Invariant Modules and being Finitely Generated?

asked 2025-04-30 18:14:19 +0200

SatiatedMoth gravatar image

This is a crosspost from MathSE

I am trying to run this code in SageMath (to get the invariant module of $\mathbb CGL_3(\mathbb F_3)$ under conjugation by the subgroup H):

G33 = GL(3, GF(3))
M1=G33([2, 0, 0, 0, 1, 0, 0, 0, 1])
M2=G33([2, 1, 0, 2, 0, 0, 0, 0, 1])
A=G33.algebra(CC)
H=G33.subgroup([M1, M2])
action = lambda g, m: m*g*m^-1 
from sage.modules.with_basis.representation import Representation
R=Representation(G33, A, action)
I=R.invariant_module()

However, when I do so I get the error that:

ValueError: General Linear Group of degree 3 over Finite Field of size 3 is not finitely generated

However, clearly $GL_3(\mathbb F_3)$ is generated by 2 matrices (SageMath returns two matrices when calling G33.gens())

Does anyone have any insight on why this is happening (why it is saying it is not finitely generated) and how I could fix the problem (what I could change about my SageMath implementation)?

Thank you, have a good one.

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