Revision history [back]

I would like to know if it is possible to define a homomorphism 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 = matrix([[0,-1],[1,-1]]) sage: mb = matrix([[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.

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

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

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

I have 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 = matrix([[0,-1],[1,-1]])
sage: mb = matrix([[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.

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, 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 = matrix([[0,-1],[1,-1]]) 
M_3([[0,-1],[1,-1]])  
sage: mb = matrix([[1,-1],[0,-1]]) 
 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.

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.

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]}

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 just informational. The elements of ´G.gens()´ 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]}

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 just informational. 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]}