On 2011-12-04 14:34, Nicolas M. Thiery wrote: > On Sun, Dec 04, 2011 at 03:56:39AM -0800, Dima Pasechnik wrote: >> unless I missed something, >> your >> http://combinat.sagemath.org/doc/thematic_tutorials/demo-symmetric-functions.html >> lacks any information as to how to take any symmetric polynomial, and >> expand it in some basis of symmetric functions. > > Indeed: this tutorial is very incomplete. Someone should stand up and > take the time to gather all the bits of docs from here and there, and > merge a nice and clean tutorial on Symmetric functions into Sage. > >> Is it even possible with the combinat functionality? > > It definitely is. Here is a story without subtitles (please ask for > those if needed: > > sage: S = SymmetricFunctions(QQ) > sage: m = S.m() > sage: e = S.e() > sage: f = e[3,2,1] > sage: f_as_poly = f.expand(3); f_as_poly > x0^3*x1^2*x2 + x0^2*x1^3*x2 + x0^3*x1*x2^2 + 3*x0^2*x1^2*x2^2 + > x0*x1^3*x2^2 + x0^2*x1*x2^3 + x0*x1^2*x2^3 > sage: f_as_poly.parent() > Multivariate Polynomial Ring in x0, x1, x2 over Rational Field > > sage: f_in_m = m.from_polynomial(f_as_poly); f_in_m > 3*m[2, 2, 2] + m[3, 2, 1] > > sage: f_in_e = e(f_in_m) > sage: f_in_e > e[3, 2, 1] - 3*e[4, 1, 1] - 2*e[4, 2] + 13*e[5, 1] - 18*e[6] > > Oops, but we don't get the original symmetric function f! Actually > this is perfectly correct since we lost information by only expanding > in 3 variables: > > sage: f_in_e.expand(3) == f.expand(3) > > If one kills all terms with parts larger than 3 (since > e_4=e_5=e_6=...=0 in three variables). So I think this illustrates exactly Dima's sentiment that Sage doesn't have functionality to write a given symmetric polynomial in terms of elementary symmetric polynomials. Maybe it's technically possible, but surely not easy!
Also, it's not clear to me even what e[3,2,1] means. How do Sage's elementary symmetric functions relate to http://en.wikipedia.org/wiki/Elementary_symmetric_polynomial#Examples -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
