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I'm very new with representations of groups.

Here it is a non-commutative group with $6$ elements defined by $D_3 = \langle p,q\ |\ p^3 = q^2 = (pq)^2 = 1\rangle$

The representation I'm interested in is the permutation matrix representation $$P = {\scriptstyle\begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\end{pmatrix}},\qquad Q = \scriptstyle\begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0\end{pmatrix}$$

From which I would like to find a matrix $T$ such that for any $X \in \langle P, Q \rangle$ I have $$T X T^{-1} = \scriptstyle\begin{pmatrix} \rho_1(X) & \\ & \rho_2(X) \\ & & \rho_3(X) \\ & & & \rho_4(X)\end{pmatrix}$$ where $\rho_i$ are the irreducible representations.

Is there a way to achieve this using Sage, Magma or Gap ?

(Note : I'm interested in the Artin L-functions of $K/\mathbb{Q}$ where $K$ is a Galois number field with Galois group $D_3$, and $T$ is supposed to help understanding how the representations could act on the field)

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1 Answer 1

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This is basically a two-stage process. First we compute (in the representation you have) the idempotents for the different irreducible representations. They are given by the formula $$ e_\chi=\frac{1}{|G|}\sum_{g\in G} \chi(g)g^{-1} $$ Multiplication with the idempotents will map into the homogeneous component for the representation, that is the row space of each idempotent is a basis for the homogeneous component for the representation.

In GAP, you would construct the matrix group (also could use homomorphism from permutations to matrices instead), and calculate the irreducible characters:

gap> m1:=PermutationMat((1,3,5)(2,6,4),6);
[ [ 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0 ],
  [ 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0 ] ]
gap> m2:=PermutationMat((1,2)(3,4)(5,6),6);
[ [ 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0 ],
  [ 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0 ] ]
gap> d3:=Group(m1,m2);Size(d3);
<matrix group with 2 generators>
6
gap> chars:=Irr(d3);
[ Character( CharacterTable( <matrix group of size 6 with 2 generators> ),
  [ 1, 1, -1 ] ), Character( CharacterTable( <matrix group of size 6 with
    2 generators> ), [ 2, -1, 0 ] ),
  Character( CharacterTable( <matrix group of size 6 with 2 generators> ),
  [ 1, 1, 1 ] ) ]
gap> List(chars,x->x[1]);
[ 1, 2, 1 ]

Next we determine the elements of the group and the classes they lie in

gap> elm:=Elements(d3);;
gap> cls:=List(elm,x->PositionProperty(ConjugacyClasses(d3),y->x in y));
[ 3, 2, 3, 2, 3, 1 ]
gap> idems:=List(chars,chi->1/Size(d3)*Sum([1..6],x->chi[cls[x]]*elm[x]^-1));

This gives us three matrices of rank $1^2$, $2^2$ and $1^2$. Their row spaces are bases for the homogeneous components for each representation. We guess that the first 4 rows of the second matrix are independent and do a base change, this gives (we also swap representations 2 and 3 to get the arrangement you want):

gap> bas:=Concatenation([idems[1][1],idems[3][1]],idems[2]{[1..4]});
[ [ 1/6, -1/6, 1/6, -1/6, 1/6, -1/6 ], [ 1/6, 1/6, 1/6, 1/6, 1/6, 1/6 ],
  [ 1/3, 0, -1/6, 0, -1/6, 0 ], [ 0, 1/3, 0, -1/6, 0, -1/6 ],
  [ -1/6, 0, 1/3, 0, -1/6, 0 ], [ 0, -1/6, 0, 1/3, 0, -1/6 ] ]
gap> RankMat(bas);
6
gap> Display(bas*m1/bas);
[ [   1,   0,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0,   0 ],
  [   0,   0,   0,   0,   1,   0 ],
  [   0,   0,   0,  -1,   0,  -1 ],
  [   0,   0,  -1,   0,  -1,   0 ],
  [   0,   0,   0,   1,   0,   0 ] ]
gap> Display(bas*m2/bas);
[ [  -1,   0,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0,   0 ],
  [   0,   0,   0,   1,   0,   0 ],
  [   0,   0,   1,   0,   0,   0 ],
  [   0,   0,   0,   0,   0,   1 ],
  [   0,   0,   0,   0,   1,   0 ] ]

Now the 1-dimensional representations are OK, but the last block has the 2-dimensional representation twice. To split this up we need to find a vector in a submodule. (The method is essentially what goes by the name of MeatAxe):

I will not describe the general method here, but just observe that an eagle-eyed view of the two matrices shows that the vector $(0,0,1,1,0,0)$ has orbit length under the group generated by the two matrices and lies in a 2-dimensional subspace. Complementing this submodule and doing the base change gives:

gap> id:=One(gp2);;
gap> v:=id[3]+id[4];
[ 0, 0, 1, 1, 0, 0 ]
gap> o:=Orbit(gp2,v);
[ [ 0, 0, 1, 1, 0, 0 ], [ 0, 0, 0, -1, 1, -1 ], [ 0, 0, -1, 0, -1, 1 ] ]
gap> RankMat(o);
2
gap> bas2:=[id[1],id[2],o[1],o[2],id[5],id[6]];
[ [ 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 1, 0, 0 ],
  [ 0, 0, 0, -1, 1, -1 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ] ]
gap> RankMat(bas2);
6
gap> Display(bas2*gp2.1/bas2);
[ [   1,   0,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0,   0 ],
  [   0,   0,   0,   1,   0,   0 ],
  [   0,   0,  -1,  -1,   0,   0 ],
  [   0,   0,  -1,  -1,   0,  -1 ],
  [   0,   0,   0,  -1,   1,  -1 ] ]
gap> Display(bas2*gp2.2/bas2);
[ [  -1,   0,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0,   0 ],
  [   0,   0,   1,   0,   0,   0 ],
  [   0,   0,  -1,  -1,   0,   0 ],
  [   0,   0,   0,   0,   0,   1 ],
  [   0,   0,   0,   0,   1,   0 ] ]

Now vectors 3 and 4 also span a submodule, but 5 and 6 do not yet. The proof of Maschke's theorem will give a general way to fix this, but as I'm lazy I observe that we could instead do the same with the vector $(0,0,0,0,1,1)$ to span another 2-dimensional subspace.

gap> v2:=id[5]+id[6];
[ 0, 0, 0, 0, 1, 1 ]
gap> o2:=Orbit(gp2,v2);;RankMat(o2);
2
gap> bas2:=[id[1],id[2],o[1],o[2],o2[1],o2[2]];
[ [ 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 1, 0, 0 ],
  [ 0, 0, 0, -1, 1, -1 ], [ 0, 0, 0, 0, 1, 1 ], [ 0, 0, -1, 1, -1, 0 ] ]
gap> RankMat(bas2);
6

To go from your original representation both base changes need to be combined, indeed it works as desired:

gap> T:=bas2*bas;
[ [ 1/6, -1/6, 1/6, -1/6, 1/6, -1/6 ], [ 1/6, 1/6, 1/6, 1/6, 1/6, 1/6 ],
  [ 1/3, 1/3, -1/6, -1/6, -1/6, -1/6 ], [ -1/6, -1/6, 1/3, -1/6, -1/6, 1/3
],
  [ -1/6, -1/6, 1/3, 1/3, -1/6, -1/6 ], [ -1/6, 1/3, -1/6, -1/6, 1/3, -1/6 ]
]
gap> Display(T*m1/T);
[ [   1,   0,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0,   0 ],
  [   0,   0,   0,   1,   0,   0 ],
  [   0,   0,  -1,  -1,   0,   0 ],
  [   0,   0,   0,   0,   0,   1 ],
  [   0,   0,   0,   0,  -1,  -1 ] ]
gap> Display(T*m2/T);
[ [  -1,   0,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0,   0 ],
  [   0,   0,   1,   0,   0,   0 ],
  [   0,   0,  -1,  -1,   0,   0 ],
  [   0,   0,   0,   0,   1,   0 ],
  [   0,   0,   0,   0,  -1,  -1 ] ]
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