Building a homomorphism from group algebra to matrix space

edit

I'm not sure why this hasn't been implemented in SageMath. It's probably an oversight rather than being due to any difficulties. It's pretty easy to implement. Here is a workaround:

F = FreeGroup(4, names='A,B,C,D')
G = GroupAlgebra(F, ZZ)
A,B,C,D = G.gens()

A1 = matrix(CC,[[0,I],[I,0]])
B1 = matrix(CC,[[I,0],[0,-I]])
C1 = matrix(CC,[[0,1],[-1,0]])

def my_im_gens_(self, codomain, im_gens, base_map=None):
    result = codomain.zero()
    for (g,c) in self._monomial_coefficients.items():
        if base_map:
            c = base_map(c)
        result += c*g(im_gens)
    return result
G.element_class._im_gens_ = my_im_gens_

Then it works:

sage: f = G.hom([A1,B1,C1,C1], check=False)
sage: f(A^2 + B^3 + C) == A1^2 + B1^3 + C1
True