Timeline for Infinite-dimensional Chevalley–Shephard–Todd theorem

5 events

when toggle format	what		by	license	comment
Jun 3, 2017 at 1:35	comment	added	jdc		True, and averaging is why we make the order restriction.
Jun 2, 2017 at 23:12	comment	added	Qiaochu Yuan		Consider the infinite symmetric group $S_{\infty}$ acting on $k[x_1, x_2, \dots ]$. The ring of invariants is just $k$, which is certainly a polynomial ring. But $S_{\infty}$ doesn't seem to me to be generated by pseudoreflections in any sense. One problem here is that away from the finite case we no longer have the ability to average over the group to produce invariants, so there's no reason there should be any.
Jun 2, 2017 at 22:06	comment	added	Noam D. Elkies		The motivating example for C-S-T is the theory of symmetric polynomials. An infinite-dimensional analogue would have to involve infinite sums, so it's no longer "just" a question of polynomial algebra. [Presumably a "pseudoreflection" is a linear transformation that acts as $1$ on some hyperplane (codimension-$1$ subspace) and a nontrivial root of unity on some $1$-dimensional complementary subspace.]
Jun 2, 2017 at 22:03	comment	added	Jim Humphreys		I'm not sure how "pseudo-reflection" would be defined in case $V$ is infinite dimensional. On the other hand, there may be some possibilities when the action of $G$ is assumed to be locally finite dimensional. I'm not sure what the constraints should be on $G$ and $V$ to get a meaningful generalization.
Jun 2, 2017 at 21:31	history	asked	jdc	CC BY-SA 3.0