I'm doing some preparation for an upcoming exam and i found his exercise here.
https://tartarus.org/gareth/maths/tripos/II/Representation_Theory.pdf It's about exercise on Page 81 Paper Section II nr. iii).(I just noticed that the are a few pages with No.81,the exercise i refer to is on Page 21 in the pdf)
It's about determining ${ X }_{ A }$,$\;$${ X }_{ S }$ from one given character. I know the two formulas that are stated in the exercise above ${ X }_{ A }(g)=\frac { 1 }{ 2 } ({ X }^{ 2 }(g)-X({ g }^{ 2 }))$ and ${ X }_{ S }(g)=\frac { 1 }{ 2 } ({ X }^{ 2 }(g)+X({ g }^{ 2 }))$. However i have real struggle to determine $X({ g }^{ 2 })$. There is also a hint given that ${ g }_{ 1 },\dots,{ g }_{ 7 }\quad $ and ${ { g }^{ 2 } }_{ 1 },\dots,{ { g }^{ 2 } }_{ 7 }$ are conjugated so we must have $X({ g }_{ 1 },\dots,{ g }_{ 7 })=X({ { g }^{ 2 } }_{ 1 },\dots,{ { g }^{ 2 } }_{ 7 })$ but if a try the calculation with these facts i get wrong results ( I know that the group stated in this Problem is $Sl(2,3)$ so i can compare).
So my question is if someone could show/explain me instructivly how to get the values of $X({ g }^{ 2 })$ with the given facts. Thanks for your help
edit: the little(and wrong)) calculatio i have by using $X({ g }_{ 1 },...,{ g }_{ 7 })=X({ { g }^{ 2 } }_{ 1 },....,{ { g }^{ 2 } }_{ 7 })$ is ${ X }_{ A }(g)=\frac { 1 }{ 2 } ({ (2,-2,0,-{ w }^{ 2 },-w,w,{ w }^{ 2 }) }^{ 2 }-((2,-2,0,-{ w }^{ 2 },-w,w,{ w }^{ 2 }))$
It says that $g_1^2$ is conjugate to $g_1$, $g_2^2$ is conjugate to $g_1$ etc.
For instance $\chi_A(g_4)=\frac12(\chi(g_4)^2-\chi(g_4^2)) =\frac12(\chi(g_4)^2-\chi(g_5))$ as $g_4^2$ is conjugate to $g_5$, and so $\chi(g_4^2)=\chi(g_5)$.
Now we can put the given values in to get $\chi_A(g_4)=\frac12((-\omega)^2-(-\omega))=\frac12(\omega+\omega)=\omega$ etc.