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Nicolas Borie
Maître de conférences en Informatique Équipe Combinatoire algébrique et calcul formel Responsable de la filière informatique Design, Architecture et Développement de l'ESIEE Paris Bureau 1B190 Université Gustave Eiffel Bâtiment Copernic, 5 Bd Descartes, 77420 Champs-sur-Marne France E-mail: or |
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Index | Recherche | Enseignement | Software | Borie Awards | PLaTon
Pictures of root system
Pictures of Type A:
Classical cases:
Appliying the patch from the ticket #4327 of the Sage trac server on your sage system, you could also produce the same pictures.
sage:
RootSystem(["A",2]).weight_space().plot(simple_reflection_hyperplanes=True,
bounding_box=[[-2,2],[-2,2]], simple_coroots=True)
sage:
RootSystem(["A",2]).ambient_space().plot()
Affine cases:
sage: RootSystem(["A", 2,
1]).weight_space().plot(simple_roots=True, fundamental_weights=True,
alcoves=[[-1..2], [-1..3]],
alcove_walks=[[0,2,1,2,0,2,1,0,2,1,2,1,2,0,2,0,1,2,0],
[2,1,2,0,2,0,2,1,2,0,1,2,1,2,1,0,1,2,0,2,0,1,2,0,2]])
sage: M = matrix(RR,
[[0,1/2,-1/2],[0,sqrt(3)/2,sqrt(3)/2]]); P = RootSystem(["A", 2,
1]).weight_space().plot(alcoves=[[-1..2], [-1..3]], projection =
lambda x : M*vector(RR, x)); P
sage: M = matrix(RR,
[[0,sqrt(3)/2,-sqrt(3)/2,0],[0,1/2,1/2,-1],[0,sqrt(2),sqrt(2),sqrt(2)]]);
sage: RootSystem(["A", 3, 1]).weight_space().plot(simple_roots=True, fundamental_weights=True, alcoves=[[0], [0], [0]], projection = lambda x: M*(vector(x)) )
sage: RootSystem(["A", 3, 1]).weight_space().plot(simple_roots=True, fundamental_weights=True, alcoves=[[0], [0], [0]], projection = lambda x: M*(vector(x)) )
sage: M = matrix(RR,
[[0,1/2,-1/2],[0,sqrt(3)/2,sqrt(3)/2]]); sage: RootSystem(["A", 2,
1]).weight_space().plot(bounding_box=[[-5,5],[-5,5]], projection =
lambda x: M*(vector(x)))
sage: M = matrix(RR, [[0,1/2,-1/2],[0,sqrt(3)/2,sqrt(3)/2]]);
sage: def projection_on_disque(x):
... V = vector(RR, x)
... C = M*V
... d = C.norm()
... return (1-((0.3*d)/(0.3*d+1)))*C
sage: RootSystem(["A", 2, 1]).weight_space().plot(alcoves=[[-4..4],[-4..4]], projection = projection_on_disque)
sage: def projection_on_disque(x):
... V = vector(RR, x)
... C = M*V
... d = C.norm()
... return (1-((0.3*d)/(0.3*d+1)))*C
sage: RootSystem(["A", 2, 1]).weight_space().plot(alcoves=[[-4..4],[-4..4]], projection = projection_on_disque)
sage: R =
RootSystem(["A",2,1]).weight_space();
sage: W = R.weyl_group();
sage: w0,w1,w2 = W.gens();
sage: w = w0*w2*w1*w0*w1; w.reduced_words()
sage: M = matrix(RR, [[0,1/2,-1/2],[0,sqrt(3)/2,sqrt(3)/2]]);
sage: R.plot(bounding_box=[[-1,2],[0,3]], projection = lambda x: M*(vector(x)), alcove_walks = w.reduced_words())
sage: W = R.weyl_group();
sage: w0,w1,w2 = W.gens();
sage: w = w0*w2*w1*w0*w1; w.reduced_words()
sage: M = matrix(RR, [[0,1/2,-1/2],[0,sqrt(3)/2,sqrt(3)/2]]);
sage: R.plot(bounding_box=[[-1,2],[0,3]], projection = lambda x: M*(vector(x)), alcove_walks = w.reduced_words())
Pictures of Type B or C:
Classical cases:
sage: RootSystem(["B",
2]).ambient_space().plot(simple_coroots=True,
simple_reflection_hyperplanes=True, bounding_box=[[-3,3],
[-3,3]])
sage: RootSystem(["C",
2]).ambient_space().plot(simple_coroots=True,
simple_reflection_hyperplanes=True, bounding_box=[[-3,3],
[-3,3]])
Affine cases:
sage:
RootSystem(["B",2,1]).coweight_space().plot(bounding_box=[[-3,4],[-3,4]],
barycentric=True)
sage: M = Matrix(RR, [[0,1,1],[0,0,1]]); sage:
RootSystem(["C",2,1]).weight_space().plot(bounding_box=[[-3,3],[-3,3]],
projection=lambda x : M*vector(x))
Pictures of Type G:
Classical cases:
sage: RootSystem(["G",
2]).weight_space().plot(simple_coroots=True,
simple_reflection_hyperplanes=True, bounding_box=[[-2,2],
[-2,2]])
Affine cases:
sage: M = matrix(RR,
[[0,1/2,0],[0,sqrt(3)/2,sqrt(3)]]); sage:
RootSystem(["G",2,1]).weight_space().plot(bounding_box=[[-1,2],[0,4]],
projection = lambda x: M*(vector(x)), alcove_walks =
[[0,2,1,2,0,1]])
sage:
RootSystem(["G",2,1]).weight_space().plot(bounding_box=[[-3,3],[-3,3],[-3,3]])
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