Opened 8 years ago

Last modified 3 years ago

#6889 new enhancement

Algebra of multivariate polynomials invariant under the action of a permutation group

Reported by: nborie Owned by: nborie
Priority: major Milestone: sage-6.4
Component: combinatorics Keywords: invariants, permutation, group, ring, orbit, evaluation
Cc: sage-combinat Merged in:
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Description

First implementation of the Algebra of multivariate polynomials invariant under the action of a permutation group.

From a permutation group and a ring, the goal is to implement an algebra on which one can ask the primary invariants, a minimal generating set and (irreducible)secondary invariants...

Using the category framework, we construct the abstract algebra of PermutationGroupInvariantRing? and two representations of it : the graded algebra of multivariate polynomials view as combination of orbit sum of monomials (here #6812 is needed) and the polynomials view as vector evaluated in a collection of points.

This is a long run work but first implementation is comming in one or two months.

sage: mupad('package("Combinat")')                                    
sage: G = mupad.Dom.PermutationGroup(3, [[[1,2,3]]])
sage: I = mupad.Dom.PermutationGroupInvariantRing(mupad.Dom.Rational, G)
sage: I

Dom::PermutationGroupInvariantRing(Dom::Rational,Dom::PermutationGroup(3, [[[1, 2, 3]]]))

sage: I.minimalGeneratingSet()
         3 = [o([1, 1, 1]), o([2, 0, 1])],
         2 = [o([1, 1, 0])],
         1 = [o([1, 0, 0])]

sage: I.basisIndices.list(3)
         [[1, 1, 1], [2, 0, 1], [2, 1, 0], [3, 0, 0]]

sage: I.HilbertSeries()

                                  2            1
                           - ---------- - ----------
                                 3                 3
                             3 (z  - 1)   3 (z - 1)

depends on #6812 and #5891

Change History (5)

comment:1 Changed 8 years ago by nborie

  • Owner changed from mhansen to nborie
  • Status changed from new to assigned

comment:2 Changed 4 years ago by jdemeyer

  • Milestone changed from sage-5.11 to sage-5.12

comment:3 Changed 4 years ago by vbraun_spam

  • Milestone changed from sage-6.1 to sage-6.2

comment:4 Changed 3 years ago by vbraun_spam

  • Milestone changed from sage-6.2 to sage-6.3

comment:5 Changed 3 years ago by vbraun_spam

  • Milestone changed from sage-6.3 to sage-6.4
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