Given a group representation $\Psi:G\rar\OO(n)$ and the table of characters of $G$, sketch a computer program that determines the isotypic projections. Of course, you may employ any computer algebra program!

We will do it utilizing sage! Beware, for the subsequent code to work the matrix entries need to be in a ring sage can handle precisely, such as $\Z$, $\Q$, $\Q(e^{2\pi i/N})$, etc.

  1. Import 'numpy':
    import numpy as np
  2. Suppose the group $G$ is generated by $g_1,\ldots,g_d$, then $\Psi(G)$ is generated by $U_1\colon=\Psi(g_1),\ldots,U_d\colon=\Psi(g_d)\in\OO(n)$ - $n$ is the dimension of the representation:
    U1=matrix(n,[...])
        $\ldots$
    Ud=matrix(n,[...])
  3. Generate the group $\Psi(G)$ - this definitely doesn't work if the entries of the matrices 'U1,...' are arbitrary real or complex numbers!
    PsiG=MatrixGroup(U1,...,Ud)
  4. Determine all characters of $\Psi(G)$:
    Chars=PsiG.irreducible_characters()
  5. For all characters $\chi$ add up all the matrices in the Projection Theorem and print the matrix of the projection onto the isotypic component featuring the character $\chi$ and the dimension of the image of the projection:
    for chi in Chars:
        P=matrix(n)
        for g in PsiG:
            mg=g.matrix()
            P=P+chi(g)*mg
        print(P,P.rank())
If you want to find a basis for the image $\im P$ type:
ImP=P.image().matrix().transpose()
The rows of the matrix 'P.image().matrix()' will form a basis for $\im P$ and thus the columns of 'ImP' form a basis of $\im P$. Finally you may orthonormalize these vectors via Gram-Schmidt, which is implicated by $QR$-factorization in 'numpy':
Q,R=np.linalg.qr(ImP)
print matrix(Q).str(rep_mapping=lambda x: str(round(x,4))
and the columns again will form an orthonormal basis for $\im P$ (precision: 3 digits).