## Posts filed under ‘group theory’

### Cardboard dodecahedron

I made a cardboard dodecahedron for the needs of a talk.

If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetric group $\mathcal A_5$: it replaces a star by a star with a different arrangement of colours.

Since there are 12 facets and 5 ways of rotating each of them, 60 colourings can be seen by rotating the dodecahedron by direct isometries. However, it is NOT true that you can see the 120 possible colourings by allowing also reflections (the full isometry group of the dodecahedron is the symmetric group on five colours). An easy reason for this is that the colouring is invariant under symmetry through the central point (which is a determinant -1 transformation). You can also argue that reflections act as double transpositions of the colours of a star.

People also talk about five tetrahedra in a isocahedron, which can also be obtained in the dodecahedron by choosing a tetrahedron in each cube in a consistent way. The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries.

### Why can we have a Fast Fourier Transform ?

The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of times each second. It is probably more, I was assuming an average computer does billions of Discrete Cosine Transforms per day for Web browsing (JPEG images and Youtube) and music listening, but how huge is the contribution of millions of people watching MPEG-2 compressed television programs on satellite or terrestrial digital broadcasting?

It is thus important to know that the Fourier transform (which is always the Fast Fourier Transform) is really fast (especially when it is dozens, hundreds of times faster than the natural algorithm). I am not yet sure about the fast versions of variants of the discrete transform (cosine transforms and its friends), but I guess they can be derived from the classical case.

### What is it ?

The classical Fourier transform takes a T-periodic function f and computes for each integer n the integral

$f_n = \frac 1 {\sqrt T} \int_0^T f(t) \exp(-ni\omega t) dt$

where $\omega = 2\pi/T$. It separates f into components having period $T/n$, and has the property that the energy $E(f) = \frac 1 T \int |f(t)|^2 dt$ is the sum $\sum |f_k|^2$.

The discrete Fourier transform takes a discrete function given by $x = (x_0, x_1, \dots, x_{n-1})$ and computes the discrete analogue

$y_p = \sum_{q = 0}^{n-1} x_q \exp(-2i\pi \frac{pq}{n})$

### Réprésentations de l’algèbre de Lie sl(2)

Alors que la géométrie mettait autrefois en avant l’importance des groupes et des transformations d’un objet (une philosophie défendue notamment par Klein), l’influence croissante de la mécanique, l’algèbre et l’analyse tend à remplacer la théorie des groupes par celle des algèbres de Lie. La théorie de Lie, développée par Borel et Chevalley permet d’étudier les représentations de certains groupes à travers leurs algèbres de Lie, et la géométrie différentielle en est également friande.

Whereas geometry in its old times would highlight the importance of groups of transformations (as in the Erlangen program introduced by Klein), modern developments in mechanics, algebra and calculus would rather use the language of Lie algebra. Lie theory was actively developed by Borel and Chevalley, allowing to understand groups through their Lie algebras, and differential geometry is closely related to this subject as well.

### Le groupe SL2

Le groupe SL2 est le groupe constitué des matrices $\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$ telles que $ad-bc = 1$. L’inverse d’une telle matrice est donné par $\begin{pmatrix} d & -b \\ -c & a\\ \end{pmatrix}$.