Let $G$ be a finite group, and let $V$ be an $n$-dimensional *real* representation of $G$. Think of $V$ as given by a homomorphism
$$ \rho_V\colon G\to O(n).$$
Write $\chi_V$ for the character of $V$.

Here are two problems.

Using

*only*the character $\chi_V$ of $V$, determine whether $\rho_V(G)\subset SO(n)$.Using

*only*the character $\chi_V$ of $V$, and assuming $\rho_V(G)\subset SO(n)$, determine whether $\rho_V$ admits a factorization through a homomorphism $\widetilde{\rho}_V:G\to Spin(n)$.

Here's one answer to 1: the identity of formal power series $$ \sum_{k\geq0} \chi_{\Lambda^kV}(g)\\,T^k = \exp\Bigl( -\sum_{k\geq1} \frac{1}{k}\chi_V(g^k)\\, (-T)^k \Bigr)$$ where $\Lambda^kV$ is the $k$-th exterior power representation of $V$, gives $\chi_{\Lambda^nV}(g)$ as a polynomial in $\chi_V(g),\dots,\chi_V(g^n)$, and $\rho_V(G)\subset SO(n)$ if and only if all $\chi_{\Lambda^nV}(g)>0$.

Is there a better answer for 1? Is there any answer in a similar spirit for 2?

everyelement of $SO(n)$ lifts (compatibly with its powers) to an element of $Spin(n)$. Instead, you need information about $\rho(g_1), \rho(g_2), \rho(g_1 g_2)$ for all pairs $(g_1, g_2)$ -- information depending probably on the conjugacy class of the pair $(g_1, g_2)$ (not individual conjugacy classes). – Marty Jan 22 '11 at 6:25