Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I need an example involving the old invariant theory that pre-dated abstract algebra.

share|cite|improve this question

Here is the example of Kleinian singularities. Let $G$ be a finite subgroup of $SU(2,{\bf C})$.

$G$ acts on ${\bf C}^2$ and for each $G$ corresponds a Klein singularity $$ S=\frac {{\bf C}^2 } {G} $$

$$ S=\{ (x,y,z) \in {\bf C}^3 \ | \ R(x,y,z)=0 \} $$

One example is the group ${\mathcal C}_{n}$, the cyclic group.

You can realize it as matrices of the form $$ \begin{pmatrix} \omega^j & 0 \cr 0 & \omega^{-j} \end{pmatrix} $$ where $\omega $ is a primitive root of unity. The invariant functions, are functions which satisfy $f(u,v)=f(au+bv, cu+dv)$

The following three functions are invariant $u^n$, $v^n$, $uv$. Set $x=u^n$, $y=v^n$ and $z=uv$. Then they satisfy the follwoing syzygy $$z^n=xy$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.