Character tables, group theory

A character table is a square array of numbers associated with a group. The group can be very large, yet its character table can be very small by comparison. For example the largest Mathieu group M24 has size 244,823,040, though its character table has only 26 rows and columns.

There is a column in the character table for each type of element in the group, technically speaking for each conjugacy class. And there is a row in the table for each space on which the group can operate irreducibly (meaning it does not stabilize any subspaces); the first number in the row is the number of dimensions of the space. The number of rows in the table equals the number of columns — hence it's a square array — but this is not obvious. It is a consequence of some beautiful theorems in representation theory—a part of mathematics with applications to physics and other sciences.

The numbers that appear in a character table are complex numbers, but of a special type. They are algebraic integers, which implies that when they are real numbers, then they are ordinary integers — a fraction such as ½ cannot appear in a character table. There are relationships between the rows and between the columns, and these help in finding the entries in the table, just as in a sudoku puzzle.

Knowing the largest subgroups of a group helps in calculating its character table, and that in turn helps in locating large subgroups. For example when a group contains a subgroup whose size is 1/n times the size of the whole group then it can permute n objects transitively, meaning it can send any one to any other one, and this leads to a representation in n dimensions. When n is small relative to the size of the group, the character table can be very useful in deciding on the existence of suitable representations, and hence of the existence of such subgroups.