Weight lattices and weight spaces¶

class sage.combinat.root_system.weight_space.WeightSpace(root_system, base_ring, extended)

INPUT:

• root_system – a root system

• base_ring – a ring $$R$$

• extended – a boolean (default: False)

The weight space (or lattice if base_ring is $$\ZZ$$) of a root system is the formal free module $$\bigoplus_i R \Lambda_i$$ generated by the fundamental weights $$(\Lambda_i)_{i\in I}$$ of the root system.

This class is also used for coweight spaces (or lattices).

EXAMPLES:

sage: Q = RootSystem(['A', 3]).weight_lattice(); Q
Weight lattice of the Root system of type ['A', 3]
sage: Q.simple_roots()
Finite family {1: 2*Lambda - Lambda, 2: -Lambda + 2*Lambda - Lambda, 3: -Lambda + 2*Lambda}

sage: Q = RootSystem(['A', 3, 1]).weight_lattice(); Q
Weight lattice of the Root system of type ['A', 3, 1]
sage: Q.simple_roots()
Finite family {0: 2*Lambda -   Lambda               -   Lambda,
1:  -Lambda + 2*Lambda -   Lambda,
2:                -Lambda + 2*Lambda -   Lambda,
3:  -Lambda               -   Lambda + 2*Lambda}

For infinite types, the Cartan matrix is singular, and therefore the embedding of the root lattice is not faithful:

sage: sum(Q.simple_roots())
0

In particular, the null root is zero:

sage: Q.null_root()
0

This can be compensated by extending the basis of the weight space and slightly deforming the simple roots to make them linearly independent, without affecting the scalar product with the coroots. This feature is currently only implemented for affine types. In that case, if extended is set, then the basis of the weight space is extended by an element $$\delta$$:

sage: Q = RootSystem(['A', 3, 1]).weight_lattice(extended = True); Q
Extended weight lattice of the Root system of type ['A', 3, 1]
sage: Q.basis().keys()
{0, 1, 2, 3, 'delta'}

And the simple root $$\alpha_0$$ associated to the special node is deformed as follows:

sage: Q.simple_roots()
Finite family {0: 2*Lambda -   Lambda               -   Lambda + delta,
1:  -Lambda + 2*Lambda -   Lambda,
2:                -Lambda + 2*Lambda -   Lambda,
3:  -Lambda               -   Lambda + 2*Lambda}

Now, the null root is nonzero:

sage: Q.null_root()
delta

Warning

By a slight notational abuse, the extra basis element used to extend the fundamental weights is called \delta in the current implementation. However, in the literature, \delta usually denotes instead the null root. Most of the time, those two objects coincide, but not for type $$BC$$ (aka. $$A_{2n}^{(2)}$$). Therefore we currently have:

sage: Q = RootSystem(["A",4,2]).weight_lattice(extended=True)
sage: Q.simple_root(0)
2*Lambda - Lambda + delta
sage: Q.null_root()
2*delta

whereas, with the standard notations from the literature, one would expect to get respectively $$2\Lambda_0 -\Lambda_1 +1/2 \delta$$ and $$\delta$$.

Other than this notational glitch, the implementation remains correct for type $$BC$$.

The notations may get improved in a subsequent version, which might require changing the index of the extra basis element. To guarantee backward compatibility in code not included in Sage, it is recommended to use the following idiom to get that index:

sage: F = Q.basis_extension(); F
Finite family {'delta': delta}
sage: index = F.keys(); index
'delta'

Then, for example, the coefficient of an element of the extended weight lattice on that basis element can be recovered with:

sage: Q.null_root()[index]
2
Element

alias of WeightSpaceElement

basis_extension()

Return the basis elements used to extend the fundamental weights

EXAMPLES:

sage: Q = RootSystem(["A",3,1]).weight_lattice()
sage: Q.basis_extension()
Family ()

sage: Q = RootSystem(["A",3,1]).weight_lattice(extended=True)
sage: Q.basis_extension()
Finite family {'delta': delta}

This method is irrelevant for finite types:

sage: Q = RootSystem(["A",3]).weight_lattice()
sage: Q.basis_extension()
Family ()
fundamental_weight(i)

Returns the $$i$$-th fundamental weight

INPUT:

• i – an element of the index set or "delta"

By a slight notational abuse, for an affine type this method also accepts "delta" as input, and returns the image of $$\delta$$ of the extended weight lattice in this realization.

fundamental_weight()

EXAMPLES:

sage: Q = RootSystem(["A",3]).weight_lattice()
sage: Q.fundamental_weight(1)
Lambda

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

Return whether this is an extended weight lattice.

is_extended()

EXAMPLES:

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

Returns the $$j^{th}$$ simple root

EXAMPLES:

sage: L = RootSystem(["C",4]).weight_lattice()
sage: L.simple_root(3)
-Lambda + 2*Lambda - Lambda

Its coefficients are given by the corresponding column of the Cartan matrix:

sage: L.cartan_type().cartan_matrix()[:,2]
[ 0]
[-1]
[ 2]
[-1]

Here are all simple roots:

sage: L.simple_roots()
Finite family {1:  2*Lambda   - Lambda,
2:   -Lambda + 2*Lambda   - Lambda,
3:   -Lambda + 2*Lambda   - Lambda,
4:               -2*Lambda + 2*Lambda}

For the extended weight lattice of an affine type, the simple root associated to the special node is deformed by adding $$\delta$$, where $$\delta$$ is the null root:

sage: L = RootSystem(["C",4,1]).weight_lattice(extended=True)
sage: L.simple_root(0)
2*Lambda - 2*Lambda + delta

In fact $$\delta$$ is really $$1/a_0$$ times the null root (see the discussion in WeightSpace) but this only makes a difference in type $$BC$$:

sage: L = RootSystem(CartanType(["BC",4,2])).weight_lattice(extended=True)
sage: L.simple_root(0)
2*Lambda - Lambda + delta
sage: L.null_root()
2*delta

to_ambient_space_morphism()

The morphism from self to its associated ambient space.

EXAMPLES:

sage: CartanType(['A',2]).root_system().weight_lattice().to_ambient_space_morphism()
Generic morphism:
From: Weight lattice of the Root system of type ['A', 2]
To:   Ambient space of the Root system of type ['A', 2]

Warning

Implemented only for finite Cartan type.

class sage.combinat.root_system.weight_space.WeightSpaceElement

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

is_dominant()

Checks whether an element in the weight space lies in the positive cone spanned by the basis elements (fundamental weights).

EXAMPLES:

sage: W = RootSystem(['A',3]).weight_space()
sage: Lambda = W.basis()
sage: w = Lambda+Lambda
sage: w.is_dominant()
True
sage: w = Lambda-Lambda
sage: w.is_dominant()
False

In the extended affine weight lattice, ‘delta’ is orthogonal to the positive coroots, so adding or subtracting it should not effect dominance

sage: P = RootSystem(['A',2,1]).weight_lattice(extended=true)
sage: Lambda = P.fundamental_weights()
sage: delta = P.null_root()
sage: w = Lambda-delta
sage: w.is_dominant()
True
scalar(lambdacheck)

The canonical scalar product between the weight lattice and the coroot lattice.

Todo

• merge with_apply_multi_module_morphism

• allow for any root space / lattice

• define properly the return type (depends on the base rings of the two spaces)

• make this robust for extended weight lattices ($$i$$ might be “delta”)

EXAMPLES:

sage: L = RootSystem(["C",4,1]).weight_lattice()
sage: Lambda     = L.fundamental_weights()
sage: alphacheck = L.simple_coroots()
sage: Lambda.scalar(alphacheck)
1
sage: Lambda.scalar(alphacheck)
0

The fundamental weights and the simple coroots are dual bases:

sage: matrix([ [ Lambda[i].scalar(alphacheck[j])
....:            for i in L.index_set() ]
....:          for j in L.index_set() ])
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]

Note that the scalar product is not yet implemented between the weight space and the coweight space; in any cases, that won’t be the job of this method:

sage: R = RootSystem(["A",3])
sage: alpha = R.weight_space().roots()
sage: alphacheck = R.coweight_space().roots()
sage: alpha.scalar(alphacheck)
Traceback (most recent call last):
...
ValueError: -Lambdacheck + 2*Lambdacheck - Lambdacheck is not in the coroot space
to_ambient()

Maps self to the ambient space.

EXAMPLES:

sage: mu = CartanType(['B',2]).root_system().weight_lattice().an_element(); mu
2*Lambda + 2*Lambda
sage: mu.to_ambient()
(3, 1)

Warning

Only implemented in finite Cartan type. Does not work for coweight lattices because there is no implemented map from the coweight lattice to the ambient space.

to_weight_space()

Map self to the weight space.

Since $$self.parent()$$ is the weight space, this map just returns self. This overrides the generic method in $$WeightSpaceRealizations$$.

EXAMPLES:

sage: mu = CartanType(['A',2]).root_system().weight_lattice().an_element(); mu
2*Lambda + 2*Lambda
sage: mu.to_weight_space()
2*Lambda + 2*Lambda