Subject: Question about invariant theory From: "Douglas C. Ravenel" Date: Wed, 29 Jun 2005 15:45:59 -0400 (EDT) Here is a question for the listserve about invariant theory. Let $m$ and $n$ be positive integers and consider the ring \begin{displaymath} R=Z[r_{i,j}: 1\le i \le m, 1\le j \le n], \end{displaymath} \noindent the polynomial ring on $nm$ variables. I am especially interesed in the case where $n$ is prime. Let the symmetric group $\Sigma_{n}$ act by permuting the second subscript. What is the structure of the invariant subring $S$? It appears to have $\binom{n+m}{m}-1$ generators, namely the coefficients of that number of mononials of degree $\le n$ in the $m$ dummy variables $y_{i}$ in the invariant expression \begin{displaymath} \prod_{j=1}^{^{n}}\left(1+\sum_{i=1}^{m}r_{i,j}y_{i} \right) =1+\sum_{K}a_{K}y^{K}. \end{displaymath} \noindent Here $K= (k_{1},\dotsc ,k_{m})$ is a sequence of $m$ nonnegative integers (not all zero) whose sum is at most $n$, and \begin{displaymath} y^{K} = y_{1}^{k_{1}}\dotsb y_{n}^{k_{n}}. \end{displaymath} \noindent On the other hand, its Krull dimension is only $nm$ (since it is contained in $R$), so it must be some relations. What are they? A related question: Is the spectrum (in the sense of algebraic geometry) of the invariant ring $S$ smooth? Doug Douglas C. Ravenel, Chair Department of Mathematics |918 Hylan Building University of Rochester |(585) 275-4413 Rochester, New York 14627 |FAX (585) 273-4655 Email: doug@math.removethis.rochester.&this.edu Personal home page: http://www.math.rochester.edu/people/faculty/doug/