InvariantRingPermutationGroup

Fiche dév Ens Sup - Recherche
  • Création ou MAJ importante : 10/09/13
  • Correction mineure : 10/09/13

InvariantRingPermutationGroup : calculs des invariants algébriques d'un groupe de permutations

Ce logiciel a été développé (ou est en cours de développement) dans la communauté de l'Enseignement Supérieur et de la Recherche. Son état peut être variable (cf champs ci-dessous) donc sans garantie de bon fonctionnement.
  • Site web
  • Système :
  • Licence(s) : GPL
  • Etat : diffusé, stable
  • Support : maintenu, développement en cours
  • Concepteur(s) : Nicolas Borie
  • Contact concepteur(s) : nicolas.borie@univ-mlv.fr
  • Laboratoire(s), service(s)... :

 

Fonctionnalités générales du logiciel

  • Ce module Sage permet de calculer les invariants secondaires de l'anneau des invariants d'un groupe de permutations. Ces invariants secondaires sont associés aux invariants primaires formés par les polynômes symétriques.

  • Ces deux familles de polynômes invariants engendrent l'anneau des invariants en tant qu'algèbre et donc caractérisent complètement la structure algébrique.

  • Ce module est disponible sous la forme d'un patch téléchargeable dans la suite de patch Sage-Combinat.

  • Exemple avec un grand groupe :

    sage: G = TransitiveGroup(14,61)
    sage: G.cardinality()
    50803200
    sage: factorial(14)/G.cardinality()
    1716
    sage: I = InvariantRingPermutationGroup(G, QQ)
    sage: I.secondary_invariants_series()
    z^48 + z^47 + 2z^46 + 2z^45 + 4z^44 + 5z^43 + 8z^42 + 9z^41 + 14z^40 + 16z^39 + 22z^38 + 25z^37 + 33z^36 + 36z^35 + 45z^34 + 48z^33 + 58z^32 + 61z^31 + 70z^30 + 71z^29 + 80z^28 + 79z^27 + 85z^26 + 82z^25 + 87z^24 + 81z^23 + 83z^22 + 75z^21 + 75z^20 + 66z^19 + 64z^18 + 54z^17 + 52z^16 + 42z^15 + 39z^14 + 30z^13 + 28z^12 + 20z^11 + 18z^10 + 12z^9 + 11z^8 + 7z^7 + 6z^6 + 3z^5 + 3*z^4 + z^3 + z^2 + 1

    sage: I.secondary_invariants(verbose=True)
    Initialiation of secondary of degree 0
      ------ 
    Secondary of degree 1 :
      We must search 0 secondaries invariants
      ------ 
    Secondary of degree 2 :
      We must search 1 secondaries invariants
          Research of product of secondaries of degree smaller
          Research now to complete with new irreducible secondaries
            New irreducible [2]
      ------ 
    Secondary of degree 3 :
      We must search 1 secondaries invariants
          Research of product of secondaries of degree smaller
          Research now to complete with new irreducible secondaries
            New irreducible [3]
      ------ 
    Secondary of degree 4 :
      We must search 3 secondaries invariants
          Research of product of secondaries of degree smaller
            Add product [2, 2]
          Research now to complete with new irreducible secondaries
            New irreducible [4]
            (3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is not a good secondary invariants
            New irreducible [5]
      ------ 
    Secondary of degree 5 :
      We must search 3 secondaries invariants
          Research of product of secondaries of degree smaller
            Add product [2, 3]
          Research now to complete with new irreducible secondaries
            New irreducible [6]
            (4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is not a good secondary invariants
            New irreducible [7]
    ------ 
    Secondary of degree 6 :
      We must search 6 secondaries invariants
          Research of product of secondaries of degree smaller
            Add product [2, 4]
            Add product [2, 5]
            Add product [3, 3]
            Register new relation : [2, 2, 2]
          Research now to complete with new irreducible secondaries
            New irreducible [8]
            (5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is not a good secondary invariants
            New irreducible [9]
            (4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is not a good secondary invariants
            (4, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is not a good secondary invariants
            (4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is not a good secondary invariants
            (4, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) is not a good secondary invariants
            New irreducible [10]
    ------ 
    ( ... FICHIER DE LOG COUPÉ )
    ------ 
    Secondary of degree 48 :
      We must search 1 secondaries invariants
        Correction by adding the space spanned by secondaries of degree 34
        Correction by adding the space spanned by secondaries of degree 20
        Correction by adding the space spanned by secondaries of degree 6
          Research of product of secondaries of degree smaller
            Add product [3, 28, 28, 28]
    ...
    sage: for i in range(49): print i," : ", I.irreducible_secondary_invariants_of_degree(i)
    ....:
    0  :  [[(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    1  :  []
    2  :  [[(1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    3  :  [[(2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    4  :  [[(3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    5  :  [[(4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    6  :  [[(5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)],
           [(3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    7  :  [[(6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)],
           [(4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    8  :  [[(7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(6, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)],
           [(5, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    9  :  [[(7, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(6, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    10  :  [[(7, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    11  :  [[(7, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    12  :  [[(7, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    13  :  [[(12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(11, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    14  :  [[(13, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], [(12, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)],
            [(11, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]
    15  :  [[(13, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]]

Contexte d’utilisation du logiciel

Le module a été développé durant la thèse de son auteur, Nicolas Borie.

Publications liées au logiciel

Nicolas Borie, Calcul des invariants de groupes de permutations par transformee de fourier. Thèse Université Paris Sud - Paris XI, 2011.