The Unapologetic Mathematician

Mathematics for the interested outsider

Some Review

Before we push on into our new topic, let’s look back at some of the background that we’ve already covered.

We’re talking about symmetric groups, which are, of course, groups. We have various ways of writing down an element of S_n, including the two-line notation and the cycle notation that are covered in our earlier description of the symmetric groups. As an example, the two-line notation

\displaystyle\left\lvert\begin{matrix}1&2&3&4\\2&1&4&3\end{matrix}\right\rvert

and the cycle notation (1\,2)(3\,4) both describe the permutation \alpha\in S_4 that sends 1 to 2, 2 back to 1, and similarly swaps 3 and 4. Similarly, the two-line notation
the composition of
\displaystyle\left\lvert\begin{matrix}1&2&3&4\\4&2&1&3\end{matrix}\right\rvert

and the cycle notation (1\,4\,3)(2) or (equivalently) (1\,4\,3) describe the permutation \beta that cycles the elements 1, 4, and 3 (in that order) and leaves 2 untouched.

We’re specifically concerned with complex representations of these groups. That is, we want to pick some complex vector space V, and for each permutation \sigma\in S_n we want to come up with some linear transformation \rho(\sigma):V\to V for which the composition of linear transformations and the composition of permutations are “the same” in the sense that given two permutations \sigma and \tau, the transportation corresponding to the composite \rho(\sigma\tau) is equal to the composite of the corresponding transformations \rho(\sigma)\rho(\tau).

We’re primarily interested in finite-dimensional representations. That is, ones for which V is a finite-dimensional complex vector space. In this case, we know that we can always just assume that V=\mathbb{C}^k — the space of k-tuples of complex numbers — and that linear transformations are described by matrices. Composition of transformations is reflected in matrix multiplication. That is, for every permutation \sigma\in S_n we want to come up with an k\times k matrix \rho(\sigma) so that the matrix \rho(\sigma\tau) corresponding to the composition of two permutations is the product \rho(\sigma)\rho(\tau) of the matrices corresponding to the two permutations. I’ll be giving some more explicit examples soon.

About these ads

September 8, 2010 - Posted by | Algebra, Group theory, Representation Theory, Representations of Symmetric Groups

5 Comments »

  1. Minor quibble: I guess the ‘n’ in $S_n$ should be (possibly) different to the dimension of the representation?

    (Long-time reader: keep up the good work!)

    Comment by Doormat | September 9, 2010 | Reply

  2. [...] « Previous | [...]

    Pingback by Cycle Type « The Unapologetic Mathematician | September 9, 2010 | Reply

  3. [...] Sample Representations As promised, we want to see some examples of matrix representations for those who might not have seen much of [...]

    Pingback by Some Sample Representations « The Unapologetic Mathematician | September 13, 2010 | Reply

  4. [...] –Por ejemplo ub blog que desconocia y que está hablando últimamente de la teoria de representaciones de grupos de permutaciones (que trata de como pasar del lenguaje de permutaciones al lenguaje de espacios vectoriales y transformaciones lineales (matrices). A  cada permutación de un grupo dado se le asigna una transformación lineal de un espacio vectorial en si mismo):  http://unapologetic.wordpress.com/2010/09/08/some-review/.   [...]

    Pingback by Sobre el orden de los grupos finitos bigenerados « US Patent Appl. 12213303: Comentarios. | September 29, 2010 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 391 other followers

%d bloggers like this: