# Questions tagged [symmetry]

Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry

272 questions
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### Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
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### Seeing symmetries

Preliminaries Let $[n] = \{0,\dots,n-1\}$ and $P([n])$ be the power set of $[n]$. Let the correlation between two subsets $x,y$ of $[n]$ be the number $\kappa(x,y) = 1 - \frac{2}{n}|x\triangle y|$ ...
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### Reduce an ODE by One Dimension

I am reading Arnold's ODE but I cannot solve this problem. This is on page 79. $\mathbf{Problem.}$ Suppose a one-parameter group of symmetries of a direction field in an $n$-dimensional domain is ...
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### Why contains the product of highest dimensional representation (with dim$\ne 1$) with itself the rotation around the main axis?

I observe that in the "standard finite point group" symmetries containing higher dimensional irreducible representations (over $\mathbb{R}$), the total product of the highest dimensional ...
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### On sums over non-symmetric sets

Definition: We say that a subset $A\subset \mathbb{N}_0\times \mathbb{N}_0$ is symmetric if it satisfies the property: $(a,b)\in A \Rightarrow (b,a)\in A.$ We say that the subset $A$ is non-symmetric ...
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### Plane shape with three axes of symmetry

If a plane shape $\phi$ has three different axes of symmetry (all belonging to the shape plane), they all interesect at the same point. This is fairly obvious but I could not find an elementary ...
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### Can we characterise concircular vector fields by their flow?

Let $M$ be a Riemannian manifold. A vector field $X$ on $M$ is called concircular if $\nabla X=h \text{Id}_{TM}$ for some $h \in C^{\infty}(M)$, where $\nabla$ is the Levi-Civita connection. Every ...
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### Determinant of special partitioned matrix in terms of submatrix determinants

I have a few determinant-related questions that I've been struggling with for at least a few days. I couldn't see a similar question on here. So, here it is: I wrote my own electromagnetics moment ...
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### Determining size of symmetry groups with orbit-stabilizer theorem and Burnside's lemma.

The classic orbit-stabilizer theorem and Burnside lemma problems tend to have the following structure: Consider some object with a symmetry group, like a cube and its rotational symmetry group. "...
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### Symmetric solutions to second order boundary value problem

Given the equation $u''=f(u)$ with symmetric boundary conditions $u(-a)=u(a)=u_0$, is there any proof that possibly relies on dynamical systems techniques to show that the solutions are symmetric (i....
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### Find the group of symmetries of the Cube

Exercise : Find the group of symmetries of the Cube. Attempt : The elements are: $3$ rotations (by $\pi/2$ or $\pi$) about the centers of $3$ pairs of opposite faces. $1$ rotation (by $\pi$) ...
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### Symmetry of an object suspended inside another object

I am from a chemistry background and I haven't taken abstract algebra, so it is possible that this question may be basic. This is the problem that I have: Suppose that I have an object A, which is a ...
$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth manifolds, $\M$ oriented and equipped with a Riemannian metric. Let $L : J^1(\M,\N) \to \mathbb R$ be smooth; We ...