Definition 13.4.1 (Character). The character of a representation is the function defined by , the trace of the linear operator .
Note. Since trace is conjugation-invariant (), the character only depends on the conjugacy class of . That is, is a class function.
Definition 13.4.2 (Class function). A class function is a function satisfying for all .
Proposition 13.4.1 (Properties of characters). Given representations of :
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(since acts by , and eigenvalues of are roots of unity, so eigenvalues of are their conjugates).
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For a permutation representation on a set :.
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13.4.1 Symmetric Polynomials (Digression)
The connection between eigenvalues and characters motivates a brief digression. To store information about complex numbers (the eigenvalues of ), up to reordering, it suffices to specify the coefficients of the polynomial . These coefficients are the elementary symmetric polynomials .
Theorem 13.4.1 (Fundamental theorem of symmetric polynomials). The subring of symmetric polynomials is isomorphic to the polynomial algebra . That is, every symmetric polynomial is uniquely a polynomial expression in the elementary symmetric polynomials.
Note. Alternatively, one can use the power sums . Then as well. In particular, specifying the unordered tuple is equivalent to specifying . For representations, this means specifying all eigenvalues of is equivalent to specifying . But since is a group, is just —which is already determined by . So the character determines the eigenvalues.