For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for instance.
However, what I can't find is a simple statement of going back the other way – which real representations come from a usual complex character table. (This is useful if one can compute a character table with e.g. Gap.) I assume that this is pretty nontrivial, otherwise I wouldn't have to wish I had my references for representation theory with me.
And yet… wouldn't it be nice if one could (say) in certain circumstances just say "oh yeah, if two complex non-real characters look a lot alike, just add them and you get the real character."
Question: Is there ever a time where one is justified in just "adding non-real complex irreducible characters" to get real irreducible characters?
If it turns out this is trivial and I didn't know it, let's just chalk that up to the fact that nearly all of us presumably learned our representation theory mostly over $\mathbb{C}$.
Best Answer
The answer is completely described by the Frobenius–Schur indicator, $$v_2(\chi) = \frac{1}{|G|}\sum_{g\in G} \chi(g^2)$$ which is covered in chapter 4 of Isaacs's Character Theory of Finite Groups. In particular, we have the Frobenius–Schur theorem, as given on page 58 of Isaacs's CToFG:
The following is Lemma 9.18 (more or less) in Isaacs's CToFG:
Proof: If $\chi$ is the character of the representation $X$, then consider the representation $X \otimes_{\mathbb{C}} \mathbb{C}_\mathbb{R} \cong X \oplus \bar X$ obtained by replacing the complex entries $a+bi$ of $X$ with the block matrices $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$. The resulting matrix has trace $\chi(g) + \bar\chi(g)$, since we replace each diagonal entry $a+bi$ with a matrix of trace $2a$. The general case is similar: just choose a $K$-basis of $F$, and write out the associated matrix representation of $f \in F$ in its action on the $K$-vector space $F$. $\square$
Since you mention GAP, I'll mention the indicators of a character table are computed using
Indicator( chartable, 2)
and that since every complex, ordinary representation of $G$ is a representation over the fieldCF(Exponent(G))
, the $\sigma$ appearing in the Galois group are given bychi -> GaloisCyc(chi,k)
for $k$ relatively prime to $G$. In particular, $k=-1$ is complex conjugation. To compute the matrices of the $K$-representation from the $F$-representation as described in the proof, you can useBlownUpMat(Basis(AsVectorSpace(K,F)),mat)
.The function
RationalizedMat
can be applied to a character table to compute some smaller sums where one needs to multiple by the so called Schur index as in Geoff Robinson's answer: for example, it would take a real character and leave it real, even if its indicator were $-1$. However, for Brauer characters (where the Schur indices are always 1), it can be very handy.At any rate, chapters 4, 9, and 10 of Isaacs's textbook are great for understanding how characters work over different fields of characteristic 0 and how that relates to the power maps of the character table.