Character Tables
For more details on this topic, see Character table.The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on the representatives of the respective conjugacy class of G. The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the trivial character, and the first column by (the conjugacy class of) the identity. The entries of the first column are the values of the irreducible characters at the identity, the degrees of the irreducible characters. Characters of degree 1 are known as linear characters.
Here is the character table of, the cyclic group with three elements and generator u:
| (1) | (u) | (u2) | |
| 1 | 1 | 1 | 1 |
| χ1 | 1 | ω | ω2 |
| χ2 | 1 | ω2 | ω |
where ω is a primitive third root of unity.
The character table is always square, because the number of irreducible representations is equal to the number of conjugacy classes. The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1 × 1 matrices containing the entry 1).
Read more about this topic: Character Theory
Famous quotes containing the words character and/or tables:
“When much intercourse with a friend has supplied us with a standard of excellence, and has increased our respect for the resources of God who thus sends a real person to outgo our ideal; when he has, moreover, become an object of thought, and, whilst his character retains all its unconscious effect, is converted in the mind into solid and sweet wisdom,—it is a sign to us that his office is closing, and he is commonly withdrawn from our sight in a short time.”
—Ralph Waldo Emerson (1803–1882)
“Hast thou named all the birds without a gun?
Loved the wood rose, and left it on its stalk?
At rich men’s tables eaten bread and pulse?
Unarmed, faced danger with a heart of trust?”
—Ralph Waldo Emerson (1803–1882)