Characters of Representations — Studies in Algebra and Group Theory

33 Characters of Representations

19 November 2025 · Denis Auroux

Last time we studied representations of via the eigenvalues of the actions of a transposition and a -cycle. We’ll soon see that, for any group , the eigenvalues of the actions of elements of go a long way towards classifying representations, but we need some way of organizing the information.

Digression: symmetric polynomials

(This is all motivation for the study of characters.)

Observe: an efficient way to store information about (complex) numbers, unordered and possibly with repetitions, is to specify the coefficients of the polynomial of which they are the roots, i.e. . These coefficients are symmetric polynomials in .
acts on the space of polynomials by permuting the variables.

Definition 33.1. A symmetric polynomial is an that is a fixed point of the -action: for all .

Remark. Equality of polynomials means, as usual, equality of coefficients, which over a finite field is a stronger condition than having equality as functions on . Of course over there is no difference.

Definition 33.2. The elementary symmetric polynomials are

Check: the coefficient of in is, up to the sign , .

Hence: the fundamental theorem of algebra gives a bijection with inverse In other terms: the coefficients of the polynomial .

Theorem 33.3. The subring of symmetric polynomials in , i.e. , is isomorphic to the polynomial algebra in variables . I.e., every symmetric polynomial is uniquely a polynomial expression in the elementary symmetric polynomials.

We won’t prove this, but to see why this works, look at the case . The vector space of symmetric polynomials has basis

Observe: any symmetric polynomial in variables can be written as and we work by induction on degree.

Remark. The theorem can be understood in terms of the representation theory of ! Namely, the space of homogeneous degree polynomials is , on which acts by the permutation representation (standard trivial), and the invariant part is , the trivial summand. Now, the homogeneous degree polynomials are , and the invariant part = the trivial summands in the decomposition of into irreducibles! (Unfortunately we haven’t studied representations of in enough depth to carry through with a proof along these lines.)

Another family of symmetric polynomials are the power sums: These make sense for all , but in fact suffice:

Theorem 33.4.

In particular, specifying an unordered tuple is equivalent to specifying .

Back to representation theory: why we care about this

We’ve seen that, to understand a representation of , we should look at the eigenvalues of for each ; but this is a lot of information. We’ve just said: to specify the eigenvalues of , it is enough to specify the power sums . But in fact !

So it’s enough to describe just the sum of the eigenvalues for every : since is a group, the trace of is also part of this.

Definition 33.5. The character of a representation is the function , .

Remark. For a -dimensional representation of , i.e. a homomorphism , the character is just the same thing, hence a (multiplicative) homomorphism. For a higher-dimensional representation, though, .

However, since trace is conjugation invariant, , so only depends on the conjugacy class of .

Definition 33.6. A class function is a function invariant under conjugation: .

Example 33.7. Given representations and :

(eigenvalues of …).
(eigenvalues of , and ).
, since acts by , and the eigenvalues are roots of unity, so , hence .
.

Example 33.8. If acts on a finite set , then there is an associated permutation representation of dimension , with basis , on which acts by permutation matrices: . Then since the ’s on the diagonal of the matrix correspond to fixed points of , and there are ’s otherwise.

Character tables

The character table of a group lists, for each irreducible representation of , the values of its character on each conjugacy class of .

Example 33.9 (). Columns are the conjugacy classes, rows the irreducible representations:

The first column is . The row for comes either from the eigenvalues ( for , for ), or from the fact that = the permutation representation, which takes the values ; then subtract .

Now we have a faster way of decomposing into irreducibles: , so takes the values . Since are linearly independent and , we get . (This is equivalent to counting eigenvalues as we did last time, but somewhat faster!)

Notes not reached in this lecture, covered at the start of {{LINK:L34}}

Some magic with characters

If is a representation of , the invariant part is .

Proposition 33.10. is a projection onto : and .

So:

If are representations of , then (the -action on is ), so but if and are irreducible, then by Schur’s lemma,

Definition 33.11. Define a Hermitian inner product on the space of class functions by

For characters of representations, by the above,

Theorem 33.12. The characters of irreducible representations of are orthonormal for .

This implies the characters of irreducible representations are linearly independent class functions!

Corollary 33.13. The number of irreducible representations of is at most the number of conjugacy classes of . (We’ll see later that they are in fact equal.)

Corollary 33.14. Every representation of is completely determined by its character: denoting the irreducible representations by , any representation , where

Corollary 33.15. For any representation , , and is irreducible iff .

This is useful because, given a representation , it gives information about the number of irreducible summands making up . E.g.:

	is
	irreducible
	a direct sum of different irreducibles
	a direct sum of different irreducibles
	either a sum of different irreducibles, or twice the same one