Weight lattice realizations

class sage.combinat.root_system.weight_lattice_realizations.WeightLatticeRealizations(base, name=None)

Bases: sage.categories.category_types.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.

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: 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)

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

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)

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

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])

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: 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])
class ElementMethods
symmetric_form(la)

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: 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: 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()
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

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

TESTS:

We check that \(A_{2n}^{(2)}\) has 3 different root lengths:

sage: P = RootSystem(['A',4,2]).weight_lattice()
sage: al = P.simple_roots()
sage: [al[i].symmetric_form(al[i]) for i in P.index_set()]
[2, 4, 8]
class WeightLatticeRealizations.ParentMethods
dynkin_diagram_automorphism_of_alcove_morphism(f)

Returns 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 dictionnary mapping elements of the index set to itself.

EXAMPLES:

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

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

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}

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

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

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: 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: 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}

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

embed_at_level(x, level=1)

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 – an integer or element of the base ring (default: 1)

EXAMPLES:

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]
fundamental_weight(i)

Returns 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: 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

TESTS:

sage: super(sage.combinat.root_system.weight_space.WeightSpace, L).fundamental_weight(1)
Traceback (most recent call last):
...
NotImplementedError: <abstract method fundamental_weight at ...>
fundamental_weights()

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

EXAMPLES:

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)]
is_extended()

Returns whether this is a realization of the extended weight lattice

EXAMPLES:

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

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

INPUT:

  • \(f\) – a linear map from self to self which preserves alcoves.

Let \(A\) be the fundamental alcove. This returns a reduced word \(i_1,...,i_k\) such that the affine Weyl group element \(w = s_{i_1} \circ \dots \circ s_{i_k}\) maps the alcove \(f(A)\) back to \(A\). In other words, the alcove walk \(i_1,...,i_k\) 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: R = RootSystem(["A",2,1]).weight_lattice()
sage: alpha = R.simple_roots()
sage: Lambda = R.fundamental_weights()

We consider first translations by elements of the root lattice:

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

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) [0, 2] sage: R.reduced_word_of_alcove_morphism(omega2.translation) [0, 1]

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

TESTS:

sage: R = RootSystem(['A',5,1]).weight_lattice()
sage: alpha = R.simple_roots()
sage: Lambda = R.fundamental_weights()
sage: omega1 = Lambda[1] - Lambda[0]
sage: R.reduced_word_of_alcove_morphism(omega1.translation)
[0, 5, 4, 3, 2]
sage: R.reduced_word_of_alcove_morphism(alpha[0].translation)
[1, 2, 3, 4, 5, 4, 3, 2, 1, 0]

sage: R = RootSystem(['C',3,1]).weight_lattice()
sage: alpha = R.simple_roots()
sage: Lambda = R.fundamental_weights()
sage: omega1 = 2*(Lambda[1] - Lambda[0])
sage: omega2 = 2*(Lambda[2] - Lambda[0])
sage: omega3 = Lambda[3] - Lambda[0]
sage: R.reduced_word_of_alcove_morphism(omega1.translation)
[0, 1, 2, 3, 2, 1]
sage: R.reduced_word_of_alcove_morphism(omega2.translation)
[0, 1, 0, 2, 1, 3, 2, 1, 3, 2]
sage: R.reduced_word_of_alcove_morphism(omega3.translation)
[0, 1, 0, 2, 1, 0]
sage: W = WeylGroup(['C',3,1])
sage: s = W.simple_reflections()
sage: w = s[0]*s[1]*s[2]*s[3]*s[2]
sage: W.from_reduced_word(R.reduced_word_of_alcove_morphism(omega2.translation)) == w*w
True
sage: w = s[0]*s[1]*s[2]*s[0]*s[1]*s[0]
sage: W.from_reduced_word(R.reduced_word_of_alcove_morphism(omega3.translation)) == w
True

sage: R = RootSystem(['D',4,1]).weight_lattice()
sage: Lambda = R.fundamental_weights()
sage: omega1 = Lambda[1] - Lambda[0]
sage: omega2 = Lambda[2] - 2*Lambda[0]
sage: omega3 = Lambda[3] - Lambda[0]
sage: omega4 = Lambda[4] - Lambda[0]
sage: R.reduced_word_of_alcove_morphism(omega1.translation)
[0, 2, 3, 4, 2, 0]
sage: R.reduced_word_of_alcove_morphism(omega2.translation)
[0, 2, 1, 3, 2, 4, 2, 1, 3, 2]
sage: R.reduced_word_of_alcove_morphism(omega3.translation)
[0, 2, 1, 4, 2, 0]
sage: R.reduced_word_of_alcove_morphism(omega4.translation)
[0, 2, 1, 3, 2, 0]
sage: W = WeylGroup(['D',4,1])
sage: s = W.simple_reflections()
sage: w = s[0]*s[2]*s[3]*s[4]*s[2]
sage: w1= s[1]*s[2]*s[3]*s[4]*s[2]
sage: W.from_reduced_word(R.reduced_word_of_alcove_morphism(omega2.translation)) == w*w1
True

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_alcove_morphism(omega1.translation)
[0, 2, 3, 4, 5, 3, 2, 0]
sage: W = WeylGroup(['D',5,1])
sage: s = W.simple_reflections()
sage: w = s[0]*s[2]*s[3]*s[4]*s[5]*s[3]*s[2]
sage: w1= s[1]*s[2]*s[3]*s[4]*s[5]*s[3]*s[2]
sage: W.from_reduced_word(R.reduced_word_of_alcove_morphism(omega2.translation)) == w*w1
True
reduced_word_of_translation(t)

Given an element of the root lattice, this returns a reduced word \(i_1,...,i_k\) such that the Weyl group element \(s_{i_1} \circ \dots \circ s_{i_k}\) implements the “translation” where \(x\) maps to \(x + level(x)*t\). In other words, the alcove walk \(i_1,...,i_k\) 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: 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)
[0, 2, 3, 4, 5, 3, 2, 0]
sage: R.reduced_word_of_translation(omega2)
[0, 2, 1, 3, 2, 4, 3, 5, 3, 2, 1, 4, 3, 2]

A non simply laced case:

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

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()

EXAMPLES:

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

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

EXAMPLES:

sage: RootSystem(['C',4,1]).weight_lattice().rho_classical()
-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())
0

Warning

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

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

Let walk = \([i_1,\dots,i_n]\) 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 [math.RT]

EXAMPLES:

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

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)

Returns 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: 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)
(1, -1, 0, 0)
sage: simple_root(L, 2)
(0, 1, -1, 0)
sage: simple_root(L, 3)
(1, 1, 2, 0)

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

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

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)

EXAMPLES:

sage: RootSystem(['A',3]).ambient_lattice().weyl_dimension([2,1,0,0])
20

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

EXAMPLES:

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

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