Suppose I am given some representation of a finite group: $\rho : G \to \text{GL}(n, \mathbb{C})$. I want to compute a unitary representation $\tau$ which is isomorphic to $\rho$.
I know about Weyl's trick, where you define a $G$-invariant inner product, $\langle v,w \rangle = \frac1{|G|}\sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle_0$ where $\langle,\rangle_0$ is the dot product (the usual inner product). Then $\rho$ is unitary with respect to the new inner product (by relabelling the sum, easy argument).
But how does this help us compute the representation of $\tau$ where the images are unitary matrices? By this I mean unitary according to the original inner product.
Here are my thoughts so far: Inner products on $\mathbb{C}^n$ differ by a linear map, $\langle v,w \rangle = \langle Av, w \rangle_0$ for some $A \in \text{GL}(n, \mathbb{C})$, positive definite. Also $\langle v,w \rangle = v^* A^* w$. Maybe at this point I can do the Cholesky decomposition of $A$: $\langle v,w \rangle = v^* A^* w = v^* L^*L w = \langle Lv, Lw \rangle_0$. So the new inner product is just the dot product after doing the linear map $L$. And since this inner product is $G$-invariant, doesn't this imply $L$ is $G$-invariant also?
The Cholesky decomposition is problematic though, since in computer algebra systems (like GAP, SageMath), arbitrary square roots in cyclotomic fields (the closest thing to $\mathbb{C}$) are not possible.
Any advice on my approach or better ideas are welcome.
Indeed, $L$ in the Cholesky decompostion does provide a change of the basis, $y:=Lx$, in which the matrix of the scalar product becomes $I$. If you want to stay in the same cyclotomic field, you can settle for a bit less: compute $LDL^*$ decompostion instead, with $L$ lower-triangular with all 1s on the diagonal, and $D$ diagonal matrix (the entries of $D$ will be positive real cyclotomic numbers).
While it is apparently true that one can go to a bigger cyclotomic field to write $D=\Lambda\Lambda^*$, with $\Lambda$ a diagonal matrix of cyclotomics - only reference I know is this Mathoverflow answer by Will Sawin, I don't know good degree bounds (and the problem of finding such a $\Lambda$ is apparently hard, there were cryptographic systems designed using the very problem of factoring cyclotomics in this way).
Note that taking square roots is even worse in a sense, you might have to leave the cyclotomic field completely. (E.g. $\sqrt{\sqrt{2}}$ is not cyclotomic, whereas $\sqrt{2}$ is).