## Friday, February 8, 2013

### The Weyl group of C4

Time for a pretty picture! It's the Weyl group of the finite Lie algebra C4.

Although C4 is 'only' 36 dimensional, its Weyl group is a bit bigger and has 384 elements. The pictures doesn't show the elements of the Weyl group though, but rather all possible connections between them. If you looks closely the black dots aren't dots -- they're points where are lots of lines meet.

## Thursday, January 31, 2013

### Coxeter planes

Here are a few nice Coxeter projections of representations of finite Lie algebras. If you don't know what a Coxeter projection is, or a finite Lie algebra for that matter, have a look at my PhD thesis (from which I took these pictures). Or, visit John Stembridge's website for more information and pictures.

Update: some of these pictures do not show the full representation, but only a subset.

The orbits of the two highest dominant weights of the 1764 representation of A5.

The orbits of the two highest dominant weights of the 5985 representation of A17.

The orbits of the two highest dominant weights of the 442 representation of B6.

## Thursday, January 3, 2013

### Feynman's ode to a flower

Here's a beautiful animation by Fraser Davidson accompanying a soundbite from Richard Feynman. The soundbite is from a 1981 episode of BBC's Horizon, "The pleasure of finding things out", which, if you haven't seen it, I can wholeheartedly recommend.

## Sunday, December 23, 2012

### xTras 1.0.6

Just in time for the holidays, a new version of xTras, my xAct Mathematica package, has hit the selves. Version 1.0.6 updates AllContractions, which should be much faster. You can now compute all possible contractions of your favorite tensors in a reasonable amount of time (provided there aren't too many indices to contract over), like so:
In:    AllContractions[ RiemannCD[a,b,c,d]RiemannCD[e,f,g,h] ];
Out: { $$R^2, R_{ab} R^{ab}, R_{abcd} R^{abcd}, R_{acbd} R^{abcd}$$ }
The last two contractions are actually not independent, but are related to each other via the Bianchi identity. But that's a multi-term symmetry, and unfortunately AllContractions doesn't take those into account.

Other functions, such as TensorCollect and SolveConstants should also see some speed improvements. And besides the usual bugfixes there are nice new functions such as SolveTensors (which can solve a system of linear tensorial equations in terms of tensors) and IndexConfigurations. The latter gives all possible independent index configurations of an expression. For example:
In:    IndexConfigurations[ metric[a,b]metric[c,d] ];
Out: { $$g^{ab} g^{cd}, g^{ac} g^{bd}, g^{ad} g^{bc}$$ }
Here's another example:
In:    IndexConfigurations[ RiemannCD[a,b,c,d] ];
Out: { $$R^{abcd}, R^{acbd}, R^{adbc}$$ }
Note that the three different index configurations of the Riemann tensor are actually not independent due to the Bianchi identity. Like AllContractions, IndexConfigurations doesn't take these multi-term symmetries into account.

xTras 1.0.6 has some more improvements; check the Changelog for details. As usual, the new version can be downloaded from its Google Code page.

## Wednesday, November 7, 2012

### xAct notebooks

I put some notebooks for xAct on my website, at http://www.aei.mpg.de/~nutma/. There's an introductory notebook, and two more advanced ones covering the Noether procedure for spin 2 and some calculations for spin 3. Go have a look!

## Sunday, March 25, 2012

### xTras for xAct

I'm a big fan of xAct, a tensor algebra package for Mathematica. While there are other tensor packages for Mathematica on the market, xAct is by far the best. It canonicalizes tensorial expressions blazingly fast, and its perturbation capabilities are state-of-the-art. If you've always wanted to do second-order perturbations of four-derivative curvature tensors but were afraid the actual calculation might take you some weeks, xAct is your man. It does it in a few seconds, and more importantly, doesn't mess up minus signs.

In fact, I like xAct so much, I wrote an additional package for. It's aptly called xTras, and its available over on Google Code. It brings some functionality that I found missing in xAct, like computing equations of motion for the metric, perturbations around AdS spaces, and Young projectors (yes, this also includes Bianchi identities).

I'll demonstrate some of the new functionality by computing the linearization of the Einstein tensor around AdS spaces. Here goes!
First, open up Mathematica, and enter the following line:
In:    <<xActxTras
This loads the xTras package (assuming you're managed to download and install it). Next we'll define a manifold and a metric:
In:    DefConstantSymbol[dimension,PrintAs->"D"]
In:    DefManifold[M,dimension,IndexRange[a,f]]
In:    DefMetric[-1,g[-a,-b],CD]
We'll be doing stuff on AdS spaces, which has a constant curvature. Hence we need to define a constant symbol to indicate that curvature:
In:    DefConstantSymbol[L]
Now we're ready to define the standard Lagrangian for gravity with a cosmological constant:
In:    lagrangian = RicciScalarCD[] -(dimension-2)(dimension-1)L
We'd like to compute the equations of motion that follow from this Lagrangian. To do so, we first write the command
In:    DefMetricVariation[g,h,eps]
This command makes it possible to do covariant metric variations. It also registers the command VarL, which varies Lagrangians:
In:    eom = VarL[g[a,b]][lagrangian] //TensorCollect
Out:    $$\frac{1}{2} (2 - 3 D + D^2) L g_{ab} + R_{ab} - \frac{1}{2} g_{ab} R$$
The equations of motion should allow for AdS spaces. To check this, we first generate a list of replacement rules for curvature tensors of the covariant derivative CD on symmetric spaces:
And indeed, the equations of motion are zero for this background:
In:    eom /. AdSrules // ToCanonical
Out:    $$0$$
This means we can perturb around this solution. So without any further ado, here's the linear perturbation of the Einstein tensor:
In:    ExpandBackground[eom, BackgroundSolution -> AdSrules] // TensorCollect
Out:    $$(l - D l) h^{1}{}_{ab} + \frac{1}{2} (-1 + D) l g_{ab} h^{1c}{}_{c} - \frac{1}{2} \triangledown_{a}\triangledown_{b}h^{1c}{}_{c} + \frac{1}{2} \triangledown_{c}\triangledown_{a}h^{1}{}_{b}{}^{c} + \frac{1}{2} \triangledown_{c}\triangledown_{b}h^{1}{}_{a}{}^{c}$$
$$- \frac{1}{2} \triangledown_{c}\triangledown^{c}h^{1}{}_{ab} - \frac{1}{2} g_{ab} \triangledown_{d}\triangledown_{c}h^{1cd} + \frac{1}{2} g_{ab} \triangledown_{d}\triangledown^{d}h^{1c}{}_{c}$$
And that's it! Granted, we could also have done this by hand. But the power of xAct is that it can do much more complicated calculations without breaking a sweat. If we would like to know the second order pertubation of the Einstein tensor, we can simply replace the above input by ExpandBackground[eom,2,BackgroundSolution->AdSrules]. Pretty cool, right?

## Monday, March 29, 2010

### Affine root systems

Currently I'm working on my PhD thesis. It isn't finished yet, but I decided the following images were worth a sneak peak:

Besides looking nice, the pictures actually convey some information. They're so-called Hasse diagrams of the root systems of a few affine Lie algebras. From left to right we have the following affine algebras: (a) A1+, (b) C2+, (c) D4+, (d) A8+, (e) D7+, and (f) E7+. But luckily you don't need to fully understand the mathematical background, which is admittedly quite complicated, to enjoy their beauty.