why nag? (2)
Now, can
we assign such an non-commutative tangent space, that is a
803077456b656e6c21a72d3dc3883fcd.gif' title='\mathbf{rep}~Q'
alt='\mathbf{rep}~Q' /> for some quiver Q, to
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' />? As
35572514a2942f20799ca4517df18e53.gif' title='\Gamma = \mathbb{Z}_2 \ast
\mathbb{Z}_3' alt='\Gamma = \mathbb{Z}_2 \ast \mathbb{Z}_3' /> we may
restrict any solution
0867769958ef42023457738a62808027.gif' title='V=(X,Y)' alt='V=(X,Y)' />
in
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' /> to the finite subgroups
aeab2a60e0268f50c685c9ed5b738caf.gif' title='\mathbb{Z}_2'
alt='\mathbb{Z}_2' /> and
0bddb2181901c5b50efce95fbe7aa4fb.gif' title='\mathbb{Z}_3'
alt='\mathbb{Z}_3' />. Now, representations of finite cyclic groups are
decomposed into eigen-spaces. For example
69893859e92c3b7f6e67d621e65b24a4.gif' title='V \downarrow_{\mathbb{Z}_2}
= V_+ \oplus V_-' alt='V \downarrow_{\mathbb{Z}_2} = V_+ \oplus V_-'
/>
where
d5f0a745d513201482dc5c17aac50adc.gif' title='V_{\pm} = \{ v \in V~|~g.v
= \pm v \}' alt='V_{\pm} = \{ v \in V~|~g.v = \pm v \}' /> with g the
generator of
aeab2a60e0268f50c685c9ed5b738caf.gif' title='\mathbb{Z}_2'
alt='\mathbb{Z}_2' />. Similarly,
1dca3fccb587c7af3b77783dc1bb40fe.gif' title='V \downarrow_{\mathbb{Z}_3}
= V_1 \oplus V_{\rho} \oplus V_{\rho^2}' alt='V
\downarrow_{\mathbb{Z}_3} = V_1 \oplus V_{\rho} \oplus V_{\rho^2}'
/>
where
d2606be4e0cd2c9a6179c8f2e3547a85.gif' title='\rho' alt='\rho' /> is a
primitive 3-rd root of unity. That is, to any solution
4aa77322a539a54b95c4bc6b175040e2.gif' title='V \in \mathbf{rep}~\Gamma'
alt='V \in \mathbf{rep}~\Gamma' /> we have found 5 vector spaces
a6aeeec24b028fbe618fdce14dda39fa.gif' title='V_+,V_-,V_1,V_{\rho}'
alt='V_+,V_-,V_1,V_{\rho}' /> and
bdd8deb26a9c04fad6653d91bef944aa.gif' title='V_{\rho^2}'
alt='V_{\rho^2}' /> so we would like them to correspond to the vertices
of our conjectured quiver Q.
What are the arrows of Q, or
equivalently, is there a natural linear map between the vertex-vector
spaces? Clearly, as
b4d25eb459a844e278bddfdb6ff9f1ac.gif' title='V_+ \oplus V_- = V = V_1
\oplus V_{\rho} \oplus V_{\rho^2}' alt='V_+ \oplus V_- = V = V_1 \oplus
V_{\rho} \oplus V_{\rho^2}' />
any choice of two bases of V (one
compatible with the left-side decomposition, the other with the
right-side decomposition) are related by a basechange matrix B which we
can decompose into six blocks (corresponding to the two decompositions
in 2 resp. 3 subspaces
de48f4f9dc33bb551e0dea3907376eee.gif' title='B = \begin{bmatrix} B_{11}
& B_{12} \\ B_{21} & B_{22} \\ B_{31} & B_{32} \end{bmatrix}' alt='B =
\begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \\ B_{31} & B_{32}
\end{bmatrix}' />
which gives us 6 linear maps between the
vertex-vector spaces. Hence, to
4aa77322a539a54b95c4bc6b175040e2.gif' title='V \in \mathbf{rep}~\Gamma'
alt='V \in \mathbf{rep}~\Gamma' /> does correspond in a natural way a
representation of dimension vector
418dd64ba3cd4289441d53fe254876d3.gif'
title='\alpha=(a_1,a_2,b_1,b_2,b_3)' alt='\alpha=(a_1,a_2,b_1,b_2,b_3)'
/> (where
ea3ec4af491dbab500dfc9d1b120a3c9.gif'
title='dim(V_+)=a_1,\ldots,dim(V_{\rho^2})=b_3'
alt='dim(V_+)=a_1,\ldots,dim(V_{\rho^2})=b_3' />) of the quiver Q which
is of the form
26a8254b3e45806da1b86a09e0fb0c95.gif' title='\xymatrix{ & & & &
\vtx{b_1} \\ \vtx{a_1} \ar[rrrru]^(.3){B_{11}} \ar[rrrrd]^(.3){B_{21}}
\ar[rrrrddd]_(.2){B_{31}} & & & & \\ & & & & \vtx{b_2} \\ \vtx{a_2}
\ar[rrrruuu]_(.7){B_{12}} \ar[rrrru]_(.7){B_{22}}
\ar[rrrrd]_(.7){B_{23}} & & & & \\ & & & & \vtx{b_3}}' alt='\xymatrix{ &
& & & \vtx{b_1} \\ \vtx{a_1} \ar[rrrru]^(.3){B_{11}}
\ar[rrrrd]^(.3){B_{21}} \ar[rrrrddd]_(.2){B_{31}} & & & & \\ & & & &
\vtx{b_2} \\ \vtx{a_2} \ar[rrrruuu]_(.7){B_{12}} \ar[rrrru]_(.7){B_{22}}
\ar[rrrrd]_(.7){B_{23}} & & & & \\ & & & & \vtx{b_3}}' />
Clearly, not every representation of
803077456b656e6c21a72d3dc3883fcd.gif' title='\mathbf{rep}~Q'
alt='\mathbf{rep}~Q' /> is obtained in this way. For starters, the
eigen-space decompositions force the numerical restriction
49ed1b19828d9d278fa694f738b3b74d.gif' title='a_1+a_2 = dim(V) =
b_1+b_2+b_3 ' alt='a_1+a_2 = dim(V) = b_1+b_2+b_3 ' />
on the
dimension vector and the square matrix constructed from the arrow-linear
maps must be invertible. However, if both these conditions are
satisfied, we can reconstruct the (isomorphism class) of the solution in
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' /> from this quiver representation by taking
40fc94be2e9fcc9669b714c372c433fd.gif' title='X = B^{-1} \begin{bmatrix}
1_{b_1} & 0 & 0 \\ 0 & \rho^2 1_{b_2} & 0 \\ 0 & 0 & \rho 1_{b_3}
\end{bmatrix} B \begin{bmatrix} 1_{a_1} & 0 \\ 0 & -1_{a_2}
\end{bmatrix}' alt='X = B^{-1} \begin{bmatrix} 1_{b_1} & 0 & 0 \\ 0 &
\rho^2 1_{b_2} & 0 \\ 0 & 0 & \rho 1_{b_3} \end{bmatrix} B
\begin{bmatrix} 1_{a_1} & 0 \\ 0 & -1_{a_2} \end{bmatrix}' />
0b143199ecf31dee12d70f6a2cd2438f.gif' title='Y = \begin{bmatrix} 1_{a_1}
& 0 \\ 0 & -1_{a_2} \end{bmatrix} B^{-1} \begin{bmatrix} 1_{b_1} & 0 & 0
\\ 0 & \rho^2 1_{b_2} & 0 \\ 0 & 0 & \rho 1_{b_3} \end{bmatrix} B'
alt='Y = \begin{bmatrix} 1_{a_1} & 0 \\ 0 & -1_{a_2} \end{bmatrix}
B^{-1} \begin{bmatrix} 1_{b_1} & 0 & 0 \\ 0 & \rho^2 1_{b_2} & 0 \\ 0 &
0 & \rho 1_{b_3} \end{bmatrix} B' />
Hence, it makes sense to
view
803077456b656e6c21a72d3dc3883fcd.gif' title='\mathbf{rep}~Q'
alt='\mathbf{rep}~Q' /> as a linearization of, or as a tangent space to,
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' />. However, though we reduced the study of
solutions of the polynomial system of equations to linear algebra, we
have not reduced the isomorphism problem in size. In fact, if we start
of with a matrix-solution
0867769958ef42023457738a62808027.gif' title='V=(X,Y)' alt='V=(X,Y)' />
of size n we end up with a quiver-representation of total dimension 2n.
So, can we construct some sort of non-commutative normal space to the
isomorphism classes? That is, is there another quiver Q whose
representations can be interpreted as normal-spaces to orbits in certain
points?