than one way?

Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...

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123
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### Conserved quantities

So, if we have system of differential equations obtained from Lagrange function, by means of Noether theoerem (if we know some one-parameter symmetry group), we can derive conserved quantity.
But how ...

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53
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### integer network flow with symmetry

Suppose we have a weighted directed graph $G=(V,E,f)$. Each $e\in E$ is associated with $f_e\in \mathbb{N}$. There is a source node $s$, which only has outgoing edges, and a sink node $t$, which only ...

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### Surprising symmetry in the Ramanujan bound

The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy
$$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$
With a ...

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### Are square configurations the only critical points of the energy on the circle?

$\newcommand{\S}{\mathbb{S}^1}$
$\newcommand{\la}{\lambda}$Let$$M=\{(x_1,x_2,x_3,x_4) \in (\S)^4\,\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$
Define $E:M \to \mathbb{R}$ by
$$E(x_1,x_2,...

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532
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### Does a compact contractible metric space have a point that is fixed by all isometries?

Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
Question: Is there a point $x\in X$ fixed by all $\phi\in\...

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### Symmetries of contractable subsets of $\Bbb R^n$

Let $K\subset\Bbb R^n$ be a non-empty compact subset of $\Bbb R^n$. A symmetry of $K$ is an isometry of $\Bbb R^n$ that fixes $K$ set-wise. Since $K$ is compact, there is always a point $x\in\Bbb R^n$ ...

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### Relation between a symmetry of a Lagrangian and a canonical transformation

In Deligne and Freed: "Classical Field Theory. Quantum Fields and Strings: A course for Mathematicians" a symmetry of a Lagrangian $L$ is defined as a vector field $\xi$ so that there is a n-...

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530
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### Combinatorics and symmetry in matrices under row and column swaps

Suppose we have a $m\times n$ matrix and a sequence of numbers with which to fill the matrix, $\{c_1,c_2 \dots c_k \}$. I like to think of the numbers as colors, hence the notation. How many unique ...

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185
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### $G_2$ as the symmetry group of a geometric object

Is there a seven-dimensional geometric object whose full group of symmetries is isomorphic to the compact Lie group $G_2$, or does the same problem as with the special orthogonal groups occur?

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### Multiplicity of Dirichlet Laplacian eigenvalues of asymmetric domains

Let $\Delta$ be the Laplacian on a smooth domain $\Omega\subset \mathbb{R}^2$ with Dirichlet boundary conditions. I am interested in whether the implication
\begin{align}
\Omega \text{ is asymmetric } ...

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### Lie groupoids as symmetries of mechanical systems?

Lie groups are well studied as symmetries of mechanical systems in symplectic/Poisson geometry. For instance, if $G$ acts freely and properly on a mechanical system modeled by a symplectic manifold $(...

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### How to find the symmetry group of the differential equation

I'm studying the following differential equation
$$
x \frac{\partial^3}{\partial x^3} P[h, x]
= \left (x^3 \frac{\partial^3}{\partial x^3} +
3x^2 h \frac{\partial^3}{\partial x^2 \partial h} +
...

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### Erlangen program for "network geometry"

The subject of network geometry (Boguna et al., Network Geometry,
Nature Reviews Physics 2021) looks at "geometric aspects" of complex networks.
This is about studying a metric on the nodes, ...

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63
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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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### 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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142
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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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### 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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123
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### 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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### 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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### 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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143
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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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### 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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339
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### 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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482
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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114
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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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### 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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287
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### 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 ...