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Building a homomorphism from group algebra to matrix space

asked 2021-06-15 19:02:04 +0200

cate15 gravatar image

I would like to define an algebra homomorphism between a group algebra over the integers and a complex matrix space. More precisely I have a free group F on 4 generators and the associated group algebra G and I would like to construct a homomorphism which sends each of the generators to a certain complex matrix. Inspired by a similar question, I have tried multiple things, but nothing seems to work

I have

F.<A,B,C,D>=FreeGroup(4)
F.inject_variables()
R=MatrixSpace(CC,2)
A1 = matrix(CC,[[0,I],[I,0]])
B1 = matrix(CC,[[I,0],[0,-I]])
C1 = matrix(CC,[[0,1],[-1,0]])
G=GroupAlgebra(F, ZZ)

I would like to define the homomorphism G->R which sends A to A1, B to B1 and both C and D to C1.

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Link to the related question: https://ask.sagemath.org/question/106...

cate15 gravatar imagecate15 ( 2021-06-15 19:02:55 +0200 )edit

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answered 2021-06-15 20:38:47 +0200

rburing gravatar image

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
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For implementing this in SageMath I opened https://trac.sagemath.org/ticket/31989

rburing gravatar imagerburing ( 2021-06-16 10:26:12 +0200 )edit

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Asked: 2021-06-15 19:02:04 +0200

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Last updated: Jun 15