Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$. 


Let $x$ be an involution in $G$. 


I'd like to ask the following


**Question 1:**



> 
> What is known about $\chi(x)?$
> 
> 
> 


1a) Are there criteria when $\chi(x)$ is positive / negative / zero ?


Of course, $1\_{\text{Aut}(V)}=\rho(x^2)=\rho(x)^2$, such that the only possible eigenvalues of $\rho(x)$ are $\pm 1$. 


Moreover, there is an article written by P.X. Gallagher with the title "Character values at involutions" (DOI: <https://doi.org/10.1090/S0002-9939-1994-1185260-1>) dealing with the case that $\int\_G\chi\_1\chi\_2\chi\_3 \neq 0$, where the integral is in the sense of the Haar measure.


**EDIT:**
The main parts of Gallagher's results are the following ones:


For an involution $\sigma$ of a finite group $G$ and an irreducible complex representation $R$ of $G$, denote by $q$ the proportion of $-1$'s among the eigenvalues of $R(\sigma)$. Then:


$(\*)$ $\frac{1}{h}\leq q \leq 1-\frac{1}{h}$, unless $q = 0$ or $1$,
where $h$ is the index of the centralizer $C$ of $\sigma$.


Moreover, if $\int\_G\chi\_1\chi\_2\chi\_3 \neq 0$, then $(\*)$ is refined to prove that the proportions of $-1$'s among the eigenvalues of $\rho\_1, \rho\_2$ and $\rho\_3$ (i.e., the corresponding representations) at $\sigma$ are the sides of a triangle on a sphere of circumference 2.


$\ $


1b) When does $\int\_G\chi\_1\chi\_2\chi\_3 \neq 0$ happen (necessary / sufficient criteria)?


1c) When does $\int\_G\chi\_1\chi\_2\chi\_3 = 0$ happen (necessary / sufficient criteria)?


1d) Are there results apart from Gallagher's result? 


1e) Can one deduce additional information, if all considered characters lie in the same 2-block?


Thanks for the help.



What is known about ordinary character values at involutions?
There are many results about the values of $\chi(t)$ when $t$ is an involution of a finite group $G$ and $\chi$ is an irreducible character: Isaacs' book on Character Theory has many such results collected from the literature, but there are many others scattered around:


For example, if $G = O^{2}(G)$ (equivalently, if $G/G^{\prime}$ has odd order), then $\chi(1) \equiv \chi(t)$ (mod $4$).


(Knörr): We have $\chi(t) = 0$ for every involution $t$ if and only if $|S|$ divides $\chi(1)$, where $S$ is a Sylow $2$-subgroup of $G.$


Regarding block theory, whenever $t $ is an involution of $G$ and $\chi$ is an irreducible character in the principal $2$-block of $G$, we have $\chi(tuv) = \chi(tu)$ whenever $u,v \in C\_{G}(t)$ have odd order and $v \in O\_{2^{\prime}}(C\_{G}(t)),$ which is a consequence of Brauer's Second and Third Main Theorems.


I could probably give several more examples if you gave further clues as to what you are looking for.


Further edit to address a question from comments: if $t$ is an involution of $G$ and $B$ is a $2$-block of $G$, then results of Brauer imply the following facts (among others):


If $B$ has defect group $D$ and $t$ is not $G$-conjugate to an element of $D$, then we have $\chi(t) = 0 $ for every complex irreducible character $\chi \in B$.


If $B$ has dfect group $D$ and some conjugate of $t$ lies in $D$, then there is an irreducible character $\chi \in B$ with $\chi(t) \neq 0$, and we have $\sum\_{ \chi \in {\rm Irr}(B)} \chi(1)\chi(t) =0,$ so there are irreducible characters in $B$ taking both positive and negative values at $t$.
Later edit: Another theorem of Brauer is that if $B$ is a $2$-block of defect $d >1$, then the number of irreducible characters in $B$ of degree exactly divisible by $2^{a-d}$ is divisible by $4$. In particular, this implies that if $|G|$ is divisible by $4$ and $t$ is an involution of $G$, then the number of irreducible characters $\chi$ in the principal $2$-block such that $\chi(t)$ is odd is a multiple of $4$.


