Progress (or lack of it)

Amended 10th April now that I've learnt how to use Math JAX

Sorry for not blogging recently I've been a bit bogged down at work. This has meant an alarming decay in my productivity vis a vis mathematics. Still I'm on target as far as the course calendars are concerned.
Finished and posted the 2nd part of the first TMA for M208. Am half way through the TMA on M337 (which doesn't need to be in until April) so so far so good. The only sticking point is the Linear Statistical modelling course I have been unable to get a licence key for the software. This sort of thing makes me paranoid I installed the software following the instructions. It then said you could register online but when I pushed the button claiming to be able to access the website nothing happened. So I was forced to use the offline procedure I sent an e-mail off to the guys who run GenStat but so far have heard nothing. I will wait till the middle of the week then start complaining. I really hate this beauracrcy and being made to feel like a criminal simply because I want to use a piece of software which I've paid for. Doesn't look like a great start to a course that I wasn't particularly interested in in the first place we'll see what happens.

On a brighter note next topic in M208 is group theory. Neil has opened up a shared activities website to share practical applications of group theory. Many many years ago as part of my physics degree I studied group theory (from a physicists) perspective as it applied to problems in quantum mechanics specifically
the application of representation theory to classify the normal vibrations of molecules and it;s application to particle physics. It seems a good opportunity to revise this interesting topic.

To start off I've looked at how to simplify the calculation of Cayley tables for dihedral groups I have poor visualisation skills so get really confused when a shape is rotated through N/2pi degrees then reflected about an obscure axis. Some people seem to be able to do this naturally  I can't fortunately algebra comes to the rescue I'll post more in a Latex document. However (and apologies to those reading this who know this already).

Dihedral groups are the groups associated with the symmetry of regular polygons. It can be shown (as I will expand further on my Latex document) that there are only two symmetry operations namely rotations and reflections.

The Cayley table splits into 4 quadrants
                         rot   refl
                rot     rot   refl
                refl   refl    rot

The elements of the group are rotations through $$\frac{2\pi}{N}$$ where N is the Number of sides of the polygon
and reflections about a given axis represented by s. A rotation through $$\frac{j * 2\pi}{N}$$ where j an integer running from 1 to N-1 is denoted by $$r^j$$.

The elements of the group are $$(e,r^1, r^2 .......    r^(N-1), s  rs   r^1s  ....... r^(N-1)s)$$
the second half being rotations of the basic reflection. reflections obey the basic rule

                                                       $$s^2 = e$$ .

The multiplication rules are really quite straightforward

A rotation through $$\frac{k*2\pi}{N}$$ followed by another rotation $$\frac{j*2\pi}{N}$$ is simply another rotation

               $$r^j * r^k  = r^{(j+k)}$$ 

where j+k is evaluated to modulo N.

A reflection followed by a rotation is simply

             $$r^j * r^{k } s   = r^{(j+k)} s.$$

  The other two operations  are a bit more complicated but it is quite straightforward to verify this magic rule

              $$r^k s    = s (r^k)^{-1}$$

(Those who are into quantum mechanics will recognise this as an anticommutation relation)

This means that for products of the form

       $$r^j * sr^k$$ we get   $$r^j * (r^k)^{-1} s$$

But $$(r^k)^{-1}$$ is simply$$ r^{N-k}$$

(so for a 5 sided polygon the inverse of $$r^2$$ is simply$$ r^{5-2} = r^3$$)

Finally a product of the form

       $$ r^j s * r^k s  = r^j s s (r^k)^{-1} = r^{N-k+j}$$ as $$s^2 = e$$

Where the second stage has been gotten from the anticommutation relations.

The point is that now it is relatively straightforward to generate the cayley table for such a dihedral group
without having to use matrices, two line symbols or get yourself confused looking at figures,

Anyone care to have a go at say 5 or 6 sides.

In the next post on this topic I'll show how these rules can also simplify the generation of conjugacy classes

(Apologies for the wide gaps still trying to master LATEX properly)

Best wishes Chris