On 2013-04-19, Simon King <simon.k...@uni-jena.de> wrote: > Hi Johannes, > > On 2013-04-18, Johannes <dajo.m...@web.de> wrote: >> Hi guys, >> >> I have the following setting: Given a finite subgroup G of GL_\C(n) of >> order k, acting on C[x_1,...,x_n] by multiplication with (potenz of a ) >> k-th root of unity. What is the best way, to translate this setting to sage? >> In the end I'm interested into the ring of invariants under G and it's >> representation as quotient. > > Towards an answer: > > Since you want to compute an invariant ring, and since CC is not exactly > a field (rounding errors), it might make sense to work over a number > field that contains a k-th root of unity. For example: > > # Create the number field > sage: F.<zeta> = NumberField(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) > sage: zeta^7 == 1 > True > # Create a 3x3 matrix that acts by multiplication with zeta > sage: MS = MatrixSpace(F, 3) > sage: g = MS(zeta) > # Create the corresponding matrix group. It has the correct order > # Note that the method "multiplicative_order" or the matrix fails! > sage: G = MatrixGroup([g]) > sage: G.order() > 7 > # Compute a minimal generating set of the invariant ring, as a sub-ring > sage: G.invariant_generators() > [x3^7, > x2*x3^6, > x1*x3^6, > x2^2*x3^5, > x1*x2*x3^5, > x1^2*x3^5, > x2^3*x3^4, > x1*x2^2*x3^4, > x1^2*x2*x3^4, > x1^3*x3^4, > x2^4*x3^3, > x1*x2^3*x3^3, > x1^2*x2^2*x3^3, > x1^3*x2*x3^3, > x1^4*x3^3, > x2^5*x3^2, > x1*x2^4*x3^2, > x1^2*x2^3*x3^2, > x1^3*x2^2*x3^2, > x1^4*x2*x3^2, > x1^5*x3^2, > x2^6*x3, > x1*x2^5*x3, > x1^2*x2^4*x3, > x1^3*x2^3*x3, > x1^4*x2^2*x3, > x1^5*x2*x3, > x1^6*x3, > x2^7, > x1*x2^6, > x1^2*x2^5, > x1^3*x2^4, > x1^4*x2^3, > x1^5*x2^2, > x1^6*x2, > x1^7] > > So, the invariant ring could be represented as a ring with not less than > 36 generators, modulo algebraic relations. Now, I am afraid I don't know > an easy way to find algebraic relations of the above sub-algebra > generators---I am afraid I can't answer how to represent it as a > quotient ring. well, x1^7, x2^7, x3^7 form a system of primary invariants, and the ring is the direct sum of y*F[x1^7, x2^7, x3^7], where y is either 1 or one of the remaining 33 generators (or perhaps some of the latter can be excluded, I didn't check). Cf. Corollary 2.7.4 of B.Sturmfels' "Algorithms in invariant theory".
Dima > > Best regards, > Simon > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
