Weight lattice realizations¶

class sage.combinat.root_system.weight_lattice_realizations.WeightLatticeRealizations(base, name=None)[source]¶

Bases: Category_over_base_ring

The category of weight lattice realizations over a given base ring.

A weight lattice realization \(L\) over a base ring \(R\) is a free module (or vector space if \(R\) is a field) endowed with an embedding of the root lattice of some root system. By restriction, this embedding defines an embedding of the root lattice of this root system, which makes \(L\) a root lattice realization.

Typical weight lattice realizations over \(\ZZ\) include the weight lattice, and ambient lattice. Typical weight lattice realizations over \(\QQ\) include the weight space, and ambient space.

To describe the embedding, a weight lattice realization must implement a method fundamental_weight`(i) returning for each `i() in the index set the image of the fundamental weight \(\Lambda_i\) under the embedding.

In order to be a proper root lattice realization, a weight lattice realization should also implement the scalar product with the coroot lattice; on the other hand, the embedding of the simple roots is given for free.

See also

EXAMPLES:

Here, we consider the root system of type \(A_7\), and embed the weight lattice element \(x = \Lambda_1 + 2 \Lambda_3\) in several root lattice realizations:

Sage

sage: R = RootSystem(["A",7])
sage: Lambda = R.weight_lattice().fundamental_weights()
sage: x = Lambda[2] + 2 * Lambda[5]

sage: L = R.weight_space()
sage: L(x)
Lambda[2] + 2*Lambda[5]

sage: L = R.ambient_lattice()
sage: L(x)
(3, 3, 2, 2, 2, 0, 0, 0)

Python

>>> from sage.all import *
>>> R = RootSystem(["A",Integer(7)])
>>> Lambda = R.weight_lattice().fundamental_weights()
>>> x = Lambda[Integer(2)] + Integer(2) * Lambda[Integer(5)]

>>> L = R.weight_space()
>>> L(x)
Lambda[2] + 2*Lambda[5]

>>> L = R.ambient_lattice()
>>> L(x)
(3, 3, 2, 2, 2, 0, 0, 0)

Sage Live

R = RootSystem(["A",7])
Lambda = R.weight_lattice().fundamental_weights()
x = Lambda[2] + 2 * Lambda[5]
L = R.weight_space()
L(x)
L = R.ambient_lattice()
L(x)

We embed the weight space element \(x = \Lambda_1 + 1/2 \Lambda_3\) in the ambient space:

Sage

sage: Lambda = R.weight_space().fundamental_weights()
sage: x = Lambda[2] + 1/2 * Lambda[5]

sage: L = R.ambient_space()
sage: L(x)
(3/2, 3/2, 1/2, 1/2, 1/2, 0, 0, 0)

Python

>>> from sage.all import *
>>> Lambda = R.weight_space().fundamental_weights()
>>> x = Lambda[Integer(2)] + Integer(1)/Integer(2) * Lambda[Integer(5)]

>>> L = R.ambient_space()
>>> L(x)
(3/2, 3/2, 1/2, 1/2, 1/2, 0, 0, 0)

Sage Live

Lambda = R.weight_space().fundamental_weights()
x = Lambda[2] + 1/2 * Lambda[5]
L = R.ambient_space()
L(x)

Of course, one can’t embed the weight space in the ambient lattice:

Sage

sage: L = R.ambient_lattice()
sage: L(x)
Traceback (most recent call last):
...
TypeError: do not know how to make x (= Lambda[2] + 1/2*Lambda[5])
an element of self (=Ambient lattice of the Root system of type ['A', 7])

Python

>>> from sage.all import *
>>> L = R.ambient_lattice()
>>> L(x)
Traceback (most recent call last):
...
TypeError: do not know how to make x (= Lambda[2] + 1/2*Lambda[5])
an element of self (=Ambient lattice of the Root system of type ['A', 7])

Sage Live

L = R.ambient_lattice()
L(x)

If \(K_1\) is a subring of \(K_2\), then one could in theory have an embedding from the weight space over \(K_1\) to any weight lattice realization over \(K_2\); this is not implemented:

Sage

sage: K1 = QQ
sage: K2 = QQ['q']
sage: L = R.ambient_space(K2)

sage: Lambda = R.weight_space(K2).fundamental_weights()
sage: L(Lambda[1])
(1, 0, 0, 0, 0, 0, 0, 0)

sage: Lambda = R.weight_space(K1).fundamental_weights()
sage: L(Lambda[1])
Traceback (most recent call last):
...
TypeError: do not know how to make x (= Lambda[1]) an element
of self (=Ambient space of the Root system of type ['A', 7])

Python

>>> from sage.all import *
>>> K1 = QQ
>>> K2 = QQ['q']
>>> L = R.ambient_space(K2)

>>> Lambda = R.weight_space(K2).fundamental_weights()
>>> L(Lambda[Integer(1)])
(1, 0, 0, 0, 0, 0, 0, 0)

>>> Lambda = R.weight_space(K1).fundamental_weights()
>>> L(Lambda[Integer(1)])
Traceback (most recent call last):
...
TypeError: do not know how to make x (= Lambda[1]) an element
of self (=Ambient space of the Root system of type ['A', 7])

Sage Live

K1 = QQ
K2 = QQ['q']
L = R.ambient_space(K2)
Lambda = R.weight_space(K2).fundamental_weights()
L(Lambda[1])
Lambda = R.weight_space(K1).fundamental_weights()
L(Lambda[1])

class ElementMethods[source]¶

Bases: object

symmetric_form(la)[source]¶

Return the symmetric form of self with la.

Return the pairing \(( | )\) on the weight lattice. See Chapter 6 in Kac, Infinite Dimensional Lie Algebras for more details.

Warning

For affine root systems, if you are not working in the extended weight lattice/space, this may return incorrect results.

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: P = RootSystem(['C',2]).weight_lattice()
sage: al = P.simple_roots()
sage: al[1].symmetric_form(al[1])
2
sage: al[1].symmetric_form(al[2])
-2
sage: al[2].symmetric_form(al[1])
-2
sage: Q = RootSystem(['C',2]).root_lattice()
sage: alQ = Q.simple_roots()
sage: all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
....:     for i in P.index_set() for j in P.index_set())
True

sage: # needs sage.graphs
sage: P = RootSystem(['C',2,1]).weight_lattice(extended=True)
sage: al = P.simple_roots()
sage: al[1].symmetric_form(al[1])
2
sage: al[1].symmetric_form(al[2])
-2
sage: al[1].symmetric_form(al[0])
-2
sage: al[0].symmetric_form(al[1])
-2
sage: Q = RootSystem(['C',2,1]).root_lattice()
sage: alQ = Q.simple_roots()
sage: all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
....:     for i in P.index_set() for j in P.index_set())
True
sage: La = P.basis()
sage: [La['delta'].symmetric_form(al) for al in P.simple_roots()]
[0, 0, 0]
sage: [La[0].symmetric_form(al) for al in P.simple_roots()]
[1, 0, 0]

sage: P = RootSystem(['C',2,1]).weight_lattice()
sage: Q = RootSystem(['C',2,1]).root_lattice()
sage: al = P.simple_roots()                                             # needs sage.graphs
sage: alQ = Q.simple_roots()                                            # needs sage.graphs
sage: all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])  # needs sage.graphs
....:     for i in P.index_set() for j in P.index_set())
True

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> P = RootSystem(['C',Integer(2)]).weight_lattice()
>>> al = P.simple_roots()
>>> al[Integer(1)].symmetric_form(al[Integer(1)])
2
>>> al[Integer(1)].symmetric_form(al[Integer(2)])
-2
>>> al[Integer(2)].symmetric_form(al[Integer(1)])
-2
>>> Q = RootSystem(['C',Integer(2)]).root_lattice()
>>> alQ = Q.simple_roots()
>>> all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
...     for i in P.index_set() for j in P.index_set())
True

>>> # needs sage.graphs
>>> P = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice(extended=True)
>>> al = P.simple_roots()
>>> al[Integer(1)].symmetric_form(al[Integer(1)])
2
>>> al[Integer(1)].symmetric_form(al[Integer(2)])
-2
>>> al[Integer(1)].symmetric_form(al[Integer(0)])
-2
>>> al[Integer(0)].symmetric_form(al[Integer(1)])
-2
>>> Q = RootSystem(['C',Integer(2),Integer(1)]).root_lattice()
>>> alQ = Q.simple_roots()
>>> all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
...     for i in P.index_set() for j in P.index_set())
True
>>> La = P.basis()
>>> [La['delta'].symmetric_form(al) for al in P.simple_roots()]
[0, 0, 0]
>>> [La[Integer(0)].symmetric_form(al) for al in P.simple_roots()]
[1, 0, 0]

>>> P = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice()
>>> Q = RootSystem(['C',Integer(2),Integer(1)]).root_lattice()
>>> al = P.simple_roots()                                             # needs sage.graphs
>>> alQ = Q.simple_roots()                                            # needs sage.graphs
>>> all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])  # needs sage.graphs
...     for i in P.index_set() for j in P.index_set())
True

Sage Live

# needs sage.graphs
P = RootSystem(['C',2]).weight_lattice()
al = P.simple_roots()
al[1].symmetric_form(al[1])
al[1].symmetric_form(al[2])
al[2].symmetric_form(al[1])
Q = RootSystem(['C',2]).root_lattice()
alQ = Q.simple_roots()
all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
    for i in P.index_set() for j in P.index_set())
# needs sage.graphs
P = RootSystem(['C',2,1]).weight_lattice(extended=True)
al = P.simple_roots()
al[1].symmetric_form(al[1])
al[1].symmetric_form(al[2])
al[1].symmetric_form(al[0])
al[0].symmetric_form(al[1])
Q = RootSystem(['C',2,1]).root_lattice()
alQ = Q.simple_roots()
all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
    for i in P.index_set() for j in P.index_set())
La = P.basis()
[La['delta'].symmetric_form(al) for al in P.simple_roots()]
[La[0].symmetric_form(al) for al in P.simple_roots()]
P = RootSystem(['C',2,1]).weight_lattice()
Q = RootSystem(['C',2,1]).root_lattice()
al = P.simple_roots()                                             # needs sage.graphs
alQ = Q.simple_roots()                                            # needs sage.graphs
all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])  # needs sage.graphs
    for i in P.index_set() for j in P.index_set())

The result of \((\Lambda_0 | \alpha_0)\) should be \(1\), however we get \(0\) because we are not working in the extended weight lattice:

Sage

sage: La = P.basis()
sage: [La[0].symmetric_form(al) for al in P.simple_roots()]             # needs sage.graphs
[0, 0, 0]

Python

>>> from sage.all import *
>>> La = P.basis()
>>> [La[Integer(0)].symmetric_form(al) for al in P.simple_roots()]             # needs sage.graphs
[0, 0, 0]

Sage Live

La = P.basis()
[La[0].symmetric_form(al) for al in P.simple_roots()]             # needs sage.graphs

to_weight_space(base_ring=None)[source]¶

Map self to the weight space.

Warning

Implemented for finite Cartan type.

EXAMPLES:

Sage

sage: b = CartanType(['B',2]).root_system().ambient_space().from_vector(vector([1,-2])); b
(1, -2)
sage: b.to_weight_space()
3*Lambda[1] - 4*Lambda[2]
sage: b = CartanType(['B',2]).root_system().ambient_space().from_vector(vector([1/2,0])); b
(1/2, 0)
sage: b.to_weight_space()
1/2*Lambda[1]
sage: b.to_weight_space(ZZ)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
sage: b = CartanType(['G',2]).root_system().ambient_space().from_vector(vector([4,-5,1])); b
(4, -5, 1)
sage: b.to_weight_space()
-6*Lambda[1] + 5*Lambda[2]

Python

>>> from sage.all import *
>>> b = CartanType(['B',Integer(2)]).root_system().ambient_space().from_vector(vector([Integer(1),-Integer(2)])); b
(1, -2)
>>> b.to_weight_space()
3*Lambda[1] - 4*Lambda[2]
>>> b = CartanType(['B',Integer(2)]).root_system().ambient_space().from_vector(vector([Integer(1)/Integer(2),Integer(0)])); b
(1/2, 0)
>>> b.to_weight_space()
1/2*Lambda[1]
>>> b.to_weight_space(ZZ)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
>>> b = CartanType(['G',Integer(2)]).root_system().ambient_space().from_vector(vector([Integer(4),-Integer(5),Integer(1)])); b
(4, -5, 1)
>>> b.to_weight_space()
-6*Lambda[1] + 5*Lambda[2]

Sage Live

b = CartanType(['B',2]).root_system().ambient_space().from_vector(vector([1,-2])); b
b.to_weight_space()
b = CartanType(['B',2]).root_system().ambient_space().from_vector(vector([1/2,0])); b
b.to_weight_space()
b.to_weight_space(ZZ)
b = CartanType(['G',2]).root_system().ambient_space().from_vector(vector([4,-5,1])); b
b.to_weight_space()

class ParentMethods[source]¶

Bases: object

dynkin_diagram_automorphism_of_alcove_morphism(f)[source]¶

Return the Dynkin diagram automorphism induced by an alcove morphism.

INPUT:

f – a linear map from self to self which preserves alcoves

This method returns the Dynkin diagram automorphism for the decomposition \(f = d w\) (see reduced_word_of_alcove_morphism()), as a dictionary mapping elements of the index set to itself.

EXAMPLES:

Sage

sage: R = RootSystem(["A",2,1]).weight_lattice()
sage: alpha = R.simple_roots()                                          # needs sage.graphs
sage: Lambda = R.fundamental_weights()

Python

>>> from sage.all import *
>>> R = RootSystem(["A",Integer(2),Integer(1)]).weight_lattice()
>>> alpha = R.simple_roots()                                          # needs sage.graphs
>>> Lambda = R.fundamental_weights()

Sage Live

R = RootSystem(["A",2,1]).weight_lattice()
alpha = R.simple_roots()                                          # needs sage.graphs
Lambda = R.fundamental_weights()

Translations by elements of the root lattice induce a trivial Dynkin diagram automorphism:

Sage

sage: # needs sage.graphs sage.libs.gap
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[0].translation)
{0: 0, 1: 1, 2: 2}
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[1].translation)
{0: 0, 1: 1, 2: 2}
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[2].translation)
{0: 0, 1: 1, 2: 2}

Python

>>> from sage.all import *
>>> # needs sage.graphs sage.libs.gap
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(0)].translation)
{0: 0, 1: 1, 2: 2}
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(1)].translation)
{0: 0, 1: 1, 2: 2}
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(2)].translation)
{0: 0, 1: 1, 2: 2}

Sage Live

# needs sage.graphs sage.libs.gap
R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[0].translation)
R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[1].translation)
R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[2].translation)

This is no more the case for translations by general elements of the (classical) weight lattice at level 0:

Sage

sage: omega1 = Lambda[1] - Lambda[0]
sage: omega2 = Lambda[2] - Lambda[0]

sage: # needs sage.graphs sage.libs.gap
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
{0: 1, 1: 2, 2: 0}
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
{0: 2, 1: 0, 2: 1}

sage: # needs sage.graphs sage.libs.gap
sage: R = RootSystem(['C',2,1]).weight_lattice()
sage: alpha = R.simple_roots()
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[1].translation)
{0: 2, 1: 1, 2: 0}

sage: # needs sage.graphs sage.libs.gap
sage: R = RootSystem(['D',5,1]).weight_lattice()
sage: Lambda = R.fundamental_weights()
sage: omega1 = Lambda[1] - Lambda[0]
sage: omega2 = Lambda[2] - 2*Lambda[0]
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
{0: 1, 1: 0, 2: 2, 3: 3, 4: 5, 5: 4}
sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5}

Python

>>> from sage.all import *
>>> omega1 = Lambda[Integer(1)] - Lambda[Integer(0)]
>>> omega2 = Lambda[Integer(2)] - Lambda[Integer(0)]

>>> # needs sage.graphs sage.libs.gap
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
{0: 1, 1: 2, 2: 0}
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
{0: 2, 1: 0, 2: 1}

>>> # needs sage.graphs sage.libs.gap
>>> R = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice()
>>> alpha = R.simple_roots()
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(1)].translation)
{0: 2, 1: 1, 2: 0}

>>> # needs sage.graphs sage.libs.gap
>>> R = RootSystem(['D',Integer(5),Integer(1)]).weight_lattice()
>>> Lambda = R.fundamental_weights()
>>> omega1 = Lambda[Integer(1)] - Lambda[Integer(0)]
>>> omega2 = Lambda[Integer(2)] - Integer(2)*Lambda[Integer(0)]
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
{0: 1, 1: 0, 2: 2, 3: 3, 4: 5, 5: 4}
>>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5}

Sage Live

omega1 = Lambda[1] - Lambda[0]
omega2 = Lambda[2] - Lambda[0]
# needs sage.graphs sage.libs.gap
R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
# needs sage.graphs sage.libs.gap
R = RootSystem(['C',2,1]).weight_lattice()
alpha = R.simple_roots()
R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[1].translation)
# needs sage.graphs sage.libs.gap
R = RootSystem(['D',5,1]).weight_lattice()
Lambda = R.fundamental_weights()
omega1 = Lambda[1] - Lambda[0]
omega2 = Lambda[2] - 2*Lambda[0]
R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)

Algorithm: computes \(w\) of the decomposition, and see how \(f\circ w^{-1}\) permutes the simple roots.

embed_at_level(x, level=1)[source]¶

Embed the classical weight \(x\) in the level level hyperplane.

This is achieved by translating the straightforward embedding of \(x\) by \(c\Lambda_0\) for \(c\) some appropriate scalar.

INPUT:

x – an element of the corresponding classical weight/ambient lattice
level – integer or element of the base ring (default: 1)

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: L = RootSystem(["B",3,1]).weight_space()
sage: L0 = L.classical()
sage: alpha = L0.simple_roots()
sage: omega = L0.fundamental_weights()
sage: L.embed_at_level(omega[1], 1)
Lambda[1]
sage: L.embed_at_level(omega[2], 1)
-Lambda[0] + Lambda[2]
sage: L.embed_at_level(omega[3], 1)
Lambda[3]
sage: L.embed_at_level(alpha[1], 1)
Lambda[0] + 2*Lambda[1] - Lambda[2]

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> L = RootSystem(["B",Integer(3),Integer(1)]).weight_space()
>>> L0 = L.classical()
>>> alpha = L0.simple_roots()
>>> omega = L0.fundamental_weights()
>>> L.embed_at_level(omega[Integer(1)], Integer(1))
Lambda[1]
>>> L.embed_at_level(omega[Integer(2)], Integer(1))
-Lambda[0] + Lambda[2]
>>> L.embed_at_level(omega[Integer(3)], Integer(1))
Lambda[3]
>>> L.embed_at_level(alpha[Integer(1)], Integer(1))
Lambda[0] + 2*Lambda[1] - Lambda[2]

Sage Live

# needs sage.graphs
L = RootSystem(["B",3,1]).weight_space()
L0 = L.classical()
alpha = L0.simple_roots()
omega = L0.fundamental_weights()
L.embed_at_level(omega[1], 1)
L.embed_at_level(omega[2], 1)
L.embed_at_level(omega[3], 1)
L.embed_at_level(alpha[1], 1)

fundamental_weight(i)[source]¶

Return the \(i\)-th fundamental weight.

INPUT:

i – an element of the index set

By a slight notational abuse, for an affine type this method should also accept 'delta' as input, and return the image of \(\delta\) of the extended weight lattice in this realization.

This should be overridden by any subclass, and typically be implemented as a cached method for efficiency.

EXAMPLES:

Sage

sage: L = RootSystem(["A",3]).ambient_lattice()
sage: L.fundamental_weight(1)
(1, 0, 0, 0)

sage: L = RootSystem(["A",3,1]).weight_lattice(extended=True)
sage: L.fundamental_weight(1)
Lambda[1]
sage: L.fundamental_weight("delta")
delta

Python

>>> from sage.all import *
>>> L = RootSystem(["A",Integer(3)]).ambient_lattice()
>>> L.fundamental_weight(Integer(1))
(1, 0, 0, 0)

>>> L = RootSystem(["A",Integer(3),Integer(1)]).weight_lattice(extended=True)
>>> L.fundamental_weight(Integer(1))
Lambda[1]
>>> L.fundamental_weight("delta")
delta

Sage Live

L = RootSystem(["A",3]).ambient_lattice()
L.fundamental_weight(1)
L = RootSystem(["A",3,1]).weight_lattice(extended=True)
L.fundamental_weight(1)
L.fundamental_weight("delta")

fundamental_weights()[source]¶

Return the family \((\Lambda_i)_{i\in I}\) of the fundamental weights.

EXAMPLES:

Sage

sage: e = RootSystem(['A',3]).ambient_lattice()
sage: f = e.fundamental_weights()
sage: [f[i] for i in [1,2,3]]
[(1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0)]

Python

>>> from sage.all import *
>>> e = RootSystem(['A',Integer(3)]).ambient_lattice()
>>> f = e.fundamental_weights()
>>> [f[i] for i in [Integer(1),Integer(2),Integer(3)]]
[(1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0)]

Sage Live

e = RootSystem(['A',3]).ambient_lattice()
f = e.fundamental_weights()
[f[i] for i in [1,2,3]]

is_extended()[source]¶

Return whether this is a realization of the extended weight lattice.

EXAMPLES:

Sage

sage: RootSystem(["A",3,1]).weight_lattice().is_extended()
False
sage: RootSystem(["A",3,1]).weight_lattice(extended=True).is_extended()
True

Python

>>> from sage.all import *
>>> RootSystem(["A",Integer(3),Integer(1)]).weight_lattice().is_extended()
False
>>> RootSystem(["A",Integer(3),Integer(1)]).weight_lattice(extended=True).is_extended()
True

Sage Live

RootSystem(["A",3,1]).weight_lattice().is_extended()
RootSystem(["A",3,1]).weight_lattice(extended=True).is_extended()

This method is irrelevant for finite root systems, since the weight lattice need not be extended to ensure that the root lattice embeds faithfully:

Sage

sage: RootSystem(["A",3]).weight_lattice().is_extended()
False

Python

>>> from sage.all import *
>>> RootSystem(["A",Integer(3)]).weight_lattice().is_extended()
False

Sage Live

RootSystem(["A",3]).weight_lattice().is_extended()

reduced_word_of_alcove_morphism(f)[source]¶

Return the reduced word of an alcove morphism.

INPUT:

f – a linear map from self to self which preserves alcoves

Let \(A\) be the fundamental alcove. This returns a reduced word \(i_1, \ldots, i_k\) such that the affine Weyl group element \(w = s_{i_1} \circ \cdots \circ s_{i_k}\) maps the alcove \(f(A)\) back to \(A\). In other words, the alcove walk \(i_1, \ldots, i_k\) brings the fundamental alcove to the corresponding translated alcove.

Let us throw in a bit of context to explain the main use case. It is customary to realize the alcove picture in the coroot or coweight lattice \(R^\vee\). The extended affine Weyl group is then the group of linear maps on \(R^\vee\) which preserve the alcoves. By [Kac “Infinite-dimensional Lie algebra”, Proposition 6.5] the affine Weyl group is the semidirect product of the associated finite Weyl group and the group of translations in the coroot lattice (the extended affine Weyl group uses the coweight lattice instead). In other words, an element of the extended affine Weyl group admits a unique decomposition of the form:

\[f = d w ,\]

where \(w\) is in the Weyl group, and \(d\) is a function which maps the fundamental alcove to itself. As \(d\) permutes the walls of the fundamental alcove, it permutes accordingly the corresponding simple roots, which induces an automorphism of the Dynkin diagram.

This method returns a reduced word for \(w\), whereas the method dynkin_diagram_automorphism_of_alcove_morphism() returns \(d\) as a permutation of the nodes of the Dynkin diagram.

Nota bene: recall that the coroot (resp. coweight) lattice is implemented as the root (resp weight) lattice of the dual root system. Hence, this method is implemented for weight lattice realizations, but in practice is most of the time used on the dual side.

EXAMPLES:

We start with type \(A\) which is simply laced; hence we do not have to worry about the distinction between the weight and coweight lattice:

Sage

sage: R = RootSystem(["A",2,1]).weight_lattice()
sage: alpha = R.simple_roots()                                          # needs sage.graphs
sage: Lambda = R.fundamental_weights()

Python

>>> from sage.all import *
>>> R = RootSystem(["A",Integer(2),Integer(1)]).weight_lattice()
>>> alpha = R.simple_roots()                                          # needs sage.graphs
>>> Lambda = R.fundamental_weights()

Sage Live

R = RootSystem(["A",2,1]).weight_lattice()
alpha = R.simple_roots()                                          # needs sage.graphs
Lambda = R.fundamental_weights()

We consider first translations by elements of the root lattice:

Sage

sage: R.reduced_word_of_alcove_morphism(alpha[0].translation)           # needs sage.graphs
[1, 2, 1, 0]
sage: R.reduced_word_of_alcove_morphism(alpha[1].translation)           # needs sage.graphs
[0, 2, 0, 1]
sage: R.reduced_word_of_alcove_morphism(alpha[2].translation)           # needs sage.graphs
[0, 1, 0, 2]

Python

>>> from sage.all import *
>>> R.reduced_word_of_alcove_morphism(alpha[Integer(0)].translation)           # needs sage.graphs
[1, 2, 1, 0]
>>> R.reduced_word_of_alcove_morphism(alpha[Integer(1)].translation)           # needs sage.graphs
[0, 2, 0, 1]
>>> R.reduced_word_of_alcove_morphism(alpha[Integer(2)].translation)           # needs sage.graphs
[0, 1, 0, 2]

Sage Live

R.reduced_word_of_alcove_morphism(alpha[0].translation)           # needs sage.graphs
R.reduced_word_of_alcove_morphism(alpha[1].translation)           # needs sage.graphs
R.reduced_word_of_alcove_morphism(alpha[2].translation)           # needs sage.graphs

We continue with translations by elements of the classical weight lattice, embedded at level \(0\):

sage: omega1 = Lambda[1] - Lambda[0] sage: omega2 = Lambda[2] - Lambda[0]

sage: R.reduced_word_of_alcove_morphism(omega1.translation) # needs sage.graphs [0, 2] sage: R.reduced_word_of_alcove_morphism(omega2.translation) # needs sage.graphs [0, 1]

The following tests ensure that the code agrees with the tables in Kashiwara’s private notes on affine quantum algebras (2008).

reduced_word_of_translation(t)[source]¶

Given an element of the root lattice, this returns a reduced word \(i_1, \ldots, i_k\) such that the Weyl group element \(s_{i_1} \circ \cdots \circ s_{i_k}\) implements the “translation” where \(x\) maps to \(x + level(x)*t\). In other words, the alcove walk \(i_1, \ldots, i_k\) brings the fundamental alcove to the corresponding translated alcove.

Note

There are some technical conditions for \(t\) to actually be a translation; those are not tested (TODO: detail).

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: R = RootSystem(["A",2,1]).weight_lattice()
sage: alpha = R.simple_roots()
sage: R.reduced_word_of_translation(alpha[1])
[0, 2, 0, 1]
sage: R.reduced_word_of_translation(alpha[2])
[0, 1, 0, 2]
sage: R.reduced_word_of_translation(alpha[0])
[1, 2, 1, 0]

sage: R = RootSystem(['D',5,1]).weight_lattice()
sage: Lambda = R.fundamental_weights()
sage: omega1 = Lambda[1] - Lambda[0]
sage: omega2 = Lambda[2] - 2*Lambda[0]
sage: R.reduced_word_of_translation(omega1)                             # needs sage.graphs
[0, 2, 3, 4, 5, 3, 2, 0]
sage: R.reduced_word_of_translation(omega2)                             # needs sage.graphs
[0, 2, 1, 3, 2, 4, 3, 5, 3, 2, 1, 4, 3, 2]

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> R = RootSystem(["A",Integer(2),Integer(1)]).weight_lattice()
>>> alpha = R.simple_roots()
>>> R.reduced_word_of_translation(alpha[Integer(1)])
[0, 2, 0, 1]
>>> R.reduced_word_of_translation(alpha[Integer(2)])
[0, 1, 0, 2]
>>> R.reduced_word_of_translation(alpha[Integer(0)])
[1, 2, 1, 0]

>>> R = RootSystem(['D',Integer(5),Integer(1)]).weight_lattice()
>>> Lambda = R.fundamental_weights()
>>> omega1 = Lambda[Integer(1)] - Lambda[Integer(0)]
>>> omega2 = Lambda[Integer(2)] - Integer(2)*Lambda[Integer(0)]
>>> R.reduced_word_of_translation(omega1)                             # needs sage.graphs
[0, 2, 3, 4, 5, 3, 2, 0]
>>> R.reduced_word_of_translation(omega2)                             # needs sage.graphs
[0, 2, 1, 3, 2, 4, 3, 5, 3, 2, 1, 4, 3, 2]

Sage Live

# needs sage.graphs
R = RootSystem(["A",2,1]).weight_lattice()
alpha = R.simple_roots()
R.reduced_word_of_translation(alpha[1])
R.reduced_word_of_translation(alpha[2])
R.reduced_word_of_translation(alpha[0])
R = RootSystem(['D',5,1]).weight_lattice()
Lambda = R.fundamental_weights()
omega1 = Lambda[1] - Lambda[0]
omega2 = Lambda[2] - 2*Lambda[0]
R.reduced_word_of_translation(omega1)                             # needs sage.graphs
R.reduced_word_of_translation(omega2)                             # needs sage.graphs

A non simply laced case:

Sage

sage: R = RootSystem(["C",2,1]).weight_lattice()
sage: Lambda = R.fundamental_weights()
sage: c = R.cartan_type().translation_factors(); c                      # needs sage.graphs
Finite family {0: 1, 1: 2, 2: 1}
sage: R.reduced_word_of_translation((Lambda[1]-Lambda[0]) * c[1])       # needs sage.graphs
[0, 1, 2, 1]
sage: R.reduced_word_of_translation((Lambda[2]-Lambda[0]) * c[2])       # needs sage.graphs
[0, 1, 0]

Python

>>> from sage.all import *
>>> R = RootSystem(["C",Integer(2),Integer(1)]).weight_lattice()
>>> Lambda = R.fundamental_weights()
>>> c = R.cartan_type().translation_factors(); c                      # needs sage.graphs
Finite family {0: 1, 1: 2, 2: 1}
>>> R.reduced_word_of_translation((Lambda[Integer(1)]-Lambda[Integer(0)]) * c[Integer(1)])       # needs sage.graphs
[0, 1, 2, 1]
>>> R.reduced_word_of_translation((Lambda[Integer(2)]-Lambda[Integer(0)]) * c[Integer(2)])       # needs sage.graphs
[0, 1, 0]

Sage Live

R = RootSystem(["C",2,1]).weight_lattice()
Lambda = R.fundamental_weights()
c = R.cartan_type().translation_factors(); c                      # needs sage.graphs
R.reduced_word_of_translation((Lambda[1]-Lambda[0]) * c[1])       # needs sage.graphs
R.reduced_word_of_translation((Lambda[2]-Lambda[0]) * c[2])       # needs sage.graphs

See also _test_reduced_word_of_translation().

Todo

Add a picture in the doc
Add a method which, given an element of the classical weight lattice, constructs the appropriate value for t

rho()[source]¶

EXAMPLES:

Sage

sage: RootSystem(['A',3]).ambient_lattice().rho()
(3, 2, 1, 0)

Python

>>> from sage.all import *
>>> RootSystem(['A',Integer(3)]).ambient_lattice().rho()
(3, 2, 1, 0)

Sage Live

RootSystem(['A',3]).ambient_lattice().rho()

rho_classical()[source]¶

Return the embedding at level 0 of \(\rho\) of the classical lattice.

EXAMPLES:

Sage

sage: RootSystem(['C',4,1]).weight_lattice().rho_classical()            # needs sage.graphs
-4*Lambda[0] + Lambda[1] + Lambda[2] + Lambda[3] + Lambda[4]
sage: L = RootSystem(['D',4,1]).weight_lattice()
sage: L.rho_classical().scalar(L.null_coroot())                         # needs sage.graphs
0

Python

>>> from sage.all import *
>>> RootSystem(['C',Integer(4),Integer(1)]).weight_lattice().rho_classical()            # needs sage.graphs
-4*Lambda[0] + Lambda[1] + Lambda[2] + Lambda[3] + Lambda[4]
>>> L = RootSystem(['D',Integer(4),Integer(1)]).weight_lattice()
>>> L.rho_classical().scalar(L.null_coroot())                         # needs sage.graphs
0

Sage Live

RootSystem(['C',4,1]).weight_lattice().rho_classical()            # needs sage.graphs
L = RootSystem(['D',4,1]).weight_lattice()
L.rho_classical().scalar(L.null_coroot())                         # needs sage.graphs

Warning

In affine type BC dual, this does not live in the weight lattice:

Sage

sage: L = CartanType(["BC",2,2]).dual().root_system().weight_space()
sage: L.rho_classical()                                             # needs sage.graphs
-3/2*Lambda[0] + Lambda[1] + Lambda[2]
sage: L = CartanType(["BC",2,2]).dual().root_system().weight_lattice()
sage: L.rho_classical()                                             # needs sage.graphs
Traceback (most recent call last):
...
ValueError: 5 is not divisible by 2

Python

>>> from sage.all import *
>>> L = CartanType(["BC",Integer(2),Integer(2)]).dual().root_system().weight_space()
>>> L.rho_classical()                                             # needs sage.graphs
-3/2*Lambda[0] + Lambda[1] + Lambda[2]
>>> L = CartanType(["BC",Integer(2),Integer(2)]).dual().root_system().weight_lattice()
>>> L.rho_classical()                                             # needs sage.graphs
Traceback (most recent call last):
...
ValueError: 5 is not divisible by 2

Sage Live

L = CartanType(["BC",2,2]).dual().root_system().weight_space()
L.rho_classical()                                             # needs sage.graphs
L = CartanType(["BC",2,2]).dual().root_system().weight_lattice()
L.rho_classical()                                             # needs sage.graphs

signs_of_alcovewalk(walk)[source]¶

Let walk = \([i_1,\ldots,i_n]\) denote an alcove walk starting from the fundamental alcove \(y_0\), crossing at step 1 the wall \(i_1\), and so on.

For each \(k\), set \(w_k = s_{i_1} \circ s_{i_k}\), and denote by \(y_k = w_k(y_0)\) the alcove reached after \(k\) steps. Then, \(y_k\) is obtained recursively from \(y_{k-1}\) by applying the following reflection:

\[y_k = s_{w_{k-1} \alpha_{i_k}} y_{k-1}.\]

The step is said positive if \(w_{k-1} \alpha_{i_k}\) is a negative root (considering \(w_{k-1}\) as element of the classical Weyl group and \(\alpha_{i_k}\) as a classical root) and negative otherwise. The algorithm implemented here use the equivalent property:

.. MATH:: \langle w_{k-1}^{-1} \rho_0, \alpha^\vee_{i_k}\rangle > 0

Where \(\rho_0\) is the sum of the classical fundamental weights embedded at level 0 in this space (see rho_classical()), and \(\alpha^\vee_{i_k}\) is the simple coroot associated to \(\alpha_{i_k}\).

This function returns a list of the form \([+1,+1,-1,...]\), where the \(k\)-th entry denotes whether the \(k\)-th step was positive or negative.

See equation 3.4, of Ram: Alcove walks …, arXiv math/0601343v1

EXAMPLES:

Sage

sage: L = RootSystem(['C',2,1]).weight_lattice()
sage: L.signs_of_alcovewalk([1,2,0,1,2,1,2,0,1,2])                      # needs sage.libs.gap
[-1, -1, 1, -1, 1, 1, 1, 1, 1, 1]

sage: L = RootSystem(['A',2,1]).weight_lattice()
sage: L.signs_of_alcovewalk([0,1,2,1,2,0,1,2,0,1,2,0])                  # needs sage.libs.gap
[1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1]

sage: L = RootSystem(['B',2,1]).coweight_lattice()
sage: L.signs_of_alcovewalk([0,1,2,0,1,2])                              # needs sage.libs.gap
[1, -1, 1, -1, 1, 1]

Python

>>> from sage.all import *
>>> L = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice()
>>> L.signs_of_alcovewalk([Integer(1),Integer(2),Integer(0),Integer(1),Integer(2),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2)])                      # needs sage.libs.gap
[-1, -1, 1, -1, 1, 1, 1, 1, 1, 1]

>>> L = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice()
>>> L.signs_of_alcovewalk([Integer(0),Integer(1),Integer(2),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2),Integer(0)])                  # needs sage.libs.gap
[1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1]

>>> L = RootSystem(['B',Integer(2),Integer(1)]).coweight_lattice()
>>> L.signs_of_alcovewalk([Integer(0),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2)])                              # needs sage.libs.gap
[1, -1, 1, -1, 1, 1]

Sage Live

L = RootSystem(['C',2,1]).weight_lattice()
L.signs_of_alcovewalk([1,2,0,1,2,1,2,0,1,2])                      # needs sage.libs.gap
L = RootSystem(['A',2,1]).weight_lattice()
L.signs_of_alcovewalk([0,1,2,1,2,0,1,2,0,1,2,0])                  # needs sage.libs.gap
L = RootSystem(['B',2,1]).coweight_lattice()
L.signs_of_alcovewalk([0,1,2,0,1,2])                              # needs sage.libs.gap

Warning

This method currently does not work in the weight lattice for type BC dual because \(\rho_0\) does not live in this lattice (but an integral multiple of it would do the job as well).

simple_root(i)[source]¶

Return the \(i\)-th simple root.

This default implementation takes the \(i\)-th simple root in the weight lattice and embeds it in self.

EXAMPLES:

Since all the weight lattice realizations in Sage currently implement a simple_root method, we have to call this one by hand:

Sage

sage: from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
sage: simple_root = WeightLatticeRealizations(QQ).parent_class.simple_root.f
sage: L = RootSystem("A3").ambient_space()
sage: simple_root(L, 1)                                                 # needs sage.graphs
(1, -1, 0, 0)
sage: simple_root(L, 2)                                                 # needs sage.graphs
(0, 1, -1, 0)
sage: simple_root(L, 3)                                                 # needs sage.graphs
(1, 1, 2, 0)

Python

>>> from sage.all import *
>>> from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
>>> simple_root = WeightLatticeRealizations(QQ).parent_class.simple_root.f
>>> L = RootSystem("A3").ambient_space()
>>> simple_root(L, Integer(1))                                                 # needs sage.graphs
(1, -1, 0, 0)
>>> simple_root(L, Integer(2))                                                 # needs sage.graphs
(0, 1, -1, 0)
>>> simple_root(L, Integer(3))                                                 # needs sage.graphs
(1, 1, 2, 0)

Sage Live

from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
simple_root = WeightLatticeRealizations(QQ).parent_class.simple_root.f
L = RootSystem("A3").ambient_space()
simple_root(L, 1)                                                 # needs sage.graphs
simple_root(L, 2)                                                 # needs sage.graphs
simple_root(L, 3)                                                 # needs sage.graphs

Note that this last root differs from the one implemented in L by a multiple of the vector (1,1,1,1):

Sage

sage: L.simple_roots()                                                  # needs sage.graphs
Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}

Python

>>> from sage.all import *
>>> L.simple_roots()                                                  # needs sage.graphs
Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}

Sage Live

L.simple_roots()                                                  # needs sage.graphs

This is a harmless artefact of the \(SL\) versus \(GL\) interpretation of type \(A\); see the thematic tutorial on Lie Methods and Related Combinatorics in Sage for details.

weyl_dimension(highest_weight)[source]¶

Return the dimension of the highest weight representation of highest weight highest_weight.

EXAMPLES:

Sage

sage: RootSystem(['A',3]).ambient_lattice().weyl_dimension([2,1,0,0])
20
sage: P = RootSystem(['C',2]).weight_lattice()
sage: La = P.basis()
sage: P.weyl_dimension(La[1]+La[2])                                     # needs sage.graphs
16

sage: type(RootSystem(['A',3]).ambient_lattice().weyl_dimension([2,1,0,0]))
<class 'sage.rings.integer.Integer'>

Python

>>> from sage.all import *
>>> RootSystem(['A',Integer(3)]).ambient_lattice().weyl_dimension([Integer(2),Integer(1),Integer(0),Integer(0)])
20
>>> P = RootSystem(['C',Integer(2)]).weight_lattice()
>>> La = P.basis()
>>> P.weyl_dimension(La[Integer(1)]+La[Integer(2)])                                     # needs sage.graphs
16

>>> type(RootSystem(['A',Integer(3)]).ambient_lattice().weyl_dimension([Integer(2),Integer(1),Integer(0),Integer(0)]))
<class 'sage.rings.integer.Integer'>

Sage Live

RootSystem(['A',3]).ambient_lattice().weyl_dimension([2,1,0,0])
P = RootSystem(['C',2]).weight_lattice()
La = P.basis()
P.weyl_dimension(La[1]+La[2])                                     # needs sage.graphs
type(RootSystem(['A',3]).ambient_lattice().weyl_dimension([2,1,0,0]))

super_categories()[source]¶

EXAMPLES:

Sage

sage: from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
sage: WeightLatticeRealizations(QQ).super_categories()
[Category of root lattice realizations over Rational Field]

Python

>>> from sage.all import *
>>> from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
>>> WeightLatticeRealizations(QQ).super_categories()
[Category of root lattice realizations over Rational Field]

Sage Live

from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
WeightLatticeRealizations(QQ).super_categories()