Finding $\chi _{\text{det}(V)} $ for two-dimensional representation

Question

So I am trying to work out what $\chi _{\text{det}(V)} $ is for the two-dimensional matrix representation of $D_8 $ where $D_8$ is the group of symmetries of the square.

I have found the character table for $D_8$ but I don’t know much about this two-dimensional representation of $D_8$, I just deduced it’s character from the other characters.

Anyway, here $\chi _{\text{det}(V)}(g)=\text{det}(\rho (g)) $ which is a one-dimensional character.

If I could find out what the matrix $\rho(g)$ is for this two-dimensional representation $\rho$ then I could just find the determinant of it and that would be easy. But as I say I don’t really know what this two-dimensional representation is.

There is an obvious two-dimensional representation of your group which is precisely the one given by its action on the square! You do know it. — Mariano Suárez-Álvarez, Mar 13 '23 at 12:24
How is this two-dimensional? Do we view the square in $\mathbb{C}^2$? — Anonmath101, Mar 13 '23 at 12:28
Wait, so for $g \in D_8$ here $\rho (g) $ is just some standard matrix. So if $a$ is the rotation through $\pi /2$ the. $\rho(a)$ is just the standard 2x2 rotation matrix? — Anonmath101, Mar 13 '23 at 12:44
Is there a way to do this without knowing explicitly what the two dimensional representation is explicitly? Can you calculate $\chi_{\text{det}(V)} $ without knowing exactly what $\rho$ is? — Anonmath101, Mar 13 '23 at 13:53
What want to compute the determinant of something without knowing it? — Mariano Suárez-Álvarez, Mar 13 '23 at 14:06
There are formulas for the character of the exterior powers of a representation V in terms of the character of V, which you can find here but you need to know the character of V. — Mariano Suárez-Álvarez, Mar 13 '23 at 14:08
(the usual action on the plane preserving a square is given by some real 2 by 2 matrices: just view them as complex matrices!) — Mariano Suárez-Álvarez, Mar 13 '23 at 14:11

Finding $\chi _{\text{det}(V)} $ for two-dimensional representation

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