The exceptional symmetry of E8

Here's a little movie I made about the Lie group E₈:

Out of all the known Lie groups, E₈ stands out as the largest and most complex exceptional group. It has 248 generating elements, which by themselves have an astounding degree of symmetry. This symmetry can only be fully grasped in 8-dimensional space. But luckily it is also possible to project E₈ onto a two-dimensional plane, chosen such that the resulting image preserves a small fraction of its total symmetry. There are different choices for these two-dimensional planes, some preserving more symmetry than others. The movie rotates through a selection of these planes in succession.

Lecture notes on Mathematica

Last week Filippo Guarneri and I gave a daylong lecture series on the venerable topic of Mathematica. We covered topics ranging from basic evaluation and programming techniques to tensor calculus. Needless to say, we've written up a nice set of lectures notes for those of you who weren't there. Feel free to have a look!

LieLink: a Mathematica interface for LiE

Ever wished that you could easily transfer you computations done in LiE to Mathematica? You now can with LieLink! LieLink is an small Mathematica package that interfaces with LiE, allowing you to directly execute LiE commands in Mathematica and get the result back. Here's an small example:

<<LieLink`

SetDefaultAlgebra["A2"]

LieTensor[{1, 0}, {1, 0}]
(* => {0,1} + {2,0} *)

You can download LieLink from github.com/teake/LieLink.

xTras 1.2

After quite a bit of hard work, I'm proud to announce xTras 1.2! The main change over the 1.1 releases is that it now has built-in documentation. That means that when you enter for example ?AllContractions, there will be a nicely formatted message that informs you how to use AllContractions:

Pressing the blue >> link at the end of the message opens up the reference page of AllContractions in the documentation center:

All of the built-in documentation can also be found online, at http://www.xact.es/xtras/documentation/.
Besides the new documentation, there's a plethora of other changes under the hood. As always, see the changelog for more details.

xTras v1.1.3

I've just posted a new version of my Mathematica package xTras. It fixes a couple of bugs and introduces some new functions. One of these functions is MakeTraceless, which takes any tensorial expression and returns its traceless version. For example:

  In:   MakeTraceless[RiemannCD[-a,-b,-c,-d]]
Out:   \( R_{abcd} + \frac{2 R \underset{1234}{Sym}(g_{ac} g_{bd})}{2 -3 d + d^2} + - \frac{4 \underset{1234}{Sym}(g_{bd} R_{ac})}{-2 + d} \)

The output uses the implicit symmetrizations of the SymManipulator package. But we can also symmetrize explicitly by expanding the symmetries:

  In:   ToCanonical@ExpandSym@MakeTraceless[RiemannCD[-a, -b, -c, -d]]
Out:   \(- \frac{g_{bd} R_{ac}}{-2 + d} + \frac{g_{bc} R_{ad}}{-2 + d} + \frac{g_{ad} R_{bc}}{-2 + d}  - \frac{g_{ac} R_{bd}}{-2 + d}  - \frac{g_{ad} g_{bc} R}{2  -3 d + d^2} + \frac{g_{ac} g_{bd} R}{2  -3 d + d^2} + R_{abcd} \)

Of course, this is just the Weyl tensor in d dimensions:

  In:   Simplification@ RiemannToWeyl@ExpandSym@MakeTraceless[RiemannCD[-a, -b, -c, -d]]
Out:  \(W_{abcd} \)

Another new function is ConstructDDIs, which construct dimensional dependent identities (DDIs). Say we have a two-dimensional manifold. We can then ask for a list of all DDIs which have one curvature tensor and two free indices:

  In:   ConstructDDIs[RiemannCD[a,b,c,d],IndexList[a,b]
Out:   \({R^{ab}  - \tfrac{1}{2} g^{ab} R} \)

It returns just one DDI, namely the vanishing of the Einstein tensor. This is well know fact, namely that gravity in two dimensions is purely topological, because the Einstein-Hilbert action is equal to the two-dimensional Euler density.

As always, you can grab the newest version of xTras from its www.xact.es/xtras, or have a look at the changelog for all the new features.

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.

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.

The adjoint representation of B₁₉.

The adjoint representation of D₁₀.