Representation theory of finite groups

Each finite group can be represented as a group of linear transformations, or in other words as a group of matrices, in many different ways. For example a group that can permute n symbols {a₁, a₂, ..., a_n} has a natural representation as a group of linear transformations in n dimensions, whose n coordinates are permuted in the same way as the symbols {a₁, a₂, ..., a_n}.

Each representation can be split into a sum of irreducible representations in a unique way (irreducible meaning it cannot be deconstructed into smaller representations), and one of the most important ways of studying a finite group is to find all its irreducible representations. This data is then presented in the form of a character table, each irreducible representation giving one row of the table.

For example the symmetric group Symm(n) — the group of all permutations of n symbols {a₁, a₂, ..., a_n} — operates naturally in n dimensions, by permuting n coordinates represented by {a₁, a₂, ..., a_n}. This representation splits into two: a one-dimensional representation (spanned by a₁+a₂+...+a_n), and an irreducible representation of dimension n‑1. For instance Symm(4) — a group of size 24 — has five irreducible representations, whose dimensions are: 1, 1, 2, 3 and 3. Notice that the squares of these numbers add up to 24, and this is a general fact: the squares of the dimensions of the irreducible representations sum to the size of the group. Another fact is that the dimension of an irreducible representation must be a divisor of the size of the group.

The Monster has 194 different irreducible representations, a very small number compared to the size of the group, which is roughly 10⁵⁴. But this is a general feature of finite simple groups—they have relatively few irreducible representations.