vuur's profile - activity

sage: R.<t> = PolynomialRing(ZZ)
sage: I = R.ideal([t^2-2])
sage: S.<z> = R.quotient_ring(I)
sage: eq_z = (z+1)^2-5; eq_z
2*z - 2
sage: SR(eq_z.lift()).variables()
(t,)
sage: SR(eq_z).variables()
()

I'd expect the last tuple to contain the variable z, what am I missing?

Is it possible to construct a direct sum of matrix spaces?

For example:

M1 = MatrixSpace(QQ, 1)
M2 = MatrixSpace(QQ, 2)
M = DirectSum(M1, M1, M2)    # doesn’t work

Thanks

Thanks for answering. I will try out what works best for me.

I don´t mind using (or calculating) the basis instead of gens, Thanks for answering. It helps alot :-)

yes, I used M_3 instead of matrix, I posted the wrong code here, sorry for that

I would like to know if it is possible to define a homomorphism (of algebras) between a group algebra over the rationals and a matrix space over the rationals.

I have

sage: S3 = SymmetricGroup(3)  
sage: QG = GroupAlgebra(S3, QQ)  
sage: QG.gens()  
Finite family {(1,2,3): (1,2,3), (1,2): (1,2)}  
sage: M_3 = MatrixSpace(QQ, 2)  
sage: ma = M_3([[0,-1],[1,-1]])  
sage: mb = M_3([[1,-1],[0,-1]])

Now I would like to map each generator of QG to a matrix. In this case I would like to map (1,2,3) to ma and (1,2) to mb and use that to define a homomorphism. But everything I tried didn't work.

Edit:
I've encountered another issue, though I can work around it. So this is only for information. The elements of G.gens() don't coincide with the reduced words of length 1.

sage: S3 = SymmetricGroup(3)
sage: S3.gens()
[(1,2,3), (1,2)]
sage: {g: g.reduced_word() for g in S3}
{(): [],
 (2,3): [2],
 (1,2): [1],
 (1,2,3): [2, 1],
 (1,3,2): [1, 2],
 (1,3): [1, 2, 1]}