Questions tagged [symmetry]

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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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8 votes
2 answers
244 views

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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1 vote
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42 views

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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5 votes
1 answer
242 views

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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2 votes
1 answer
179 views

$G_2$ 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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2 votes
1 answer
88 views

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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5 votes
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75 views

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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2 votes
1 answer
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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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0 votes
0 answers
82 views

$0/1$ permanent symmetries

Assume matrix $A\in\{0,1\}^{n\times n}$ satisfies condition either $per(A)=0$ or $per(A)=1$. Under what conditions is the value of $per(A+A')>per(A)$ where $'$ is transpose? Under what conditions ...
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3 votes
0 answers
126 views

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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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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2 votes
0 answers
137 views

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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7 votes
1 answer
213 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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2 votes
0 answers
94 views

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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0 answers
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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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6 votes
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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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2 votes
0 answers
108 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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1 vote
2 answers
278 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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1 vote
0 answers
106 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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0 votes
0 answers
106 views

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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1 vote
1 answer
98 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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8 votes
1 answer
322 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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9 votes
1 answer
442 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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1 vote
1 answer
269 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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3 votes
0 answers
33 views

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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4 votes
2 answers
263 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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3 votes
0 answers
96 views

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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1 vote
1 answer
334 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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6 votes
3 answers
529 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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0 votes
0 answers
29 views

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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1 vote
0 answers
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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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1 vote
0 answers
38 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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3 votes
1 answer
154 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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2 votes
1 answer
91 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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4 votes
0 answers
101 views

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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6 votes
1 answer
143 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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2 votes
0 answers
95 views

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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2 votes
1 answer
111 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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9 votes
0 answers
88 views

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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4 votes
0 answers
41 views

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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4 votes
1 answer
252 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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2 votes
0 answers
184 views

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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2 votes
1 answer
83 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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3 votes
1 answer
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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3 votes
0 answers
86 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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3 votes
0 answers
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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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1 vote
0 answers
113 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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2 votes
0 answers
60 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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1 vote
2 answers
594 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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