Character table

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Definition

Let $G$ be a finite group, and $k$ a sufficiently large field for $G$ . The character table of $G$ is a matrix whose rows are indexed by the irreducible representations of $G$ , and columns by the conjugacy classes in $G$ , where the entry in row $\rho$ and column $c$ is the character of the representation $\rho$ on the conjugacy class $c$ .

Facts

The character table is a square matrix

Further information: Number of irreducible representations equals number of conjugacy classes

There as many conjugacy classes as irreducible representations, so the matrix is a square matrix.

Orthogonality of the rows

Define the weighted character table as the character table with each column multipled by the size of the conjugacy class.

The rows of the weighted character table are orthogonal to each other. Further, the inner product of each row with itself is the cardinality of the group. This in particular shows that the (weighted) characters form an orthogonal basis for the row space and hence for the space of all function spaces. In particular, any class function can be written in a unique way as a linear combination of characters.

Orthogonality of the columns

Since the rows are orthogonal and the inner product of each with itself is the cardinality of the group, the product of the matrix and its transpose is a scalar matrix. Thus, the columns of the weighted character table are orthogonal, which in turn shows that the columns of the original character table are orthogonal.

No ordering of the rows and the columns

As such, there is no canonical way in which we can order the rows (viz irreducible representations) or the columns (viz conjugacy classes). Thus the character table is ambiguous upto both pre- and post-multiplication by permutation matrices.

However, if we are given a conjugacy class-representation bijection, we can use that to identify rows and columns, and hence we can arrange rows and columns in the same order. Now, the character table is ambiguous upto conjugation by a permutation matrix.

This, for instance, is what happens in the case of the symmetric group.

Rows sum to non-negative integer for finite groups

Further information: Sum of elements in row of character table of finite group is non-negative integer

For complex representations, the sum of each row of the character table of a finite group sums to a non-negative integer.

Does not classify groups up to isomorphism

Further information: Groups with same character table need not be isomorphic, Character table-equivalent groups

Two groups with the same values in their respective character tables need not be isomorphic.

Applications

Reading off normal subgroups

Further information: Correspondence between normal subgroups and the kernels of characters of a finite group

It turns out (see page linked) that normal subgroups of a finite group are precisely those which can be written as a kernel of some character of a complex linear representation of the group, equivalently the intersection of kernels of some irreducible characters of that group. Thus, computing the character table of a group makes it very easy to read off the normal subgroups of a group.

Calculating the character table

Tricks and intuitions by hand

Often when calculating a character table of a group by hand one can use human intuition alongside a few tricks to do it.

At any point, when applying these methods and tricks, once you have a square character table of distinct irreducible characters, that is the number of said characters obtained is equal to the number of conjugacy classes, you are done.

We of course always have the trivial linear representation. Write down that first.

Are there any obvious representations? For example, if you are working with a symmetric group, there are the sign representation and the standard representation, or with a dihedral group, there is the 2 dimensional representation corresponding to the rotations and reflections that it performa on the relevant regular polygon. Calculate their characters and add them to a row in the table.

Once some representations are written down, calculating the tensor product of representations as well as the symmetric and alternating-squares may provide more. They will not necessarily be irreducible, so make sure to decompose them into irreducible parts, which may lead to several new characters.

If you know a subgroup of the group, and know the irreducible representations of that subgroup, using representation induction may yield some of the original group.

When most (especially if all but one) of the irreducible characters are found, one may use row and column orthogonality to make finding the remaining characters reduced to a problem of linear algebra.

More computational algorithms

If one is calculating a character table of a group that they do not have intuition for, or is perhaps performing the calculation via a computer, there are algorithms that compute the character table.

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