Opened 13 years ago

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

Reported by: Owned by: nborie nborie major sage-8.4 combinatorics invariants, permutation, group, ring, orbit, evaluation sage-combinat, tscrim N/A

### 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: 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

### comment:1 Changed 13 years ago by nborie

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

### comment:2 Changed 9 years ago by jdemeyer

• Milestone changed from sage-5.11 to sage-5.12

### comment:3 Changed 8 years ago by vbraun_spam

• Milestone changed from sage-6.1 to sage-6.2

### comment:4 Changed 8 years ago by vbraun_spam

• Milestone changed from sage-6.2 to sage-6.3

### comment:5 Changed 8 years ago by vbraun_spam

• Milestone changed from sage-6.3 to sage-6.4

### comment:6 Changed 4 years ago by SimonKing

• Report Upstream set to N/A

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).