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. 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.
