Opened 13 years ago
Last modified 4 years ago
#6889 new enhancement
Algebra of multivariate polynomials invariant under the action of a permutation group
Reported by: | nborie | Owned by: | nborie |
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Priority: | major | Milestone: | sage-8.4 |
Component: | combinatorics | Keywords: | invariants, permutation, group, ring, orbit, evaluation |
Cc: | sage-combinat, tscrim | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
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)
Change History (7)
comment:1 Changed 13 years ago by
- Owner changed from mhansen to nborie
- Status changed from new to assigned
comment:2 Changed 9 years ago by
- Milestone changed from sage-5.11 to sage-5.12
comment:3 Changed 8 years ago by
- Milestone changed from sage-6.1 to sage-6.2
comment:4 Changed 8 years ago by
- Milestone changed from sage-6.2 to sage-6.3
comment:5 Changed 8 years ago by
- Milestone changed from sage-6.3 to sage-6.4
comment:6 Changed 4 years ago by
- Report Upstream set to N/A
comment:7 Changed 4 years ago by
- Cc tscrim added
- Milestone changed from sage-6.4 to sage-8.4
Note that Sage (via Singular) can compute minimal generating sets for invariant rings of permutation groups. But the result is not implemented as a ring on its own (i.e., it is a method that returns a list of generators).