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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.

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

  2. 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?

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I guess there is a cohomological obstruction for 2 in $H^2(G,\mathbb{Z}/2\mathbb{Z})$. Although, I have no idea how to express this in character theory. –  Donu Arapura Jan 20 '11 at 17:06
I don't think that any such answer exists for (2). The obstruction to lifting is called the second Steifel-Whitney class of the orthogonal real representation $\rho_V$, written $w_2(\rho_V)$. It's difficult to compute and quite important for Galois representations (related to $\epsilon$-factors). Such results lie beyond simple character calculuations, I think. Compare, for instance, the problem of deciding whether a rep. into $SO(3)$ arises as $Sym^2$ of a rep into $SU(2)$. Can you tell by the character? –  Marty Jan 21 '11 at 0:26
(I suppose that the $SU(2)$ and $SO(3)$ case is just the case $n = 3$ of the question...) –  Marty Jan 21 '11 at 20:13
Okay. But I don't see why, a priori, having a cohomological obstruction means you can't compute it from the character; my question 1 is really about the vanishing of $w_1(\rho_V)$, after all. On the other hand, I could certainly believe that such a problem could be intractible. –  Charles Rezk Jan 21 '11 at 22:20
Here's another reason why such an answer shouldn't exist for (2). Let's say you know all the eigenvalues for $\rho(g)$, $\rho(g^2)$, etc., information given by the character. Well, that information tells you nothing about whether $\rho$ factors through the 2-fold Spin cover, because every element 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
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