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The following well known theorem can be found in many books on character theory:

Let $\chi$ be a faithful character of a finite group $G$ and suppose that $\chi(g)$ takes on exactly $m$ different values for $g \in G$. Then every $\psi \in \operatorname{Irr}(G)$ is a constituent of one the characters $\chi^j$, for $0 ≤ j < m$.

This particular theorem gives no information about the multiplicities of the constituents of the $\chi^j$.

My question is, are there any results which give more detailed information (dimension, etc... ) about the subspace of the space of class functions spanned by powers of a faithful irreducible character $\chi$, perhaps given some conditions on $G$? I'm interested in the question in general, but especially in the case where $G$ is a $p$-group.


As an example of what I mean, the group $S_4$ has 5 conjugacy classes, and has a faithful irreducible character $\chi$ which takes on the values $[3,-1,-1,0,1]$. Fixing an ordering for the conjugacy classes of $S_4$, I'm thinking of this as a vector in $\mathbb C^5$. The powers of $\chi$ are of the form $\chi^j = [3^j,(-1)^j,(-1)^j,0,1]$. Now $1,\chi,\chi^2,\chi^3$ span a 4 dimensional subspace of $\mathbb C^5$, and if my calculations are right, all $\chi^j$, $j >3$ can be expressed as linear combinations of lower powers of $\chi$.

Another example I tried was an extraspecial group of order 27. Here I found 11 conjugacy classes, and a faithful irreducible character $\chi$ of degree 3 whose powers only spanned a subspace of dimension 4.

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The multiplicities can be computed using the orthogonality relations. What exactly are you asking about? – Qiaochu Yuan Jul 9 '12 at 1:59
@Dane: the regular character is faithful, and its powers span a space of dimension 1 (2 if you count the trivial character). So very every group there is at least one very bad character to start with. – Jack Schmidt Jul 9 '12 at 2:24
Oops. I meant for $\chi$ to be irreducible. – Dane Jul 9 '12 at 2:33
Only certain $p$-groups have faithful irreducibles (cyclic center). Looking at such groups of order at most 100, it appears the span always has dimension of the form $p^n+1$ if the group is non-abelian with cyclic center. I'm not sure how to predict $n$. – Jack Schmidt Jul 9 '12 at 2:50
up vote 3 down vote accepted

Claim: Let $S$ be a finite set, $k$ be a field, and $f : S \to k$ be a function. The dimension of the subspace of the space of functions $S \to k$ spanned by the (positive) powers of $f$ is the number of distinct nonzero values taken by $f$.

Proof. Clearly the number of distinct nonzero values $d$ is an upper bound. To prove that this upper bound is achieved, let $x_1, ... x_d$ be the distinct nonzero values taken by $f$. The linear transformation $$(c_1, ... c_d) \mapsto \sum_{i=1}^d c_i f^i$$

has matrix given in an appropriate basis by essentially the Vandermonde matrix, which is invertible. The conclusion follows.

I don't know anything about the minimal number of distinct nonzero values a faithful irreducible character can take, though.

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