Matrices of irreducible representations of common groups

I was wondering where can one find the matrices (and not just the character tables) of the irreducible representations of the most common groups (alternating, symmetric, octahedral, etc..) ?

• Many computer algebras can do this. For example, the GAP system is particularly suited for this. – Geoff Robinson Jul 21 '11 at 10:07

Some remarks on how to find matrices if you are fluent with MAGMA or GAP or some such (rather than where to find them):

• You can generate the matrices of all the representations of symmetric groups using the theory of Young tableaux. See e.g. the book by Fulton and Harris, "Representation theory, a first course", Graduate Texts in Mathematics 129, or James and Kerber, "The representation theory of the symmetric group", Encyclopedia of Mathematics and its Applications 16. Alternating groups work very similarly.
• For a general group $G$ and a general representation $\rho$ with character $\chi$, you can easily get the matrices for $\rho^{\oplus \chi(1)}$ if you just know $\chi$. Indeed, the representation is given by $e_\rho \mathbb{C}[G]$, where $e_\chi$ is the idempotent corresponding to $\rho$, $$e_\chi = \frac{1}{|G|}\sum_{g\in G} \chi(1)\chi(g^{-1})g.$$
• If you know the matrices for all representations of a group $G$ and if $A$ is an abelian group, then you can also find the matrices for all representations of any semi-direct product $A\rtimes G$. The procedure is described in another answer of mine. Note that although that answer always talks about characters, you actually get the matrices this way, you just have to know how to get matrices of an induced representation from matrices of the original thing (if you don't know, feel free to ask).
• Finally, for the most general case, please refer to an answer of mine on MO and the reference therein.

There's a little book called Group Tables, by Thomas and Wood, that has this information up to order 40 or so (excluding order 32).

EDIT: See also Simon Jon Nickerson, An atlas of characteristic zero representations, http://www.maths.qmw.ac.uk/~raw/SJNphd.pdf Also, you can get representations of many groups from http://brauer.maths.qmul.ac.uk/Atlas/v3/

This is an excellent question, but it can get quite complicated, see for instance the paper of Eric Kuisch who worked on this. By the way, I still find it fascinating that in (complex/characteristic 0) representation theory you throw away most of the information (the entries) of the matrices in question, except for the trace - the character. And a famous theorem of Georg Frobenius asserts that this doesn't rise a problem - up to similarity, representations are determined by their characters! For representations over fields with characteristic $p > 0$ the situation is different.

• groups are not determined up to isomorphism by their characters, I think you mean (complex) reps – Matthew Towers Jul 21 '11 at 10:40
• Yes of course, I mean the representations are similar over characteristic zero iff the characters are the same. Thanks. Will edit my text. – Nicky Hekster Jul 21 '11 at 22:08