Quotation from “The marriage of heaven and hell”

Those who restrain desire, do so because theirs is weak enough
MHH5;   E34|        to be restrained; and the restrainer or reason usurps its place &
MHH5;   E34|        governs the unwilling.
MHH5;   E34|        And being restraind it by degrees becomes passive till it is
MHH5;   E34|        only the shadow of desire.

Aside

I think I proved the representation ring of sl_{4} is isomorphic to so_{6} today. I have no idea how to prove the general case for the representation ring of so_{2n}, partly because the Weyl group isomorphic to (Z/2)^{n-1}\rtimes S_{n} has an order of 2^{n-1}*n!. This is a huge number and it is not clear to me how to find a basis manually without computer programming. I shall try Jim’s method if that works. Since the Weyl group is huge, the index theory method would fail for the same reason since we need to Weyl translate every product into the positive Weyl chamber then minor \rho, which is not very clear when |W|=192 or 1920, etc. The most imminent task at hand is to re-learn spinor representation and prove messy proof of n=3. For then the general case’s proof would be much clearer and easier to understand because we can tell how the fundamental representations associated with the fundamental weight vectors along the edges of the Weyl chamber. 

$R(h_{so_{6}})$ is a a free module over $R(so_{6})$

 

We need to prove that $\Z[x,y,z,x^{-1},y^{-1},z^{-1}]$ is a free module over $\Z[x+y+z+x^{-1}+y^{-1}+z^{-1},x^{\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}}, x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}}]$.

 

We shall prove that $\Z[x,y,z,x^{-1},y^{-1},z^{-1}]\cong \Z[x,y,z,w]/(xyzw-1)$ as well as $\Z[x+y+z+x^{-1}+y^{-1}+z^{-1},x^{\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}}, x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}}]$ is isomorphic to $\Z[x+y+z+w,xy+yz+zx+xw+wy+wz,xyz+yzw+wzx+yzw]/(xyzw-1)$.

 

The first proof $\Z[x,y,z,x^{-1},y^{-1},z^{-1}]\cong \Z[x,y,z,w]/(xyzw-1)$ is easy since $w=x^{-1}y^{-1}z^{-1}$ implies $x^{-1},y^{-1},z^{-1}\in \Z[x,y,z,w]/(xyzw-1)$.

 

The second proof $\Z[x+y+z+x^{-1}+y^{-1}+z^{-1},x^{\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}}, x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}}]$ is isomorphic to $\Z[x+y+z+w,xy+yz+zx+xw+wy+wz,xyz+yzw+wzx+yzw]/(xyzw-1)$ is also straightforward.