#6889: Algebra of multivariate polynomials invariant under the action of a
permutation group
---------------------------+------------------------------------------------
Reporter: nborie | Owner: mhansen
Type: enhancement | Status: new
Priority: major | Milestone: sage-combinat
Component: combinatorics | Keywords: invariants, permutation, group,
ring, orbit, evaluation
Reviewer: | Author:
Merged: |
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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
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6889>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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