Subject: Symmetric powers (final remark) Date: Thu, 31 May 2001 10:47:21 +0200 (CEST) From: Rade Zivaljevic To: dmd1@lehigh.edu Topology discussion group (from Rade \v Zivaljevi\' c; Belgrade) Final remark related to the symmetric powers of Riemann (and other) surfaces. Thanks to all (including those who sent e-mail messages) who kindly provided relevant information about symmetric powers. Here is a final remark: Sign(SP^m_G(M)) := Sign(M^m/G) = Z(G;t,e,t,e,...) where Z(G;x_1,...,x_m) is a universal polynomial (the cycle index of G) t = Sign(M) and e = Euler(M) of an even dimensional M. This is indeed a neat formula. It is implicit in Zagier LNM 290 volume and possibly well known to anyone who applied Atiyah-Singer G-signature theorem to (g,M^m), g = cyclic permutation. It is a pleasure to see an "old friend" Z(G) from enumerative combinatorics in such a nice place and perhaps this is the "right way" to state this result. Rade Zivaljevic, Belgrade