Point of post: In this post we discuss the notion of the character table of a finite group and discuss several techinques used in their construction.
Some people would consider the fact that I have waited this long to talk about characters tables despicable. Admittedly, character tables are perhaps one of the most ‘practical’ application of representation theory to pure group theory. Namely, the character table shall be a numerical array of numbers (a matrix really) which using our past information about the kernel of a character, the center of a character, etc. to shall tell us most of things one may initally like to know about a group. The interesting part, is the using the relations between characters we’ve previously derived we can often get the character table ‘for free’ in the sense that some of the entries are obvious, and the rest can be ascertained from the orthogonality relations between characters, etc.
Let be a finite group with conjugacy classes and the elements of listed as . We define the character table of to be the matrix with where is the single value , being a class function, takes on . So, in general the character table of a group should look like
As a general we remark on some general relations between the rows and colums of the character table of a group . Namely:
Theorem (CT 1): Every element of the character table is an algebraic integer. In particular, one cannot have a non-integer rational element of the character table.
Proof: This follows from the fact that the values of every character of a group are algebraic integers.
Theorem (CT 2): Let be a finite group with a character table. Then, where and are the elements of rows and respectively.
Proof: This is just a restatement about the first orthogonality relations of the irreducible characters.
Theorem (CT 3): Let be a finite group with a character table. Then,
Proof: This is just a restatement of the second orthongonality relation for irreducible characters.
Theorem (CT 4): Let be a finite group with character table. Then, for each one has that divides .
Proof: This is just a restatement of the strong version of the dimension theorem.
Theorem (CT 5): Let be a finite group then
Proof: This follows from our work with the kernels of characters.
Theorem (CT 6): Let be a finite group with a character table. Then, .
Proof: This follows from our discussion of the center of a character.
Theorem (CT 7): Let be a finite group with a character table. Then, is conjugate to if and only if is real for where is the conjugacy class containing .
Proof: Clearly if is conjugate to then the entire column corresponding to is real since
for each .
Conversely, suppose that is real for each . Then, . We note then by the second orthogonality relation that is zero if and only if and aren’t conjugate. But, since it clearly can’t be zero, it follows that and are conjugate.
Theorem (CT 8): More generally, the columns containing and are conjugate for each .
Theorem (CT 9): Let be a character of the finie group , if then for every .
Proof: Let be any irrep which admits as its character. Since we know that . Thus, it clearly follows that the endormorphism algebra being dimensional must be equal to . In particular, since we know that for some . Note though that if is the inner product on then by unitarity of we have that
So, choosing a non-zero allows us to conclude from the above (since ) that and so . Noting then that and the fact that was arbitrary allows us to follow.
1. Isaacs, I. Martin. Character Theory of Finite Groups. New York: Academic, 1976. Print.
2. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.