Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p \neq 2$.


Let $V$ be a finite-dimensional irreducible $kG$-module. If $V \cong V^\*$, then $V$ admits a nonzero $G$-invariant bilinear form $(-,-)$, unique up to scalar, such that $(-,-)$ is alternating or symmetric. Is there a formula or a general method to determine whether $(-,-)$ is going to be alternating or symmetric?


If $k = \mathbb{C}$, then the answer is yes: for a $\mathbb{C}G$-module $V$ with irreducible character $\chi$ we have the Frobenius-Schur indicator $$\nu\_2(\chi) = \frac{1}{|G|} \sum\_{g \in G} \chi(g^2)$$


which satisfies $\nu\_2(\chi) \in \{-1, 0, 1\}$ and


* $\nu\_2(\chi) \neq 0$ iff $V \cong V^\*$.
* $\nu\_2(\chi) = 1$ if there is a nonzero $G$-invariant orthogonal bilinear form on $V$.
* $\nu\_2(\chi) = -1$ if there is a nonzero $G$-invariant alternating bilinear form on $V$.


I think similar things should be true when $p \not \mid |G|$. What about in general? Is this an open problem?



Formula for the Frobenius-Schur indicator of a finite group?
For one answer, here is a theorem due to Thompson and Willems (*Bilinear forms in characteristic $p$ and the Frobenius-Schur indicator*, Lecture Notes in Mathematics 1185, pg. 221-230).


For an irreducible self-dual $kG$-module $V$, set $\varepsilon(V) = 1$ if $G$ preserves a nonzero symmetric bilinear form on $V$, and $\varepsilon(V) = -1$ if $G$ preserves a nonzero alternating form on $V$. 



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> **Theorem:** Let $V$ be an irreducible self-dual $kG$-module with Brauer character $\phi$. There exists an ordinary irreducible character $\chi$ such that 
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> > (1) $\chi$ is real valued.
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> > (2) The decomposition number $d(\chi, \phi)$ is odd.
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> Furthermore, any ordinary irreducible character $\chi$ satisfying (1) and (2) has the property that $\nu\_2(\chi) = \varepsilon(V)$.
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So assuming you know the ordinary character table of $G$, and the Frobenius-Schur indicators and decomposition numbers of all irreducible characters; you can give an answer. I don't know if there are any other general results.


