Groups Lecture 20

in which we rotate and reflect in two and three dimensions, and meet the unitary group.

Today we investigated \mathrm{O}_2 and \mathrm{O}_3 in more detail. We already proved last time that \mathrm{SO}_2 consist of all rotations of \mathbb{R}^2 around 0, and today we added that any matrix in \mathrm{O}_2 is a rotation or a reflection (in a line through 0). We made use of the fact that \mathrm{O}_2 is partitioned into the two cosets \mathrm{SO}_2 and \left(\begin{smallmatrix}1 & 0\\ 0& -1\end{smallmatrix}\right)\mathrm{SO}_2.

Going to three dimensions, every matrix in \mathrm{SO}_3 is a rotation around some axis, which we proved using some eigenvalue-knowledge from V+M and the two-dimensional situation. Using cosets again, we easily concluded that any matrix in \mathrm{O}_3 is the product of at most three reflections. We really do need three for -I, which shows that we can’t quite think of \mathrm{O}_3 as “rotations and reflections”, at least not if we mean just reflections in planes.

The complex equivalent of orthogonal matrices are unitary matrices, satisfying A^\dagger A=I. They form the unitary group. The determinant of a unitary matrix is a complex number on the unit circle, and the kernel of the determinant homomorphism is the special unitary group. The equivalent to orthogonal matrices being isometries is that unitary matrices preserve complex dot product.

After these matrix groups we had another quick look at the rotations of a cube, and found that they form a group isomorphic to S_4. We proved this by acting on the four diagonals of the cube and being able to generate all of S_4 as the image. The group of all symmetries of a cube is S_4\times C_2, because the “reflection in the centre point” symmetry commutes with all other symmetries. And by duality of regular polyhedra, this tells us the group of symmetries of an octahedron as well (we’ll repeat that next time).

Understanding today’s lecture

Today we pulled together quite a lot of results from previous chapters to prove these facts about specific groups. It will probably help you understand the proofs if you look back to those results. But the other way round, seeing those results applied may also help you to get more of a feeling and a deeper understanding of the results themselves.

For the part on rotations of a cube, do make yourself a Taylor-Cube and try it all out, I find that helps tremendously with understanding what is going on.

Preparing for Lecture 21

We will be entering into the territory of Möbius maps next time. Come equipped with stamina for fraction manipulation (with lots of variables), and an open mind about infinity.

Going a little deeper

I have given you very algebraic proofs of these essentially geometric facts about \mathrm{O}_2 and \mathrm{O}_3. I made a lot of use of cosets partitioning the group and such things. We did have to use some geometric knowledge, such as eigenvectors and eigenvalues and what rotations do to an orthonormal basis. But you can prove these results in a much more geometric way if you like. Beardon does this (for slightly more general results including not just linear maps). You will benefit from trying to consolidate these two different viewpoints with each other. Some people find it easier to think geometrically, some people find it easier to think algebraically. But in any case, you should develop both sides so that you get the best of both worlds.

Beardon proves some more general results: he includes any isometry in the three-dimensional case, which means you actually need up to four reflections. He also touches on the general case of n dimensions. Do read up about it if you’re interested. It is in Chapter 11, mostly 11.2 Orthogonal Maps and a bit in 11.3 Isometries of Euclidian n-space.


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