# Tagged Questions

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### Rotations in higher dimensions vs. Spheres

Where my questions stem from: When we study the rotations in a plane or of some specific higher dimensions, there exists a neat approach to represent all the rotations as a spheres $\mathbb S^i$, for ...
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### Is $SU(2)\times SU(2)=SU(4)$ true? [on hold]

I was wondering if $SU(2)\times SU(2)=SU(4)$ In other terms if $Spin(4)=SU(4)$ Thanks!
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### When do two configurations of points belong to the same Euler Equivalence Class?

When can I say, of two or more configurations of points in a plane, that they belong to the same Euler Equivalence Class? From Euler's rotation theorem, I gather that two configurations of points are ...
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### Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
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### Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
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### Dimension of $SO_n(\mathbb{R})$

Is there a simple proof that the dimension of $SO_n(\mathbb{R})$, a.k.a the group of rotations in $n$-dimensional space is $(n-1)n/2$? It would be great to see some proofs based only on the ...
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### How to Visualize Diagonally Opposite Vertices

Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the ...
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### Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$

Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$. I think $S^2$ means a 2-D sphere and $SO(3)$ is the usual $SO(3)$ group. I'm unsure how to prove that $SO(3)$ acts ...
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### Writing a rotation as a product of translation and rotation about origin

In Artin's Algebra 2011 we have Lemma 6.3.5: "An isometry $f$ that has the form $m=t_a\rho_\theta$, with $\theta\neq 0$, is a rotation through the angle $\theta$ about a point in the plane." Earlier ...
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### Representation Theory of the Dihedral Group $D_{2n}$

So I'm pretty new into Representation Theory having so far covered only a couple of example sheets. I'm thinking about the following question: Suppose we have the group $D_{2n}$ (for clarity this is ...
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### Why does knowing where two adjacent vertices of regular $n$-gon move under rigid motion determines the motion?

I am reading the book Abstract Algebra by Dummit and Foote. In the section about the group $D_{2n}$ (of order $2n$) the authors claim that knowing where two adjacent vertices move to, completely ...
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### Could someone explain chirality from a group theory point of view?

While answering this question my interest in the rotation/reflection group was piqued. I personally know very basic group theory, not much more than what a group really is. I understand that the ...
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### What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...
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I would like to calculate the Pauli spin operator rotation $$U^{\dagger } \overset{\rightharpoonup }{\sigma } U$$ where $$\overset{\rightharpoonup }{\sigma }=\sigma _x \overset{\rightharpoonup ... 2answers 154 views ### The rotation of dice on a grid Imagine you have a plane, flat surface with a square grid drawn on it. You have a standard cubic die which is placed flat on the surface. Its length is the same as the length of side of each grid ... 2answers 98 views ### A mathematical model for rotations of a die I have a normal 6-sided die and I would like to find a model for how it transforms under rotatins (north, south, east, west), that can help me determine which side is up. Just to make sure we are ... 1answer 34 views ### Rotation of a vector and number of parameters needed From mathematics we know that rotations of of a vector in three-dimensional Euclidean space are described by representations of SO(3) group, which can be parametrized using three angles (one for each ... 1answer 85 views ### How does one create a rotation about a given axis in R^{3} from rotations about the other axes? I was told that you can rotate a vector about a given axis in Cartesian space by combining rotations about the other two axes. I found a quick method for 90 degree rotations but I'm unsure how to ... 1answer 225 views ### Relation between irreps of SO(2) and SO(3)? Is there a relation between the irreducible representations of SO(2) and SO(3)? For instance, consider an n-by-n matrix representation of SO(3), G, if I restrict all of my rotations to have a common ... 0answers 95 views ### What are the other elements that stabilize a body diagonal of a cube besides the group of rotations about that diagonal? I am trying to understand why the order of the stabilizer group of a body diagonal of a cube is 6 rather than 3. It is clear to me that rotations about that diagonal stabilize the diagonal, by ... 1answer 221 views ### What transforms under SU(2) as a matrix under SO(3)? A vector \boldsymbol{r} in \mathbb{R}^3 transforms under rotation \boldsymbol{A} to \boldsymbol{r}'=\boldsymbol{Ar}. It is equivalent to an SU(2) "rotation" as$$\left( ...
In crystallography we define a "misorientation", $M_{AB/A}$, as the rotation required to bring crystal A into coincidence with crystal B, as measured with respect to the reference frame of crystal A. ...