When is a class function on a group G (finite abelian) into the rational numbers Q an element of the rational representation ring of G? Given a class function $f: G \to \mathbb Q$, where $G$ is a finite abelian group, is there an easy way to decide whether $f$ is an element of the rational representation ring $R_{\mathbb Q}(G)$, i.e. whether $f$ is a virtual character of some representations of $G$?
If it makes things easier you could also assume that $G$ is a $p$-group and the class function has values in $\mathbb Z$. 
 A: This essentially boils down to the case of a cyclic group.  For a cyclic group of order n, the irreducible representations correspond to the action on $\mathbb Q[\omega_d]$ where $\omega_d$ is a primitive $d^{th}$-root of unity where $d$ divides $n$.  So one can easily produce the rational character table and check if your function is a non-negative linear combination of these irreducible characters.
Added: You can now use the orthogonality relations over $\mathbb Q$ to see if something is a character of a rep.  One knows the endomorphism algebra of $\mathbb Q[\omega_d]$ as a module over the cyclic group $G$ is $\mathbb Q[\omega_d]$ which has degree $\phi(d)$.  So the orthogonality relations (valid over any field of characteristic 0, but in the non-algebraically closed case involves the dimensions of the endomorphism division algebras) can be used to decompose class functions in terms of irreducible characters.
Answer 2 Alternatively, one can build the character table of a finite abelian group $G$ over $\mathbb Q$ in the following way (without reducing to the case of a cyclic group).  The Schur index of a degree one character is $1$.  So the irreducible characters for $G$ are obtained as follows.  Take a complex character $\chi$ and let $\Gamma(\chi)=Gal(\mathbb Q[\chi]:\mathbb Q)$ where $\mathbb Q[\chi]$ is the field generated by the values of $\chi$.  Then the sum of the orbit of $\chi$ under $\Gamma$ is a $\mathbb Q$-irreducible character and all such characters are obtained in this fashion.  So this describes an orthogonal basis for class functions over $\mathbb Q$ and hence all class functions.
supplement This may be off, but I think a rational-valued class function $f$ is a rational character iff for each complex character $\chi$, one has the inner product $\langle f,\chi\rangle$ is an integer and this integer is constant on the orbit of $\chi$ under $\Gamma(\chi)$.
A: A necessary criterion is that, if $g \in G$ has order $m$, and $k$ is relatively prime to $m$, then $f(g) = f(g^k)$. 
Proof: Let $\rho$ be a rational representation. Let the eigenvalues of $\rho(g)$ be $\omega_1$, $\omega_2$, ..., $\omega_N$. Then $f(g) = \sum \omega_i$ and $f(g^k) = \sum \omega_i^k$. 
But the $\omega_i$ are all $m$-th roots of unity, and there is a Galois symmetry $\sigma$ of $\mathbb{Q}(e^{2 \pi i/m})$ which acts on $m$-th roots of unity by $\omega \mapsto \omega^k$. So $f(g^k)$ is the image of $f(g)$ under this symmetry. Since $f(g)$ is in $\mathbb{Q}$, this shows that $f(g) = f(g^k)$. QED 
One can show that any rational class function as above is a $\mathbb{Q}$-linear combination of rational characters. If you want to know about $\mathbb{Z}$-linear combinations, that sounds harder; I don't know a simple rule.
