3
votes
0answers
21 views

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 ...
2
votes
2answers
54 views

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 ...
5
votes
2answers
109 views

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 ...
1
vote
3answers
78 views

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 ...
1
vote
0answers
57 views

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 ...
1
vote
1answer
139 views

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 ...
0
votes
1answer
19 views

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 ...
5
votes
1answer
95 views

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 ...
2
votes
1answer
53 views

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 ...
0
votes
1answer
27 views

Pauli Spin Operator General Rotation

I would like to calculate the Pauli spin operator rotation $$ U^{\dagger } \overset{\rightharpoonup }{\sigma } U$$ where $$\overset{\rightharpoonup }{\sigma }=\sigma _x \overset{\rightharpoonup ...
8
votes
2answers
119 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 ...
2
votes
2answers
86 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 ...
0
votes
1answer
33 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 ...
1
vote
1answer
82 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 ...
1
vote
1answer
178 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 ...
1
vote
0answers
86 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 ...
0
votes
1answer
199 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( ...
9
votes
3answers
367 views

Commutative Rotations

In three dimensions, I know that in general rotations on the unit sphere are non-commutative, but I was wondering if there is a subset/subgroup of rotations that are commutative, and what this type of ...
3
votes
1answer
391 views

What is the difference between a “change of basis” and a “similarity transformation”?

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. ...