Weight lattices and weight spaces#
- class sage.combinat.root_system.weight_space.WeightSpace(root_system, base_ring, extended)[source]#
Bases:
CombinatorialFreeModuleINPUT:
root_system– a root systembase_ring– a ring \(R\)extended– a boolean (default:False)
The weight space (or lattice if
base_ringis \(\ZZ\)) 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).
See also
EXAMPLES:
sage: Q = RootSystem(['A', 3]).weight_lattice(); Q Weight lattice of the Root system of type ['A', 3] sage: Q.simple_roots() # needs sage.graphs Finite family {1: 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[2] + 2*Lambda[3]} sage: Q = RootSystem(['A', 3, 1]).weight_lattice(); Q Weight lattice of the Root system of type ['A', 3, 1] sage: Q.simple_roots() # needs sage.graphs Finite family {0: 2*Lambda[0] - Lambda[1] - Lambda[3], 1: -Lambda[0] + 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[0] - Lambda[2] + 2*Lambda[3]}
>>> from sage.all import * >>> Q = RootSystem(['A', Integer(3)]).weight_lattice(); Q Weight lattice of the Root system of type ['A', 3] >>> Q.simple_roots() # needs sage.graphs Finite family {1: 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[2] + 2*Lambda[3]} >>> Q = RootSystem(['A', Integer(3), Integer(1)]).weight_lattice(); Q Weight lattice of the Root system of type ['A', 3, 1] >>> Q.simple_roots() # needs sage.graphs Finite family {0: 2*Lambda[0] - Lambda[1] - Lambda[3], 1: -Lambda[0] + 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[0] - Lambda[2] + 2*Lambda[3]}
For infinite types, the Cartan matrix is singular, and therefore the embedding of the root lattice is not faithful:
sage: sum(Q.simple_roots()) # needs sage.graphs 0
>>> from sage.all import * >>> sum(Q.simple_roots()) # needs sage.graphs 0
In particular, the null root is zero:
sage: Q.null_root() # needs sage.graphs 0
>>> from sage.all import * >>> Q.null_root() # needs sage.graphs 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
extendedis 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'}
>>> from sage.all import * >>> Q = RootSystem(['A', Integer(3), Integer(1)]).weight_lattice(extended=True); Q Extended weight lattice of the Root system of type ['A', 3, 1] >>> 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() # needs sage.graphs Finite family {0: 2*Lambda[0] - Lambda[1] - Lambda[3] + delta, 1: -Lambda[0] + 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[0] - Lambda[2] + 2*Lambda[3]}
>>> from sage.all import * >>> Q.simple_roots() # needs sage.graphs Finite family {0: 2*Lambda[0] - Lambda[1] - Lambda[3] + delta, 1: -Lambda[0] + 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[0] - Lambda[2] + 2*Lambda[3]}
Now, the null root is nonzero:
sage: Q.null_root() # needs sage.graphs delta
>>> from sage.all import * >>> Q.null_root() # needs sage.graphs delta
Warning
By a slight notational abuse, the extra basis element used to extend the fundamental weights is called
\deltain the current implementation. However, in the literature,\deltausually 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) # needs sage.graphs 2*Lambda[0] - Lambda[1] + delta sage: Q.null_root() # needs sage.graphs 2*delta
>>> from sage.all import * >>> Q = RootSystem(["A",Integer(4),Integer(2)]).weight_lattice(extended=True) >>> Q.simple_root(Integer(0)) # needs sage.graphs 2*Lambda[0] - Lambda[1] + delta >>> Q.null_root() # needs sage.graphs 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()[0]; index 'delta'
>>> from sage.all import * >>> F = Q.basis_extension(); F Finite family {'delta': delta} >>> index = F.keys()[Integer(0)]; 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] # needs sage.graphs 2
>>> from sage.all import * >>> Q.null_root()[index] # needs sage.graphs 2
- Element[source]#
alias of
WeightSpaceElement
- basis_extension()[source]#
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}
>>> from sage.all import * >>> Q = RootSystem(["A",Integer(3),Integer(1)]).weight_lattice() >>> Q.basis_extension() Family () >>> Q = RootSystem(["A",Integer(3),Integer(1)]).weight_lattice(extended=True) >>> 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 ()
>>> from sage.all import * >>> Q = RootSystem(["A",Integer(3)]).weight_lattice() >>> Q.basis_extension() Family ()
- fundamental_weight(i)[source]#
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.See also
fundamental_weight()EXAMPLES:
sage: Q = RootSystem(["A",3]).weight_lattice() sage: Q.fundamental_weight(1) Lambda[1] sage: Q = RootSystem(["A",3,1]).weight_lattice(extended=True) sage: Q.fundamental_weight(1) Lambda[1] sage: Q.fundamental_weight("delta") delta
>>> from sage.all import * >>> Q = RootSystem(["A",Integer(3)]).weight_lattice() >>> Q.fundamental_weight(Integer(1)) Lambda[1] >>> Q = RootSystem(["A",Integer(3),Integer(1)]).weight_lattice(extended=True) >>> Q.fundamental_weight(Integer(1)) Lambda[1] >>> Q.fundamental_weight("delta") delta
- is_extended()[source]#
Return whether this is an extended weight lattice.
See also
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
>>> 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
- simple_root(j)[source]#
Returns the \(j^{th}\) simple root
EXAMPLES:
sage: L = RootSystem(["C",4]).weight_lattice() sage: L.simple_root(3) # needs sage.graphs -Lambda[2] + 2*Lambda[3] - Lambda[4]
>>> from sage.all import * >>> L = RootSystem(["C",Integer(4)]).weight_lattice() >>> L.simple_root(Integer(3)) # needs sage.graphs -Lambda[2] + 2*Lambda[3] - Lambda[4]
Its coefficients are given by the corresponding column of the Cartan matrix:
sage: L.cartan_type().cartan_matrix()[:,2] # needs sage.graphs [ 0] [-1] [ 2] [-1]
>>> from sage.all import * >>> L.cartan_type().cartan_matrix()[:,Integer(2)] # needs sage.graphs [ 0] [-1] [ 2] [-1]
Here are all simple roots:
sage: L.simple_roots() # needs sage.graphs Finite family {1: 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[2] + 2*Lambda[3] - Lambda[4], 4: -2*Lambda[3] + 2*Lambda[4]}
>>> from sage.all import * >>> L.simple_roots() # needs sage.graphs Finite family {1: 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2] - Lambda[3], 3: -Lambda[2] + 2*Lambda[3] - Lambda[4], 4: -2*Lambda[3] + 2*Lambda[4]}
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) # needs sage.graphs 2*Lambda[0] - 2*Lambda[1] + delta
>>> from sage.all import * >>> L = RootSystem(["C",Integer(4),Integer(1)]).weight_lattice(extended=True) >>> L.simple_root(Integer(0)) # needs sage.graphs 2*Lambda[0] - 2*Lambda[1] + 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) # needs sage.graphs 2*Lambda[0] - Lambda[1] + delta sage: L.null_root() # needs sage.graphs 2*delta
>>> from sage.all import * >>> L = RootSystem(CartanType(["BC",Integer(4),Integer(2)])).weight_lattice(extended=True) >>> L.simple_root(Integer(0)) # needs sage.graphs 2*Lambda[0] - Lambda[1] + delta >>> L.null_root() # needs sage.graphs 2*delta
See also
CartanType.col_annihilator()
- to_ambient_space_morphism()[source]#
The morphism from
selfto 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]
>>> from sage.all import * >>> CartanType(['A',Integer(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[source]#
Bases:
IndexedFreeModuleElement- is_dominant()[source]#
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[1] + Lambda[3] sage: w.is_dominant() True sage: w = Lambda[1] - Lambda[2] sage: w.is_dominant() False
>>> from sage.all import * >>> W = RootSystem(['A',Integer(3)]).weight_space() >>> Lambda = W.basis() >>> w = Lambda[Integer(1)] + Lambda[Integer(3)] >>> w.is_dominant() True >>> w = Lambda[Integer(1)] - Lambda[Integer(2)] >>> w.is_dominant() False
In the extended affine weight lattice, ‘delta’ is orthogonal to the positive coroots, so adding or subtracting it should not affect dominance
sage: P = RootSystem(['A',2,1]).weight_lattice(extended=true) sage: Lambda = P.fundamental_weights() sage: delta = P.null_root() # needs sage.graphs sage: w = Lambda[1] - delta # needs sage.graphs sage: w.is_dominant() # needs sage.graphs True
>>> from sage.all import * >>> P = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice(extended=true) >>> Lambda = P.fundamental_weights() >>> delta = P.null_root() # needs sage.graphs >>> w = Lambda[Integer(1)] - delta # needs sage.graphs >>> w.is_dominant() # needs sage.graphs True
- scalar(lambdacheck)[source]#
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[1].scalar(alphacheck[1]) 1 sage: Lambda[1].scalar(alphacheck[2]) 0
>>> from sage.all import * >>> L = RootSystem(["C",Integer(4),Integer(1)]).weight_lattice() >>> Lambda = L.fundamental_weights() >>> alphacheck = L.simple_coroots() >>> Lambda[Integer(1)].scalar(alphacheck[Integer(1)]) 1 >>> Lambda[Integer(1)].scalar(alphacheck[Integer(2)]) 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]
>>> from sage.all import * >>> 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() # needs sage.graphs sage: alphacheck = R.coweight_space().roots() # needs sage.graphs sage: alpha[1].scalar(alphacheck[1]) # needs sage.graphs Traceback (most recent call last): ... ValueError: -Lambdacheck[1] + 2*Lambdacheck[2] - Lambdacheck[3] is not in the coroot space
>>> from sage.all import * >>> R = RootSystem(["A",Integer(3)]) >>> alpha = R.weight_space().roots() # needs sage.graphs >>> alphacheck = R.coweight_space().roots() # needs sage.graphs >>> alpha[Integer(1)].scalar(alphacheck[Integer(1)]) # needs sage.graphs Traceback (most recent call last): ... ValueError: -Lambdacheck[1] + 2*Lambdacheck[2] - Lambdacheck[3] is not in the coroot space
- to_ambient()[source]#
Maps
selfto the ambient space.EXAMPLES:
sage: mu = CartanType(['B',2]).root_system().weight_lattice().an_element(); mu 2*Lambda[1] + 2*Lambda[2] sage: mu.to_ambient() (3, 1)
>>> from sage.all import * >>> mu = CartanType(['B',Integer(2)]).root_system().weight_lattice().an_element(); mu 2*Lambda[1] + 2*Lambda[2] >>> 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()[source]#
Map
selfto 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[1] + 2*Lambda[2] sage: mu.to_weight_space() 2*Lambda[1] + 2*Lambda[2]
>>> from sage.all import * >>> mu = CartanType(['A',Integer(2)]).root_system().weight_lattice().an_element(); mu 2*Lambda[1] + 2*Lambda[2] >>> mu.to_weight_space() 2*Lambda[1] + 2*Lambda[2]