Real representations of finite groups

In these pages we give the character tables for some of the basic finite groups, listed in the menu on the left. In particular, these tables are for the real representations, which are not so easily found elsewhere, although after each table are some comments on the relation between the real irreducibles and the complex ones. The theory describing real representations and their characters is described in the page on real characters (see the link in the menu).

Briefly, given a character \(\chi\) of a real representation, one defines 2 quantities:

  • \(\|\chi\|^2 := \frac1{|G|}\; \sum |\,\chi(g)|^2\)
  • \(\nu(\chi) := \frac1{|G|} \; \sum \chi(g^2)\)

The sums are over all elements \(g\in G\).

A complex representation is irreducible if and only if \(\|\chi\|^2 = 1\).

A real representation is irreducible if there is only one invariant quadratic form, up to scalar multiples. This is equivalent to \(\|\chi\|^2 + \nu(\chi) = 2\), as we explain on this page.

Notation

Character tables are a very important tool in both physics and chemistry (eg in molecular spectoscopy), who have given all the representations names, such as A1 or T2g. These are called the Mulliken symbols. I have adhered mostly to their notation, except that I denote the trivial representation by A0 rather than A1. However, the same abstract group may have different Mulliken symbols according to how the representation arises, whereas in these tables each (abstract) group has just one version of its character table (compare the character tables of Cs, Ci, C2 on the mathworld site, all three of these groups have 2 elements).

In each table of characters,

  • 1st row is: name of group, followed by a representative of each conjugacy class
  • 2nd row is: number of elements in the respective conjugacy class
  • remaining rows: the characters of the different (real) irreducible representations

See also the Bilbao crystallographic server for a list of character tables and other information.

Further details

Character theory - all(?) text books treat the beautiful theory of complex representations, but give little insight to the representations over the reals. So here we summarize the (slightly less elegant) theory of real representations.