On Thu, Dec 8, 2011 at 10:56 AM, Jeroen Demeyer <jdeme...@cage.ugent.be> wrote: > 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 You can get the symmetric polynomials on that page by expanding Sage's elementary symmetric functions out in a finite number of variables. sage: e = SymmetricFunctions(QQ).e() sage: e[1].expand(4) x0 + x1 + x2 + x3 sage: e[2].expand(4) x0*x1 + x0*x2 + x1*x2 + x0*x3 + x1*x3 + x2*x3 sage: e[3].expand(4) x0*x1*x2 + x0*x1*x3 + x0*x2*x3 + x1*x2*x3 sage: e[4].expand(4) x0*x1*x2*x3 e[3,2,1] represents the product e[3]*e[2]*[1]: sage: e[3]*e[2]*e[1] e[3, 2, 1] To get a symmetric function from a polynomial, you use the "from_polynomial" method as Nicolas mentioned. The only awkward part is that this method only exists for the the monomial basis. (It would be easy to add a method to the other bases which converted the polynomial to the monomial basis and then to itself, but that doesn't exist yet.) sage: m = SymmetricFunctions(QQ).m() sage: f = e[2].expand(4); f x0*x1 + x0*x2 + x1*x2 + x0*x3 + x1*x3 + x2*x3 sage: f_in_e = e(m.from_polynomial(f)); f_in_e e[2] sage: f_in_e.expand(4) x0*x1 + x0*x2 + x1*x2 + x0*x3 + x1*x3 + x2*x3 --Mike -- 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
