home *** CD-ROM | disk | FTP | other *** search
- Newsgroups: sci.math
- Path: sparky!uunet!destroyer!sol.ctr.columbia.edu!sol.ctr.columbia.edu!sjp
- From: sjp@occs.cs.oberlin.edu (Sue Patterson)
- Subject: reference in invariant theory wanted
- Sender: nobody@ctr.columbia.edu
- Organization: Oberlin College Computer Science
- Date: Wed, 11 Nov 1992 20:09:51 GMT
- Message-ID: <SJP.92Nov11150951@occs.cs.oberlin.edu>
- X-Posted-From: occs.cs.oberlin.edu
- NNTP-Posting-Host: sol.ctr.columbia.edu
- Lines: 39
-
- I'm looking at a problem in invariant theory of finite groups. I know that
- _somebody_ must have thought about it before! Can anybody give me a reference?
-
- Sorry if my exposition is too elementary, I've only looked at one undergrad
- text on the subject, and I don't know how much of their notation is standard.
-
- The situation is as follows: Given a finite matrix group G, we find the
- subring of C[x_1, ... x_n] consisting of polynomials invariant under G.
- Call this subring R. We can find polynomials in n variables f_1, ... f_m
- so that R is the set of polynomials over C in the f_i, so R = C[f_1, .. f_m].
- This can always be done.
-
- In some cases, any invariant polynomial can be written uniquely as a polynomial
- in these generating invariants. However, this isn't always the case, and
- so we get interested in nontrivial expressions of 0 as a polynomial in the
- generating invariants. In C[y_1, ... y_m] we look at the syzygy ideal
- consisting of polynomials g so that g(f_1, ... f_m) = 0. If this ideal
- isn't 0, then we have no unique way of writing our invariants in terms of the
- generators.
-
- My question is, when can we go backwards? In other words, if we have a
- prime ideal, can we find a group such that this ideal is a syzygy ideal
- of its invariants?
-
- If anybody has any references on this subject, I'd appreciate it.
-
- Please reply via e-mail, and I'll post responses to the group if there's
- interest.
-
- --Sue
- sjp@occs.cs.oberlin.edu
- --
- ---------------------------------------------------------------------------
- | "To have beauty is to have only that, | Sue Patterson |
- | but to have goodness | patterso@occs.cs.oberlin.edu |
- | is to be beautiful, too |
- |
- | -- Sappho | #include <std.disclaimer> |
- ---------------------------------------------------------------------------
-