Nicolas Borie Maître de conférences en Informatique Équipe Combinatoire algébrique et calcul formel Bureau 4B03R Université Paris-Est Marne-la-Vallée Cité Descartes, Bât Copernic 77454 Marne-la-Vallée Cedex 2 France E-mail:

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# 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: 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: 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())

## 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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