Littelmann paths

AUTHORS:

• Mark Shimozono, Anne Schilling (2012): Initial version

• Anne Schilling (2013): Implemented CrystalOfProjectedLevelZeroLSPaths

• Travis Scrimshaw (2016): Implemented InfinityCrystalOfLSPaths

class sage.combinat.crystals.littelmann_path.CrystalOfLSPaths(starting_weight, starting_weight_parent)

Bases: UniqueRepresentation, Parent

Crystal graph of LS paths generated from the straight-line path to a given weight.

INPUT:

• cartan_type – (optional) the Cartan type

• starting_weight – a weight; if cartan_type is given, then the weight should be given as a list of coefficients of the fundamental weights, otherwise it should be given in the weight_space basis; for affine highest weight crystals, one needs to use the extended weight space

The crystal class of piecewise linear paths in the weight space, generated from a straight-line path from the origin to a given element of the weight lattice.

EXAMPLES:

Sage

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space(extended = True).basis()
sage: B = crystals.LSPaths(La[2]-La[0]); B
The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]

sage: C = crystals.LSPaths(['A',2,1],[-1,0,1]); C
The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]
sage: B == C
True
sage: c = C.module_generators[0]; c
(-Lambda[0] + Lambda[2],)
sage: [c.f(i) for i in C.index_set()]
[None, None, (Lambda[1] - Lambda[2],)]

sage: R = C.R; R
Root system of type ['A', 2, 1]
sage: Lambda = R.weight_space().basis(); Lambda
Finite family {0: Lambda[0], 1: Lambda[1], 2: Lambda[2]}
sage: b=C(tuple([-Lambda[0]+Lambda[2]]))
sage: b==c
True
sage: b.f(2)
(Lambda[1] - Lambda[2],)

Python

>>> from sage.all import *
>>> R = RootSystem(['A',Integer(2),Integer(1)])
>>> La = R.weight_space(extended = True).basis()
>>> B = crystals.LSPaths(La[Integer(2)]-La[Integer(0)]); B
The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]

>>> C = crystals.LSPaths(['A',Integer(2),Integer(1)],[-Integer(1),Integer(0),Integer(1)]); C
The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]
>>> B == C
True
>>> c = C.module_generators[Integer(0)]; c
(-Lambda[0] + Lambda[2],)
>>> [c.f(i) for i in C.index_set()]
[None, None, (Lambda[1] - Lambda[2],)]

>>> R = C.R; R
Root system of type ['A', 2, 1]
>>> Lambda = R.weight_space().basis(); Lambda
Finite family {0: Lambda[0], 1: Lambda[1], 2: Lambda[2]}
>>> b=C(tuple([-Lambda[Integer(0)]+Lambda[Integer(2)]]))
>>> b==c
True
>>> b.f(Integer(2))
(Lambda[1] - Lambda[2],)

Sage Live

R = RootSystem(['A',2,1])
La = R.weight_space(extended = True).basis()
B = crystals.LSPaths(La[2]-La[0]); B
C = crystals.LSPaths(['A',2,1],[-1,0,1]); C
B == C
c = C.module_generators[0]; c
[c.f(i) for i in C.index_set()]
R = C.R; R
Lambda = R.weight_space().basis(); Lambda
b=C(tuple([-Lambda[0]+Lambda[2]]))
b==c
b.f(2)

For classical highest weight crystals, we can also compare the results with the tableaux implementation:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: sorted(C, key=str)
[(-2*Lambda[1] + Lambda[2],), (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2]),
 (-Lambda[1] + 2*Lambda[2],), (-Lambda[1] - Lambda[2],),
 (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]), (2*Lambda[1] - Lambda[2],),
 (Lambda[1] + Lambda[2],), (Lambda[1] - 2*Lambda[2],)]
sage: C.cardinality()
8
sage: B = crystals.Tableaux(['A',2],shape=[2,1])
sage: B.cardinality()
8
sage: B.digraph().is_isomorphic(C.digraph())
True

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> sorted(C, key=str)
[(-2*Lambda[1] + Lambda[2],), (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2]),
 (-Lambda[1] + 2*Lambda[2],), (-Lambda[1] - Lambda[2],),
 (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]), (2*Lambda[1] - Lambda[2],),
 (Lambda[1] + Lambda[2],), (Lambda[1] - 2*Lambda[2],)]
>>> C.cardinality()
8
>>> B = crystals.Tableaux(['A',Integer(2)],shape=[Integer(2),Integer(1)])
>>> B.cardinality()
8
>>> B.digraph().is_isomorphic(C.digraph())
True

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
sorted(C, key=str)
C.cardinality()
B = crystals.Tableaux(['A',2],shape=[2,1])
B.cardinality()
B.digraph().is_isomorphic(C.digraph())

Make sure you use the weight space and not the weight lattice for your weights:

Sage

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_lattice(extended = True).basis()
sage: B = crystals.LSPaths(La[2]); B
Traceback (most recent call last):
...
ValueError: use the weight space, rather than weight lattice for your weights

Python

>>> from sage.all import *
>>> R = RootSystem(['A',Integer(2),Integer(1)])
>>> La = R.weight_lattice(extended = True).basis()
>>> B = crystals.LSPaths(La[Integer(2)]); B
Traceback (most recent call last):
...
ValueError: use the weight space, rather than weight lattice for your weights

Sage Live

R = RootSystem(['A',2,1])
La = R.weight_lattice(extended = True).basis()
B = crystals.LSPaths(La[2]); B

REFERENCES:

• [Li1995b]

class Element

Bases: ElementWrapper

A Littelmann path (crystal element).

compress()

Merge consecutive positively parallel steps present in self.

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: Lambda = C.R.weight_space().fundamental_weights(); Lambda
Finite family {1: Lambda[1], 2: Lambda[2]}
sage: c = C(tuple([1/2*Lambda[1]+1/2*Lambda[2], 1/2*Lambda[1]+1/2*Lambda[2]]))
sage: c.compress()
(Lambda[1] + Lambda[2],)

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> Lambda = C.R.weight_space().fundamental_weights(); Lambda
Finite family {1: Lambda[1], 2: Lambda[2]}
>>> c = C(tuple([Integer(1)/Integer(2)*Lambda[Integer(1)]+Integer(1)/Integer(2)*Lambda[Integer(2)], Integer(1)/Integer(2)*Lambda[Integer(1)]+Integer(1)/Integer(2)*Lambda[Integer(2)]]))
>>> c.compress()
(Lambda[1] + Lambda[2],)

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
Lambda = C.R.weight_space().fundamental_weights(); Lambda
c = C(tuple([1/2*Lambda[1]+1/2*Lambda[2], 1/2*Lambda[1]+1/2*Lambda[2]]))
c.compress()

dualize()

Return the dualized path of self.

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: for c in C:
....:     print("{} {}".format(c, c.dualize()))
(Lambda[1] + Lambda[2],) (-Lambda[1] - Lambda[2],)
(-Lambda[1] + 2*Lambda[2],) (Lambda[1] - 2*Lambda[2],)
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]) (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
(Lambda[1] - 2*Lambda[2],) (-Lambda[1] + 2*Lambda[2],)
(-Lambda[1] - Lambda[2],) (Lambda[1] + Lambda[2],)
(2*Lambda[1] - Lambda[2],) (-2*Lambda[1] + Lambda[2],)
(-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2]) (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2])
(-2*Lambda[1] + Lambda[2],) (2*Lambda[1] - Lambda[2],)

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> for c in C:
...     print("{} {}".format(c, c.dualize()))
(Lambda[1] + Lambda[2],) (-Lambda[1] - Lambda[2],)
(-Lambda[1] + 2*Lambda[2],) (Lambda[1] - 2*Lambda[2],)
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]) (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
(Lambda[1] - 2*Lambda[2],) (-Lambda[1] + 2*Lambda[2],)
(-Lambda[1] - Lambda[2],) (Lambda[1] + Lambda[2],)
(2*Lambda[1] - Lambda[2],) (-2*Lambda[1] + Lambda[2],)
(-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2]) (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2])
(-2*Lambda[1] + Lambda[2],) (2*Lambda[1] - Lambda[2],)

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
for c in C:
    print("{} {}".format(c, c.dualize()))

e(i, power=1, to_string_end=False, length_only=False)

Return the \(i\)-th crystal raising operator on self.

INPUT:

• i – element of the index set of the underlying root system

• power – positive integer (default: 1); specifies the power of the raising operator to be applied

• to_string_end – boolean (default: False); if True, returns the dominant end of the \(i\)-string of self

• length_only – boolean; if True, returns the distance to the dominant end of the \(i\)-string of self

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: c = C[2]; c
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
sage: c.e(1)
sage: c.e(2)
(-Lambda[1] + 2*Lambda[2],)
sage: c.e(2,to_string_end=True)
(-Lambda[1] + 2*Lambda[2],)
sage: c.e(1,to_string_end=True)
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
sage: c.e(1,length_only=True)
0

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> c = C[Integer(2)]; c
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
>>> c.e(Integer(1))
>>> c.e(Integer(2))
(-Lambda[1] + 2*Lambda[2],)
>>> c.e(Integer(2),to_string_end=True)
(-Lambda[1] + 2*Lambda[2],)
>>> c.e(Integer(1),to_string_end=True)
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
>>> c.e(Integer(1),length_only=True)
0

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
c = C[2]; c
c.e(1)
c.e(2)
c.e(2,to_string_end=True)
c.e(1,to_string_end=True)
c.e(1,length_only=True)

endpoint()

Compute the endpoint of self.

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: b = C.module_generators[0]
sage: b.endpoint()
Lambda[1] + Lambda[2]
sage: b.f_string([1,2,2,1])
(-Lambda[1] - Lambda[2],)
sage: b.f_string([1,2,2,1]).endpoint()
-Lambda[1] - Lambda[2]
sage: b.f_string([1,2])
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
sage: b.f_string([1,2]).endpoint()
0
sage: b = C([])
sage: b.endpoint()
0

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> b = C.module_generators[Integer(0)]
>>> b.endpoint()
Lambda[1] + Lambda[2]
>>> b.f_string([Integer(1),Integer(2),Integer(2),Integer(1)])
(-Lambda[1] - Lambda[2],)
>>> b.f_string([Integer(1),Integer(2),Integer(2),Integer(1)]).endpoint()
-Lambda[1] - Lambda[2]
>>> b.f_string([Integer(1),Integer(2)])
(1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
>>> b.f_string([Integer(1),Integer(2)]).endpoint()
0
>>> b = C([])
>>> b.endpoint()
0

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
b = C.module_generators[0]
b.endpoint()
b.f_string([1,2,2,1])
b.f_string([1,2,2,1]).endpoint()
b.f_string([1,2])
b.f_string([1,2]).endpoint()
b = C([])
b.endpoint()

epsilon(i)

Return the distance to the beginning of the \(i\)-string.

This method overrides the generic implementation in the category of crystals since this computation is more efficient.

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: [c.epsilon(1) for c in C]
[0, 1, 0, 0, 1, 0, 1, 2]
sage: [c.epsilon(2) for c in C]
[0, 0, 1, 2, 1, 1, 0, 0]

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> [c.epsilon(Integer(1)) for c in C]
[0, 1, 0, 0, 1, 0, 1, 2]
>>> [c.epsilon(Integer(2)) for c in C]
[0, 0, 1, 2, 1, 1, 0, 0]

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
[c.epsilon(1) for c in C]
[c.epsilon(2) for c in C]

f(i, power=1, to_string_end=False, length_only=False)

Return the \(i\)-th crystal lowering operator on self.

INPUT:

• i – element of the index set of the underlying root system

• power – positive integer (default: 1); specifies the power of the lowering operator to be applied

• to_string_end – boolean (default: False); if True, returns the anti-dominant end of the \(i\)-string of self

• length_only – boolean; if True, returns the distance to the anti-dominant end of the \(i\)-string of self

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: c = C.module_generators[0]
sage: c.f(1)
(-Lambda[1] + 2*Lambda[2],)
sage: c.f(1,power=2)
sage: c.f(2)
(2*Lambda[1] - Lambda[2],)
sage: c.f(2,to_string_end=True)
(2*Lambda[1] - Lambda[2],)
sage: c.f(2,length_only=True)
1

sage: C = crystals.LSPaths(['A',2,1],[-1,-1,2])
sage: c = C.module_generators[0]
sage: c.f(2,power=2)
(Lambda[0] + Lambda[1] - 2*Lambda[2],)

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> c = C.module_generators[Integer(0)]
>>> c.f(Integer(1))
(-Lambda[1] + 2*Lambda[2],)
>>> c.f(Integer(1),power=Integer(2))
>>> c.f(Integer(2))
(2*Lambda[1] - Lambda[2],)
>>> c.f(Integer(2),to_string_end=True)
(2*Lambda[1] - Lambda[2],)
>>> c.f(Integer(2),length_only=True)
1

>>> C = crystals.LSPaths(['A',Integer(2),Integer(1)],[-Integer(1),-Integer(1),Integer(2)])
>>> c = C.module_generators[Integer(0)]
>>> c.f(Integer(2),power=Integer(2))
(Lambda[0] + Lambda[1] - 2*Lambda[2],)

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
c = C.module_generators[0]
c.f(1)
c.f(1,power=2)
c.f(2)
c.f(2,to_string_end=True)
c.f(2,length_only=True)
C = crystals.LSPaths(['A',2,1],[-1,-1,2])
c = C.module_generators[0]
c.f(2,power=2)

phi(i)

Return the distance to the end of the \(i\)-string.

This method overrides the generic implementation in the category of crystals since this computation is more efficient.

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: [c.phi(1) for c in C]
[1, 0, 0, 1, 0, 2, 1, 0]
sage: [c.phi(2) for c in C]
[1, 2, 1, 0, 0, 0, 0, 1]

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> [c.phi(Integer(1)) for c in C]
[1, 0, 0, 1, 0, 2, 1, 0]
>>> [c.phi(Integer(2)) for c in C]
[1, 2, 1, 0, 0, 0, 0, 1]

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
[c.phi(1) for c in C]
[c.phi(2) for c in C]

reflect_step(which_step, i)

Apply the \(i\)-th simple reflection to the indicated step in self.

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: b = C.module_generators[0]
sage: b.reflect_step(0,1)
(-Lambda[1] + 2*Lambda[2],)
sage: b.reflect_step(0,2)
(2*Lambda[1] - Lambda[2],)

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> b = C.module_generators[Integer(0)]
>>> b.reflect_step(Integer(0),Integer(1))
(-Lambda[1] + 2*Lambda[2],)
>>> b.reflect_step(Integer(0),Integer(2))
(2*Lambda[1] - Lambda[2],)

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
b = C.module_generators[0]
b.reflect_step(0,1)
b.reflect_step(0,2)

s(i)

Compute the reflection of self along the \(i\)-string.

This method is more efficient than the generic implementation since it uses powers of \(e\) and \(f\) in the Littelmann model directly.

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: c = C.module_generators[0]
sage: c.s(1)
(-Lambda[1] + 2*Lambda[2],)
sage: c.s(2)
(2*Lambda[1] - Lambda[2],)

sage: C = crystals.LSPaths(['A',2,1],[-1,0,1])
sage: c = C.module_generators[0]; c
(-Lambda[0] + Lambda[2],)
sage: c.s(2)
(Lambda[1] - Lambda[2],)
sage: c.s(1)
(-Lambda[0] + Lambda[2],)
sage: c.f(2).s(1)
(Lambda[0] - Lambda[1],)

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> c = C.module_generators[Integer(0)]
>>> c.s(Integer(1))
(-Lambda[1] + 2*Lambda[2],)
>>> c.s(Integer(2))
(2*Lambda[1] - Lambda[2],)

>>> C = crystals.LSPaths(['A',Integer(2),Integer(1)],[-Integer(1),Integer(0),Integer(1)])
>>> c = C.module_generators[Integer(0)]; c
(-Lambda[0] + Lambda[2],)
>>> c.s(Integer(2))
(Lambda[1] - Lambda[2],)
>>> c.s(Integer(1))
(-Lambda[0] + Lambda[2],)
>>> c.f(Integer(2)).s(Integer(1))
(Lambda[0] - Lambda[1],)

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
c = C.module_generators[0]
c.s(1)
c.s(2)
C = crystals.LSPaths(['A',2,1],[-1,0,1])
c = C.module_generators[0]; c
c.s(2)
c.s(1)
c.f(2).s(1)

split_step(which_step, r)

Split the indicated step into two parallel steps of relative lengths \(r\) and \(1-r\).

INPUT:

• which_step – a position in the tuple self

• r – a rational number between 0 and 1

EXAMPLES:

Sage

sage: C = crystals.LSPaths(['A',2],[1,1])
sage: b = C.module_generators[0]
sage: b.split_step(0,1/3)
(1/3*Lambda[1] + 1/3*Lambda[2], 2/3*Lambda[1] + 2/3*Lambda[2])

Python

>>> from sage.all import *
>>> C = crystals.LSPaths(['A',Integer(2)],[Integer(1),Integer(1)])
>>> b = C.module_generators[Integer(0)]
>>> b.split_step(Integer(0),Integer(1)/Integer(3))
(1/3*Lambda[1] + 1/3*Lambda[2], 2/3*Lambda[1] + 2/3*Lambda[2])

Sage Live

C = crystals.LSPaths(['A',2],[1,1])
b = C.module_generators[0]
b.split_step(0,1/3)

weight()

Return the weight of self.

EXAMPLES:

Sage

sage: B = crystals.LSPaths(['A',1,1],[1,0])
sage: b = B.highest_weight_vector()
sage: b.f(0).weight()
-Lambda[0] + 2*Lambda[1] - delta

Python

>>> from sage.all import *
>>> B = crystals.LSPaths(['A',Integer(1),Integer(1)],[Integer(1),Integer(0)])
>>> b = B.highest_weight_vector()
>>> b.f(Integer(0)).weight()
-Lambda[0] + 2*Lambda[1] - delta

Sage Live

B = crystals.LSPaths(['A',1,1],[1,0])
b = B.highest_weight_vector()
b.f(0).weight()

weight_lattice_realization()

Return weight lattice realization of self.

EXAMPLES:

Sage

sage: B = crystals.LSPaths(['B',3],[1,1,0])
sage: B.weight_lattice_realization()
Weight space over the Rational Field of the Root system of type ['B', 3]
sage: B = crystals.LSPaths(['B',3,1],[1,1,1,0])
sage: B.weight_lattice_realization()
Extended weight space over the Rational Field of the Root system of type ['B', 3, 1]

Python

>>> from sage.all import *
>>> B = crystals.LSPaths(['B',Integer(3)],[Integer(1),Integer(1),Integer(0)])
>>> B.weight_lattice_realization()
Weight space over the Rational Field of the Root system of type ['B', 3]
>>> B = crystals.LSPaths(['B',Integer(3),Integer(1)],[Integer(1),Integer(1),Integer(1),Integer(0)])
>>> B.weight_lattice_realization()
Extended weight space over the Rational Field of the Root system of type ['B', 3, 1]

Sage Live

B = crystals.LSPaths(['B',3],[1,1,0])
B.weight_lattice_realization()
B = crystals.LSPaths(['B',3,1],[1,1,1,0])
B.weight_lattice_realization()

class sage.combinat.crystals.littelmann_path.CrystalOfProjectedLevelZeroLSPaths(starting_weight, starting_weight_parent)

Bases: CrystalOfLSPaths

Crystal of projected level zero LS paths.

INPUT:

• weight – a dominant weight of the weight space of an affine Kac-Moody root system

When weight is just a single fundamental weight \(\Lambda_r\), this crystal is isomorphic to a Kirillov-Reshetikhin (KR) crystal, see also sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinFromLSPaths(). For general weights, it is isomorphic to a tensor product of single-column KR crystals.

EXAMPLES:

Sage

sage: R = RootSystem(['C',3,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
sage: LS.cardinality()
84
sage: GLS = LS.digraph()

sage: K1 = crystals.KirillovReshetikhin(['C',3,1],1,1)
sage: K3 = crystals.KirillovReshetikhin(['C',3,1],3,1)
sage: T = crystals.TensorProduct(K3,K1)
sage: T.cardinality()
84
sage: GT = T.digraph() # long time
sage: GLS.is_isomorphic(GT, edge_labels = True) # long time
True

Python

>>> from sage.all import *
>>> R = RootSystem(['C',Integer(3),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)]+La[Integer(3)])
>>> LS.cardinality()
84
>>> GLS = LS.digraph()

>>> K1 = crystals.KirillovReshetikhin(['C',Integer(3),Integer(1)],Integer(1),Integer(1))
>>> K3 = crystals.KirillovReshetikhin(['C',Integer(3),Integer(1)],Integer(3),Integer(1))
>>> T = crystals.TensorProduct(K3,K1)
>>> T.cardinality()
84
>>> GT = T.digraph() # long time
>>> GLS.is_isomorphic(GT, edge_labels = True) # long time
True

Sage Live

R = RootSystem(['C',3,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
LS.cardinality()
GLS = LS.digraph()
K1 = crystals.KirillovReshetikhin(['C',3,1],1,1)
K3 = crystals.KirillovReshetikhin(['C',3,1],3,1)
T = crystals.TensorProduct(K3,K1)
T.cardinality()
GT = T.digraph() # long time
GLS.is_isomorphic(GT, edge_labels = True) # long time

class Element

Bases: Element

Element of a crystal of projected level zero LS paths.

energy_function()

Return the energy function of self.

The energy function \(D(\pi)\) of the level zero LS path \(\pi \in \mathbb{B}_\mathrm{cl}(\lambda)\) requires a series of definitions; for simplicity the root system is assumed to be untwisted affine.

The LS path \(\pi\) is a piecewise linear map from the unit interval \([0,1]\) to the weight lattice. It is specified by “times” \(0 = \sigma_0 < \sigma_1 < \dotsm < \sigma_s = 1\) and “direction vectors” \(x_u \lambda\) where \(x_u \in W / W_J\) for \(1 \le u \le s\), and \(W_J\) is the stabilizer of \(\lambda\) in the finite Weyl group \(W\). Precisely,

\[\pi(t) = \sum_{u'=1}^{u-1} (\sigma_{u'}-\sigma_{u'-1}) x_{u'} \lambda + (t-\sigma_{u-1}) x_{u} \lambda\]

for \(1 \le u \le s\) and \(\sigma_{u-1} \le t \le \sigma_{u}\).

For any \(x,y \in W / W_J\), let

\[d: x = w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1} \stackrel{\beta_{2}}{\leftarrow} \cdots \stackrel{\beta_{n}}{\leftarrow} w_{n}=y\]

be a shortest directed path in the parabolic quantum Bruhat graph. Define

\[\begin{split}\mathrm{wt}(d) := \sum_{\substack{1 \le k \le n \\ \ell(w_{k-1}) < \ell(w_k)}} \beta_{k}^{\vee}.\end{split}\]

It can be shown that \(\mathrm{wt}(d)\) depends only on \(x,y\); call its value \(\mathrm{wt}(x,y)\). The energy function \(D(\pi)\) is defined by

\[D(\pi) = -\sum_{u=1}^{s-1} (1-\sigma_{u}) \langle \lambda, \mathrm{wt}(x_u,x_{u+1}) \rangle.\]

For more information, see [LNSSS2013].

Note

In the dual-of-untwisted case the parabolic quantum Bruhat graph that is used is obtained by exchanging the roles of roots and coroots. Moreover, in the computation of the pairing the short roots must be doubled (or tripled for type \(G\)). This factor is determined by the translation factor of the corresponding root. Type \(BC\) is viewed as untwisted type, whereas the dual of \(BC\) is viewed as twisted. Except for the untwisted cases, these formulas are currently still conjectural.

EXAMPLES:

Sage

sage: R = RootSystem(['C',3,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
sage: b = LS.module_generators[0]
sage: c = b.f(1).f(3).f(2)
sage: c.energy_function()
0
sage: c=b.e(0)
sage: c.energy_function()
1

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
sage: b = LS.module_generators[0]
sage: c = b.e(0)
sage: c.energy_function()
1
sage: for c in sorted(LS, key=str):
....:     print("{} {}".format(c,c.energy_function()))
(-2*Lambda[0] + 2*Lambda[1],)                    0
(-2*Lambda[1] + 2*Lambda[2],)                    0
(-Lambda[0] + Lambda[1], -Lambda[1] + Lambda[2]) 1
(-Lambda[0] + Lambda[1], Lambda[0] - Lambda[2])  1
(-Lambda[1] + Lambda[2], -Lambda[0] + Lambda[1]) 0
(-Lambda[1] + Lambda[2], Lambda[0] - Lambda[2])  1
(2*Lambda[0] - 2*Lambda[2],)                     0
(Lambda[0] - Lambda[2], -Lambda[0] + Lambda[1])  0
(Lambda[0] - Lambda[2], -Lambda[1] + Lambda[2])  0

Python

>>> from sage.all import *
>>> R = RootSystem(['C',Integer(3),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)]+La[Integer(3)])
>>> b = LS.module_generators[Integer(0)]
>>> c = b.f(Integer(1)).f(Integer(3)).f(Integer(2))
>>> c.energy_function()
0
>>> c=b.e(Integer(0))
>>> c.energy_function()
1

>>> R = RootSystem(['A',Integer(2),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(Integer(2)*La[Integer(1)])
>>> b = LS.module_generators[Integer(0)]
>>> c = b.e(Integer(0))
>>> c.energy_function()
1
>>> for c in sorted(LS, key=str):
...     print("{} {}".format(c,c.energy_function()))
(-2*Lambda[0] + 2*Lambda[1],)                    0
(-2*Lambda[1] + 2*Lambda[2],)                    0
(-Lambda[0] + Lambda[1], -Lambda[1] + Lambda[2]) 1
(-Lambda[0] + Lambda[1], Lambda[0] - Lambda[2])  1
(-Lambda[1] + Lambda[2], -Lambda[0] + Lambda[1]) 0
(-Lambda[1] + Lambda[2], Lambda[0] - Lambda[2])  1
(2*Lambda[0] - 2*Lambda[2],)                     0
(Lambda[0] - Lambda[2], -Lambda[0] + Lambda[1])  0
(Lambda[0] - Lambda[2], -Lambda[1] + Lambda[2])  0

Sage Live

R = RootSystem(['C',3,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
b = LS.module_generators[0]
c = b.f(1).f(3).f(2)
c.energy_function()
c=b.e(0)
c.energy_function()
R = RootSystem(['A',2,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
b = LS.module_generators[0]
c = b.e(0)
c.energy_function()
for c in sorted(LS, key=str):
    print("{} {}".format(c,c.energy_function()))

The next test checks that the energy function is constant on classically connected components:

Sage

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
sage: G = LS.digraph(index_set=[1,2])
sage: C = G.connected_components(sort=False)
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True, True]

sage: R = RootSystem(['D',4,2])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[2])
sage: J = R.cartan_type().classical().index_set()
sage: hw = [x for x in LS if x.is_highest_weight(J)]
sage: [(x.weight(), x.energy_function()) for x in hw]
[(-2*Lambda[0] + Lambda[2], 0), (-2*Lambda[0] + Lambda[1], 1), (0, 2)]
sage: G = LS.digraph(index_set=J)
sage: C = G.connected_components(sort=False)
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True]

sage: R = RootSystem(CartanType(['G',2,1]).dual())
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
sage: G = LS.digraph(index_set=[1,2])
sage: C = G.connected_components(sort=False)
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]

sage: ct = CartanType(['BC',2,2]).dual()
sage: R = RootSystem(ct)
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
sage: G = LS.digraph(index_set=R.cartan_type().classical().index_set())
sage: C = G.connected_components(sort=False)
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
[True, True, True, True, True, True, True, True, True, True, True]

sage: R = RootSystem(['BC',2,2])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
sage: G = LS.digraph(index_set=R.cartan_type().classical().index_set())
sage: C = G.connected_components(sort=False)
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]

Python

>>> from sage.all import *
>>> R = RootSystem(['A',Integer(2),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(Integer(2)*La[Integer(1)]+La[Integer(2)])
>>> G = LS.digraph(index_set=[Integer(1),Integer(2)])
>>> C = G.connected_components(sort=False)
>>> [all(c[Integer(0)].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True, True]

>>> R = RootSystem(['D',Integer(4),Integer(2)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(2)])
>>> J = R.cartan_type().classical().index_set()
>>> hw = [x for x in LS if x.is_highest_weight(J)]
>>> [(x.weight(), x.energy_function()) for x in hw]
[(-2*Lambda[0] + Lambda[2], 0), (-2*Lambda[0] + Lambda[1], 1), (0, 2)]
>>> G = LS.digraph(index_set=J)
>>> C = G.connected_components(sort=False)
>>> [all(c[Integer(0)].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True]

>>> R = RootSystem(CartanType(['G',Integer(2),Integer(1)]).dual())
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)]+La[Integer(2)])
>>> G = LS.digraph(index_set=[Integer(1),Integer(2)])
>>> C = G.connected_components(sort=False)
>>> [all(c[Integer(0)].energy_function()==a.energy_function() for a in c) for c in C] # long time
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]

>>> ct = CartanType(['BC',Integer(2),Integer(2)]).dual()
>>> R = RootSystem(ct)
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(Integer(2)*La[Integer(1)]+La[Integer(2)])
>>> G = LS.digraph(index_set=R.cartan_type().classical().index_set())
>>> C = G.connected_components(sort=False)
>>> [all(c[Integer(0)].energy_function()==a.energy_function() for a in c) for c in C] # long time
[True, True, True, True, True, True, True, True, True, True, True]

>>> R = RootSystem(['BC',Integer(2),Integer(2)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(Integer(2)*La[Integer(1)]+La[Integer(2)])
>>> G = LS.digraph(index_set=R.cartan_type().classical().index_set())
>>> C = G.connected_components(sort=False)
>>> [all(c[Integer(0)].energy_function()==a.energy_function() for a in c) for c in C] # long time
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]

Sage Live

R = RootSystem(['A',2,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
G = LS.digraph(index_set=[1,2])
C = G.connected_components(sort=False)
[all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
R = RootSystem(['D',4,2])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[2])
J = R.cartan_type().classical().index_set()
hw = [x for x in LS if x.is_highest_weight(J)]
[(x.weight(), x.energy_function()) for x in hw]
G = LS.digraph(index_set=J)
C = G.connected_components(sort=False)
[all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
R = RootSystem(CartanType(['G',2,1]).dual())
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
G = LS.digraph(index_set=[1,2])
C = G.connected_components(sort=False)
[all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
ct = CartanType(['BC',2,2]).dual()
R = RootSystem(ct)
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
G = LS.digraph(index_set=R.cartan_type().classical().index_set())
C = G.connected_components(sort=False)
[all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
R = RootSystem(['BC',2,2])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
G = LS.digraph(index_set=R.cartan_type().classical().index_set())
C = G.connected_components(sort=False)
[all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time

scalar_factors()

Return the scalar factors for self.

Each LS path (or self) can be written as a piecewise linear map

\[\pi(t) = \sum_{u'=1}^{u-1} (\sigma_{u'} - \sigma_{u'-1}) \nu_{u'} + (t-\sigma_{u-1}) \nu_{u}\]

for \(0 < \sigma_1 < \sigma_2 < \cdots < \sigma_s=1\) and \(\sigma_{u-1} \le t \le \sigma_{u}\) and \(1 \le u \le s\). This method returns the tuple of \((\sigma_1,\ldots,\sigma_s)\).

EXAMPLES:

Sage

sage: R = RootSystem(['C',3,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
sage: b = LS.module_generators[0]
sage: b.scalar_factors()
[1]
sage: c = b.f(1).f(3).f(2)
sage: c.scalar_factors()
[1/3, 1]

Python

>>> from sage.all import *
>>> R = RootSystem(['C',Integer(3),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)]+La[Integer(3)])
>>> b = LS.module_generators[Integer(0)]
>>> b.scalar_factors()
[1]
>>> c = b.f(Integer(1)).f(Integer(3)).f(Integer(2))
>>> c.scalar_factors()
[1/3, 1]

Sage Live

R = RootSystem(['C',3,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
b = LS.module_generators[0]
b.scalar_factors()
c = b.f(1).f(3).f(2)
c.scalar_factors()

weyl_group_representation()

Transform the weights in the LS path self to elements in the Weyl group.

Each LS path can be written as the piecewise linear map:

\[\pi(t) = \sum_{u'=1}^{u-1} (\sigma_{u'} - \sigma_{u'-1}) \nu_{u'} + (t-\sigma_{u-1}) \nu_{u}\]

for \(0 < \sigma_1 < \sigma_2 < \cdots < \sigma_s = 1\) and \(\sigma_{u-1} \le t \le \sigma_{u}\) and \(1 \le u \le s\). Each weight \(\nu_u\) is also associated to a Weyl group element. This method returns the list of Weyl group elements associated to the \(\nu_u\) for \(1\le u\le s\).

EXAMPLES:

Sage

sage: R = RootSystem(['C',3,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
sage: b = LS.module_generators[0]
sage: c = b.f(1).f(3).f(2)
sage: c.weyl_group_representation()
[s2*s1*s3, s1*s3]

Python

>>> from sage.all import *
>>> R = RootSystem(['C',Integer(3),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)]+La[Integer(3)])
>>> b = LS.module_generators[Integer(0)]
>>> c = b.f(Integer(1)).f(Integer(3)).f(Integer(2))
>>> c.weyl_group_representation()
[s2*s1*s3, s1*s3]

Sage Live

R = RootSystem(['C',3,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
b = LS.module_generators[0]
c = b.f(1).f(3).f(2)
c.weyl_group_representation()

classically_highest_weight_vectors()

Return the classically highest weight vectors of self.

EXAMPLES:

Sage

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
sage: LS.classically_highest_weight_vectors()
((-2*Lambda[0] + 2*Lambda[1],),
 (-Lambda[0] + Lambda[1], -Lambda[1] + Lambda[2]))

Python

>>> from sage.all import *
>>> R = RootSystem(['A',Integer(2),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(Integer(2)*La[Integer(1)])
>>> LS.classically_highest_weight_vectors()
((-2*Lambda[0] + 2*Lambda[1],),
 (-Lambda[0] + Lambda[1], -Lambda[1] + Lambda[2]))

Sage Live

R = RootSystem(['A',2,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
LS.classically_highest_weight_vectors()

is_perfect(level=1)

Check whether the crystal self is perfect (of level level).

INPUT:

• level – (default: 1) positive integer

A crystal \(\mathcal{B}\) is perfect of level \(\ell\) if:

1. \(\mathcal{B}\) is isomorphic to the crystal graph of a finite-dimensional \(U_q^{'}(\mathfrak{g})\)-module.

2. \(\mathcal{B}\otimes \mathcal{B}\) is connected.

3. There exists a \(\lambda\in X\), such that \(\mathrm{wt}(\mathcal{B}) \subset \lambda + \sum_{i\in I} \ZZ_{\le 0} \alpha_i\) and there is a unique element in \(\mathcal{B}\) of classical weight \(\lambda\).

4. For all \(b \in \mathcal{B}\), \(\mathrm{level}(\varepsilon (b)) \geq \ell\).

5. For all \(\Lambda\) dominant weights of level \(\ell\), there exist unique elements \(b_{\Lambda}, b^{\Lambda} \in \mathcal{B}\), such that \(\varepsilon (b_{\Lambda}) = \Lambda = \varphi(b^{\Lambda})\).

Points (1)-(3) are known to hold. This method checks points (4) and (5).

EXAMPLES:

Sage

sage: C = CartanType(['C',2,1])
sage: R = RootSystem(C)
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1])
sage: LS.is_perfect()
False
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[2])
sage: LS.is_perfect()
True

sage: C = CartanType(['E',6,1])
sage: R = RootSystem(C)
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1])
sage: LS.is_perfect()
True
sage: LS.is_perfect(2)
False

sage: C = CartanType(['D',4,1])
sage: R = RootSystem(C)
sage: La = R.weight_space().basis()
sage: all(crystals.ProjectedLevelZeroLSPaths(La[i]).is_perfect() for i in [1,2,3,4])
True

sage: C = CartanType(['A',6,2])
sage: R = RootSystem(C)
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
sage: LS.is_perfect()
True
sage: LS.is_perfect(2)
False

Python

>>> from sage.all import *
>>> C = CartanType(['C',Integer(2),Integer(1)])
>>> R = RootSystem(C)
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)])
>>> LS.is_perfect()
False
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(2)])
>>> LS.is_perfect()
True

>>> C = CartanType(['E',Integer(6),Integer(1)])
>>> R = RootSystem(C)
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)])
>>> LS.is_perfect()
True
>>> LS.is_perfect(Integer(2))
False

>>> C = CartanType(['D',Integer(4),Integer(1)])
>>> R = RootSystem(C)
>>> La = R.weight_space().basis()
>>> all(crystals.ProjectedLevelZeroLSPaths(La[i]).is_perfect() for i in [Integer(1),Integer(2),Integer(3),Integer(4)])
True

>>> C = CartanType(['A',Integer(6),Integer(2)])
>>> R = RootSystem(C)
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(La[Integer(1)]+La[Integer(2)])
>>> LS.is_perfect()
True
>>> LS.is_perfect(Integer(2))
False

Sage Live

C = CartanType(['C',2,1])
R = RootSystem(C)
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1])
LS.is_perfect()
LS = crystals.ProjectedLevelZeroLSPaths(La[2])
LS.is_perfect()
C = CartanType(['E',6,1])
R = RootSystem(C)
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1])
LS.is_perfect()
LS.is_perfect(2)
C = CartanType(['D',4,1])
R = RootSystem(C)
La = R.weight_space().basis()
all(crystals.ProjectedLevelZeroLSPaths(La[i]).is_perfect() for i in [1,2,3,4])
C = CartanType(['A',6,2])
R = RootSystem(C)
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
LS.is_perfect()
LS.is_perfect(2)

maximal_vector()

Return the maximal vector of self.

EXAMPLES:

Sage

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
sage: LS.maximal_vector()
(-3*Lambda[0] + 2*Lambda[1] + Lambda[2],)

Python

>>> from sage.all import *
>>> R = RootSystem(['A',Integer(2),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(Integer(2)*La[Integer(1)]+La[Integer(2)])
>>> LS.maximal_vector()
(-3*Lambda[0] + 2*Lambda[1] + Lambda[2],)

Sage Live

R = RootSystem(['A',2,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
LS.maximal_vector()

one_dimensional_configuration_sum(q=None, group_components=True)

Compute the one-dimensional configuration sum.

INPUT:

• q – (default: None) a variable or None; if None, a variable q is set in the code

• group_components – boolean (default: True); if True, then the terms are grouped by classical component

The one-dimensional configuration sum is the sum of the weights of all elements in the crystal weighted by the energy function. For untwisted types it uses the parabolic quantum Bruhat graph, see [LNSSS2013]. In the dual-of-untwisted case, the parabolic quantum Bruhat graph is defined by exchanging the roles of roots and coroots (which is still conjectural at this point).

EXAMPLES:

Sage

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
sage: LS.one_dimensional_configuration_sum() # long time
B[-2*Lambda[1] + 2*Lambda[2]] + (q+1)*B[-Lambda[1]]
 + (q+1)*B[Lambda[1] - Lambda[2]] + B[2*Lambda[1]]
 + B[-2*Lambda[2]] + (q+1)*B[Lambda[2]]
sage: R.<t> = ZZ[]
sage: LS.one_dimensional_configuration_sum(t, False) # long time
B[-2*Lambda[1] + 2*Lambda[2]] + (t+1)*B[-Lambda[1]]
 + (t+1)*B[Lambda[1] - Lambda[2]] + B[2*Lambda[1]]
 + B[-2*Lambda[2]] + (t+1)*B[Lambda[2]]

Python

>>> from sage.all import *
>>> R = RootSystem(['A',Integer(2),Integer(1)])
>>> La = R.weight_space().basis()
>>> LS = crystals.ProjectedLevelZeroLSPaths(Integer(2)*La[Integer(1)])
>>> LS.one_dimensional_configuration_sum() # long time
B[-2*Lambda[1] + 2*Lambda[2]] + (q+1)*B[-Lambda[1]]
 + (q+1)*B[Lambda[1] - Lambda[2]] + B[2*Lambda[1]]
 + B[-2*Lambda[2]] + (q+1)*B[Lambda[2]]
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> LS.one_dimensional_configuration_sum(t, False) # long time
B[-2*Lambda[1] + 2*Lambda[2]] + (t+1)*B[-Lambda[1]]
 + (t+1)*B[Lambda[1] - Lambda[2]] + B[2*Lambda[1]]
 + B[-2*Lambda[2]] + (t+1)*B[Lambda[2]]

Sage Live

R = RootSystem(['A',2,1])
La = R.weight_space().basis()
LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
LS.one_dimensional_configuration_sum() # long time
R.<t> = ZZ[]
LS.one_dimensional_configuration_sum(t, False) # long time

class sage.combinat.crystals.littelmann_path.InfinityCrystalOfLSPaths(cartan_type)

Bases: UniqueRepresentation, Parent

LS path model for \(\mathcal{B}(\infty)\).

Elements of \(\mathcal{B}(\infty)\) are equivalence classes of paths \([\pi]\) in \(\mathcal{B}(k\rho)\) for \(k\gg 0\), where \(\rho\) is the Weyl vector. A canonical representative for an element of \(\mathcal{B}(\infty)\) is chosen by taking \(k\) to be minimal such that the endpoint of \(\pi\) is strictly dominant but its representative in \(\mathcal{B}((k-1)\rho)\) is on the wall of the dominant chamber.

REFERENCES:

• [LZ2011]

class Element

Bases: Element

e(i, power=1, length_only=False)

Return the \(i\)-th crystal raising operator on self.

INPUT:

• i – element of the index set

• power – (default: 1) positive integer; specifies the power of the lowering operator to be applied

• length_only – boolean (default: False); if True, then return the distance to the anti-dominant end of the \(i\)-string of self

EXAMPLES:

Sage

sage: B = crystals.infinity.LSPaths(['B',3,1])
sage: mg = B.module_generator()
sage: mg.e(0)
sage: mg.e(1)
sage: mg.e(2)
sage: x = mg.f_string([1,0,2,1,0,2,1,1,0])
sage: all(x.f(i).e(i) == x for i in B.index_set())
True
sage: all(x.e(i).f(i) == x for i in B.index_set() if x.epsilon(i) > 0)
True

Python

>>> from sage.all import *
>>> B = crystals.infinity.LSPaths(['B',Integer(3),Integer(1)])
>>> mg = B.module_generator()
>>> mg.e(Integer(0))
>>> mg.e(Integer(1))
>>> mg.e(Integer(2))
>>> x = mg.f_string([Integer(1),Integer(0),Integer(2),Integer(1),Integer(0),Integer(2),Integer(1),Integer(1),Integer(0)])
>>> all(x.f(i).e(i) == x for i in B.index_set())
True
>>> all(x.e(i).f(i) == x for i in B.index_set() if x.epsilon(i) > Integer(0))
True

Sage Live

B = crystals.infinity.LSPaths(['B',3,1])
mg = B.module_generator()
mg.e(0)
mg.e(1)
mg.e(2)
x = mg.f_string([1,0,2,1,0,2,1,1,0])
all(x.f(i).e(i) == x for i in B.index_set())
all(x.e(i).f(i) == x for i in B.index_set() if x.epsilon(i) > 0)

f(i, power=1, length_only=False)

Return the \(i\)-th crystal lowering operator on self.

INPUT:

• i – element of the index set

• power – (default: 1) positive integer; specifies the power of the lowering operator to be applied

• length_only – boolean (default: False); if True, then return the distance to the anti-dominant end of the \(i\)-string of self

EXAMPLES:

Sage

sage: B = crystals.infinity.LSPaths(['D',3,2])
sage: mg = B.highest_weight_vector()
sage: mg.f(1)
(3*Lambda[0] - Lambda[1] + 3*Lambda[2],
 2*Lambda[0] + 2*Lambda[1] + 2*Lambda[2])
sage: mg.f(2)
(Lambda[0] + 2*Lambda[1] - Lambda[2],
 2*Lambda[0] + 2*Lambda[1] + 2*Lambda[2])
sage: mg.f(0)
(-Lambda[0] + 2*Lambda[1] + Lambda[2] - delta,
 2*Lambda[0] + 2*Lambda[1] + 2*Lambda[2])

Python

>>> from sage.all import *
>>> B = crystals.infinity.LSPaths(['D',Integer(3),Integer(2)])
>>> mg = B.highest_weight_vector()
>>> mg.f(Integer(1))
(3*Lambda[0] - Lambda[1] + 3*Lambda[2],
 2*Lambda[0] + 2*Lambda[1] + 2*Lambda[2])
>>> mg.f(Integer(2))
(Lambda[0] + 2*Lambda[1] - Lambda[2],
 2*Lambda[0] + 2*Lambda[1] + 2*Lambda[2])
>>> mg.f(Integer(0))
(-Lambda[0] + 2*Lambda[1] + Lambda[2] - delta,
 2*Lambda[0] + 2*Lambda[1] + 2*Lambda[2])

Sage Live

B = crystals.infinity.LSPaths(['D',3,2])
mg = B.highest_weight_vector()
mg.f(1)
mg.f(2)
mg.f(0)

phi(i)

Return \(\varphi_i\) of self.

Let \(\pi \in \mathcal{B}(\infty)\). Define

\[\varphi_i(\pi) := \varepsilon_i(\pi) + \langle h_i, \mathrm{wt}(\pi) \rangle,\]

where \(h_i\) is the \(i\)-th simple coroot and \(\mathrm{wt}(\pi)\) is the weight() of \(\pi\).

INPUT:

• i – element of the index set

EXAMPLES:

Sage

sage: B = crystals.infinity.LSPaths(['D',4])
sage: mg = B.highest_weight_vector()
sage: x = mg.f_string([1,3,4,2,4,3,2,1,4])
sage: [x.phi(i) for i in B.index_set()]
[-1, 4, -2, -3]

Python

>>> from sage.all import *
>>> B = crystals.infinity.LSPaths(['D',Integer(4)])
>>> mg = B.highest_weight_vector()
>>> x = mg.f_string([Integer(1),Integer(3),Integer(4),Integer(2),Integer(4),Integer(3),Integer(2),Integer(1),Integer(4)])
>>> [x.phi(i) for i in B.index_set()]
[-1, 4, -2, -3]

Sage Live

B = crystals.infinity.LSPaths(['D',4])
mg = B.highest_weight_vector()
x = mg.f_string([1,3,4,2,4,3,2,1,4])
[x.phi(i) for i in B.index_set()]

weight()

Return the weight of self.

Todo

This is a generic algorithm. We should find a better description and implement it.

EXAMPLES:

Sage

sage: B = crystals.infinity.LSPaths(['E',6])
sage: mg = B.highest_weight_vector()
sage: f_seq = [1,4,2,6,4,2,3,1,5,5]
sage: x = mg.f_string(f_seq)
sage: x.weight()
-3*Lambda[1] - 2*Lambda[2] + 2*Lambda[3] + Lambda[4] - Lambda[5]

sage: al = B.cartan_type().root_system().weight_space().simple_roots()
sage: x.weight() == -sum(al[i] for i in f_seq)
True

Python

>>> from sage.all import *
>>> B = crystals.infinity.LSPaths(['E',Integer(6)])
>>> mg = B.highest_weight_vector()
>>> f_seq = [Integer(1),Integer(4),Integer(2),Integer(6),Integer(4),Integer(2),Integer(3),Integer(1),Integer(5),Integer(5)]
>>> x = mg.f_string(f_seq)
>>> x.weight()
-3*Lambda[1] - 2*Lambda[2] + 2*Lambda[3] + Lambda[4] - Lambda[5]

>>> al = B.cartan_type().root_system().weight_space().simple_roots()
>>> x.weight() == -sum(al[i] for i in f_seq)
True

Sage Live

B = crystals.infinity.LSPaths(['E',6])
mg = B.highest_weight_vector()
f_seq = [1,4,2,6,4,2,3,1,5,5]
x = mg.f_string(f_seq)
x.weight()
al = B.cartan_type().root_system().weight_space().simple_roots()
x.weight() == -sum(al[i] for i in f_seq)

module_generator()

Return the module generator (or highest weight element) of self.

The module generator is the unique path \(\pi_\infty\colon t \mapsto t\rho\), for \(t \in [0,\infty)\).

EXAMPLES:

Sage

sage: B = crystals.infinity.LSPaths(['A',6,2])
sage: mg = B.module_generator(); mg
(Lambda[0] + Lambda[1] + Lambda[2] + Lambda[3],)
sage: mg.weight()
0

Python

>>> from sage.all import *
>>> B = crystals.infinity.LSPaths(['A',Integer(6),Integer(2)])
>>> mg = B.module_generator(); mg
(Lambda[0] + Lambda[1] + Lambda[2] + Lambda[3],)
>>> mg.weight()
0

Sage Live

B = crystals.infinity.LSPaths(['A',6,2])
mg = B.module_generator(); mg
mg.weight()

weight_lattice_realization()

Return the weight lattice realization of self.

EXAMPLES:

Sage

sage: B = crystals.infinity.LSPaths(['C',4])
sage: B.weight_lattice_realization()
Weight space over the Rational Field of the Root system of type ['C', 4]

Python

>>> from sage.all import *
>>> B = crystals.infinity.LSPaths(['C',Integer(4)])
>>> B.weight_lattice_realization()
Weight space over the Rational Field of the Root system of type ['C', 4]

Sage Live

B = crystals.infinity.LSPaths(['C',4])
B.weight_lattice_realization()

sage.combinat.crystals.littelmann_path.positively_parallel_weights(v, w)

Check whether the vectors v and w are positive scalar multiples of each other.

EXAMPLES:

Sage

sage: from sage.combinat.crystals.littelmann_path import positively_parallel_weights
sage: La = RootSystem(['A',5,2]).weight_space(extended=True).fundamental_weights()
sage: rho = sum(La)
sage: positively_parallel_weights(rho, 4*rho)
True
sage: positively_parallel_weights(4*rho, rho)
True
sage: positively_parallel_weights(rho, -rho)
False
sage: positively_parallel_weights(rho, La[1] + La[2])
False

Python

>>> from sage.all import *
>>> from sage.combinat.crystals.littelmann_path import positively_parallel_weights
>>> La = RootSystem(['A',Integer(5),Integer(2)]).weight_space(extended=True).fundamental_weights()
>>> rho = sum(La)
>>> positively_parallel_weights(rho, Integer(4)*rho)
True
>>> positively_parallel_weights(Integer(4)*rho, rho)
True
>>> positively_parallel_weights(rho, -rho)
False
>>> positively_parallel_weights(rho, La[Integer(1)] + La[Integer(2)])
False

Sage Live

from sage.combinat.crystals.littelmann_path import positively_parallel_weights
La = RootSystem(['A',5,2]).weight_space(extended=True).fundamental_weights()
rho = sum(La)
positively_parallel_weights(rho, 4*rho)
positively_parallel_weights(4*rho, rho)
positively_parallel_weights(rho, -rho)
positively_parallel_weights(rho, La[1] + La[2])

Make live