Subject: Invariant theory question From: Amnon Neeman Date: Fri, 8 Jul 2005 12:36:20 +1000 (EST) I'm not sure whether this elaboration is of any interest to the entire topology list; I'll let Don decide. When I answered Doug's question I answered it narrowly, speaking only about the particular generators he gave. They are of interest because they give the map from the symmetric power to the Chow variety, and one way to restate the results I cited is that this map is an isomorphism in charateristic 0 but not in characteristic p. In my original paper I was not after optimal bounds. Recently Drinfeld and Gaitsgory became interested in the question and asked whether I could give sharp bounds for the integers $m$ and $n$, and with their encouragement I thought about the problem some more and the bounds I gave in the previous email (with no reference) are the answer I found for the Drinfeld/Gaitsgory question. They are obtained by essentially the methods of my old paper. The fact that the generators are algebraically independent in the inverse limit, given by Tom below, may be found in the recent paper to which I am a coauthor. What is not discussed at all in my previous email is the very extensive literature dealing with other sets of generators. You can begin with Hermann Weyl's book on the classical groups @book {Weyl39, AUTHOR = {Weyl, Hermann}, TITLE = {The {C}lassical {G}roups. {T}heir {I}nvariants and {R}epresentations}, PUBLISHER = {Princeton University Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1939}, PAGES = {xii+302}, } For example, in II.3 you will find mn generators for the quotient field. The existence means that the field is rational. Anyway, there is a brief discussion of the literature at the end of section 3 of the most recent paper I cited in the email of a few weeks ago, which I remind you is @ARTICLE{Elmore-Hall-Neeman05, AUTHOR = {Elmore, Ryan and Hall, Peter and Neeman, Amnon}, TITLE = {An application of classical invariant theory to identifiability in nonparametric mixtures}, JOURNAL = {Ann. Inst. Fourier (Grenoble)}, VOLUME = {55}, PAGES = {1--28}, YEAR = {2005}, } Yours, Amnon