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""" 

Cartan matrices 

 

AUTHORS: 

 

- Travis Scrimshaw (2012-04-22): Nicolas M. Thiery moved matrix creation to 

:class:`CartanType` to prepare :func:`cartan_matrix()` for deprecation. 

- Christian Stump, Travis Scrimshaw (2013-04-13): Created :class:`CartanMatrix`. 

""" 

#***************************************************************************** 

# Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>, 

# Copyright (C) 2012,2013 Travis Scrimshaw <tscrim at ucdavis.edu>, 

# Copyright (C) 2013 Christian Stump, 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from six.moves import range 

from six import add_metaclass 

 

from sage.misc.cachefunc import cached_method 

from sage.matrix.constructor import matrix 

from sage.structure.element import is_Matrix 

from sage.matrix.matrix_space import MatrixSpace 

from sage.misc.inherit_comparison import InheritComparisonClasscallMetaclass 

from sage.misc.classcall_metaclass import typecall 

from sage.misc.misc import powerset 

from sage.matrix.matrix_integer_sparse import Matrix_integer_sparse 

from sage.rings.all import ZZ 

from sage.combinat.root_system.cartan_type import CartanType, CartanType_abstract 

from sage.combinat.root_system.root_system import RootSystem 

from sage.sets.family import Family 

from sage.graphs.digraph import DiGraph 

 

 

@add_metaclass(InheritComparisonClasscallMetaclass) 

class CartanMatrix(Matrix_integer_sparse, CartanType_abstract): 

r""" 

A (generalized) Cartan matrix. 

 

A matrix `A = (a_{ij})_{i,j \in I}` for some index set `I` is a 

generalized Cartan matrix if it satisfies the following properties: 

 

- `a_{ii} = 2` for all `i`, 

- `a_{ij} \leq 0` for all `i \neq j`, 

- `a_{ij} = 0` if and only if `a_{ji} = 0` for all `i \neq j`. 

 

Additionally some reference assume that a Cartan matrix is 

*symmetrizable* (see :meth:`is_symmetrizable`). However following Kac, we 

do not make that assumption here. 

 

INPUT: 

 

Can be anything which is accepted by ``CartanType`` or a matrix. 

 

If given a matrix, one can also use the keyword ``cartan_type`` when giving 

a matrix to explicitly state the type. Otherwise this will try to check the 

input matrix against possible standard types of Cartan matrices. To disable 

this check, use the keyword ``cartan_type_check = False``. 

 

EXAMPLES:: 

 

sage: CartanMatrix(['A', 4]) 

[ 2 -1 0 0] 

[-1 2 -1 0] 

[ 0 -1 2 -1] 

[ 0 0 -1 2] 

sage: CartanMatrix(['B', 6]) 

[ 2 -1 0 0 0 0] 

[-1 2 -1 0 0 0] 

[ 0 -1 2 -1 0 0] 

[ 0 0 -1 2 -1 0] 

[ 0 0 0 -1 2 -1] 

[ 0 0 0 0 -2 2] 

sage: CartanMatrix(['C', 4]) 

[ 2 -1 0 0] 

[-1 2 -1 0] 

[ 0 -1 2 -2] 

[ 0 0 -1 2] 

sage: CartanMatrix(['D', 6]) 

[ 2 -1 0 0 0 0] 

[-1 2 -1 0 0 0] 

[ 0 -1 2 -1 0 0] 

[ 0 0 -1 2 -1 -1] 

[ 0 0 0 -1 2 0] 

[ 0 0 0 -1 0 2] 

sage: CartanMatrix(['E',6]) 

[ 2 0 -1 0 0 0] 

[ 0 2 0 -1 0 0] 

[-1 0 2 -1 0 0] 

[ 0 -1 -1 2 -1 0] 

[ 0 0 0 -1 2 -1] 

[ 0 0 0 0 -1 2] 

sage: CartanMatrix(['E',7]) 

[ 2 0 -1 0 0 0 0] 

[ 0 2 0 -1 0 0 0] 

[-1 0 2 -1 0 0 0] 

[ 0 -1 -1 2 -1 0 0] 

[ 0 0 0 -1 2 -1 0] 

[ 0 0 0 0 -1 2 -1] 

[ 0 0 0 0 0 -1 2] 

sage: CartanMatrix(['E', 8]) 

[ 2 0 -1 0 0 0 0 0] 

[ 0 2 0 -1 0 0 0 0] 

[-1 0 2 -1 0 0 0 0] 

[ 0 -1 -1 2 -1 0 0 0] 

[ 0 0 0 -1 2 -1 0 0] 

[ 0 0 0 0 -1 2 -1 0] 

[ 0 0 0 0 0 -1 2 -1] 

[ 0 0 0 0 0 0 -1 2] 

sage: CartanMatrix(['F', 4]) 

[ 2 -1 0 0] 

[-1 2 -1 0] 

[ 0 -2 2 -1] 

[ 0 0 -1 2] 

 

This is different from MuPAD-Combinat, due to different node 

convention? 

 

:: 

 

sage: CartanMatrix(['G', 2]) 

[ 2 -3] 

[-1 2] 

sage: CartanMatrix(['A',1,1]) 

[ 2 -2] 

[-2 2] 

sage: CartanMatrix(['A', 3, 1]) 

[ 2 -1 0 -1] 

[-1 2 -1 0] 

[ 0 -1 2 -1] 

[-1 0 -1 2] 

sage: CartanMatrix(['B', 3, 1]) 

[ 2 0 -1 0] 

[ 0 2 -1 0] 

[-1 -1 2 -1] 

[ 0 0 -2 2] 

sage: CartanMatrix(['C', 3, 1]) 

[ 2 -1 0 0] 

[-2 2 -1 0] 

[ 0 -1 2 -2] 

[ 0 0 -1 2] 

sage: CartanMatrix(['D', 4, 1]) 

[ 2 0 -1 0 0] 

[ 0 2 -1 0 0] 

[-1 -1 2 -1 -1] 

[ 0 0 -1 2 0] 

[ 0 0 -1 0 2] 

sage: CartanMatrix(['E', 6, 1]) 

[ 2 0 -1 0 0 0 0] 

[ 0 2 0 -1 0 0 0] 

[-1 0 2 0 -1 0 0] 

[ 0 -1 0 2 -1 0 0] 

[ 0 0 -1 -1 2 -1 0] 

[ 0 0 0 0 -1 2 -1] 

[ 0 0 0 0 0 -1 2] 

sage: CartanMatrix(['E', 7, 1]) 

[ 2 -1 0 0 0 0 0 0] 

[-1 2 0 -1 0 0 0 0] 

[ 0 0 2 0 -1 0 0 0] 

[ 0 -1 0 2 -1 0 0 0] 

[ 0 0 -1 -1 2 -1 0 0] 

[ 0 0 0 0 -1 2 -1 0] 

[ 0 0 0 0 0 -1 2 -1] 

[ 0 0 0 0 0 0 -1 2] 

sage: CartanMatrix(['E', 8, 1]) 

[ 2 0 0 0 0 0 0 0 -1] 

[ 0 2 0 -1 0 0 0 0 0] 

[ 0 0 2 0 -1 0 0 0 0] 

[ 0 -1 0 2 -1 0 0 0 0] 

[ 0 0 -1 -1 2 -1 0 0 0] 

[ 0 0 0 0 -1 2 -1 0 0] 

[ 0 0 0 0 0 -1 2 -1 0] 

[ 0 0 0 0 0 0 -1 2 -1] 

[-1 0 0 0 0 0 0 -1 2] 

sage: CartanMatrix(['F', 4, 1]) 

[ 2 -1 0 0 0] 

[-1 2 -1 0 0] 

[ 0 -1 2 -1 0] 

[ 0 0 -2 2 -1] 

[ 0 0 0 -1 2] 

sage: CartanMatrix(['G', 2, 1]) 

[ 2 0 -1] 

[ 0 2 -3] 

[-1 -1 2] 

 

.. NOTE:: 

 

Since this is a matrix, :meth:`row()` and :meth:`column()` will return 

the standard row and column respectively. To get the row with the 

indices as in Dynkin diagrams/Cartan types, use 

:meth:`row_with_indices()` and :meth:`column_with_indices()` 

respectively. 

""" 

@staticmethod 

def __classcall_private__(cls, data=None, index_set=None, 

cartan_type=None, cartan_type_check=True): 

""" 

Normalize input so we can inherit from sparse integer matrix. 

 

.. NOTE:: 

 

To disable the Cartan type check, use the optional argument 

``cartan_type_check = False``. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['A',1,1]) 

sage: C2 = CartanMatrix([[2, -2], [-2, 2]]) 

sage: C3 = CartanMatrix(matrix([[2, -2], [-2, 2]]), [0, 1]) 

sage: C == C2 and C == C3 

True 

 

TESTS: 

 

Check that :trac:`15740` is fixed:: 

 

sage: d = DynkinDiagram() 

sage: d.add_edge('a', 'b', 2) 

sage: d.index_set() 

('a', 'b') 

sage: cm = CartanMatrix(d) 

sage: cm.index_set() 

('a', 'b') 

""" 

# Special case with 0 args and kwds has Cartan type 

if cartan_type is not None and data is None: 

data = CartanType(cartan_type) 

 

if data is None: 

data = [] 

n = 0 

index_set = tuple() 

cartan_type = None 

subdivisions = None 

elif isinstance(data, CartanMatrix): 

if index_set is not None: 

d = {a: index_set[i] for i,a in enumerate(data.index_set())} 

return data.relabel(d) 

return data 

else: 

dynkin_diagram = None 

subdivisions = None 

 

from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class 

if isinstance(data, DynkinDiagram_class): 

dynkin_diagram = data 

cartan_type = data._cartan_type 

else: 

try: 

cartan_type = CartanType(data) 

dynkin_diagram = cartan_type.dynkin_diagram() 

except (TypeError, ValueError): 

pass 

 

if dynkin_diagram is not None: 

n = dynkin_diagram.rank() 

index_set = dynkin_diagram.index_set() 

oir = dynkin_diagram.odd_isotropic_roots() 

reverse = {a: i for i,a in enumerate(index_set)} 

data = {(i, i): 2 if index_set[i] not in oir else 0 

for i in range(n)} 

for (i,j,l) in dynkin_diagram.edge_iterator(): 

data[(reverse[j], reverse[i])] = -l 

else: 

M = matrix(data) 

if not is_generalized_cartan_matrix(M): 

raise ValueError("the input matrix is not a generalized Cartan matrix") 

n = M.ncols() 

data = M.dict() 

subdivisions = M._subdivisions 

 

if index_set is None: 

index_set = tuple(range(n)) 

else: 

index_set = tuple(index_set) 

 

if len(index_set) != n and len(set(index_set)) != n: 

raise ValueError("the given index set is not valid") 

 

# We can do the Cartan type initialization later as this is not 

# a unique representation 

mat = typecall(cls, MatrixSpace(ZZ, n, sparse=True), data, False, True) 

# FIXME: We have to initialize the CartanMatrix part separately because 

# of the __cinit__ of the matrix. We should get rid of this workaround 

mat._CM_init(cartan_type, index_set, cartan_type_check) 

mat._subdivisions = subdivisions 

return mat 

 

def matrix_space(self, nrows=None, ncols=None, sparse=None): 

r""" 

Return a matrix space over the integers. 

 

INPUT: 

 

- ``nrows`` - number of rows 

 

- ``ncols`` - number of columns 

 

- ``sparse`` - (boolean) sparseness 

 

EXAMPLES:: 

 

sage: cm = CartanMatrix(['A', 3]) 

sage: cm.matrix_space() 

Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring 

sage: cm.matrix_space(2, 2) 

Full MatrixSpace of 2 by 2 sparse matrices over Integer Ring 

sage: cm[:2,1:] # indirect doctest 

[-1 0] 

[ 2 -1] 

""" 

if nrows is None: 

nrows = self.nrows() 

if ncols is None: 

ncols = self.ncols() 

if sparse is None: 

sparse = True 

 

if nrows == self.nrows() and ncols == self.ncols() and sparse: 

return self.parent() 

else: 

from sage.matrix.matrix_space import MatrixSpace 

return MatrixSpace(ZZ, nrows, ncols, sparse is None or bool(sparse)) 

 

def _CM_init(self, cartan_type, index_set, cartan_type_check): 

""" 

Initialize ``self`` as a Cartan matrix. 

 

TESTS:: 

 

sage: C = CartanMatrix(['A',1,1]) # indirect doctest 

sage: TestSuite(C).run(skip=["_test_category", "_test_change_ring"]) 

""" 

self._index_set = index_set 

self.set_immutable() 

 

if cartan_type is not None: 

cartan_type = CartanType(cartan_type) 

elif self.nrows() == 1: 

cartan_type = CartanType(['A', 1]) 

elif cartan_type_check: 

# Placeholder so we don't have to reimplement creating a 

# Dynkin diagram from a Cartan matrix 

self._cartan_type = None 

cartan_type = find_cartan_type_from_matrix(self) 

 

self._cartan_type = cartan_type 

 

def __reduce__(self): 

""" 

Used for pickling. 

 

TESTS:: 

 

sage: CM = CartanMatrix(['A',4]) 

sage: x = loads(dumps(CM)) 

sage: x._index_set 

(1, 2, 3, 4) 

""" 

if self._cartan_type: 

return (CartanMatrix, (self._cartan_type,)) 

return (CartanMatrix, (self.dynkin_diagram(),)) 

 

def root_system(self): 

""" 

Return the root system corresponding to ``self``. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['A',3]) 

sage: C.root_system() 

Root system of type ['A', 3] 

""" 

if self._cartan_type is not None: 

return RootSystem(self._cartan_type) 

return self.dynkin_diagram().root_system() 

 

def root_space(self): 

""" 

Return the root space corresponding to ``self``. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['A',3]) 

sage: C.root_space() 

Root space over the Rational Field of the Root system of type ['A', 3] 

""" 

return self.root_system().root_space() 

 

def reflection_group(self, type="matrix"): 

""" 

Return the reflection group corresponding to ``self``. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['A',3]) 

sage: C.reflection_group() 

Weyl Group of type ['A', 3] (as a matrix group acting on the root space) 

""" 

RS = self.root_space() 

 

if type == "matrix": 

return RS.weyl_group() 

 

if type == "permutation": 

if not self.is_finite(): 

raise ValueError("only works for finite types") 

Phi = RS.roots() 

gens = {} 

from sage.groups.perm_gps.permgroup_named import SymmetricGroup 

S = SymmetricGroup(len(Phi)) 

for i in self.index_set(): 

pi = S([ Phi.index( beta.simple_reflection(i) ) + 1 for beta in Phi ]) 

gens[i] = pi 

return S.subgroup( gens[i] for i in gens ) 

 

raise ValueError("The reflection group is only available as a matrix group or as a permutation group.") 

 

def symmetrizer(self): 

""" 

Return the symmetrizer of ``self``. 

 

EXAMPLES:: 

 

sage: cm = CartanMatrix([[2,-5],[-2,2]]) 

sage: cm.symmetrizer() 

Finite family {0: 2, 1: 5} 

 

TESTS: 

 

Check that the symmetrizer computed from the Cartan matrix agrees 

with the values given by the Cartan type:: 

 

sage: ct = CartanType(['B',4,1]) 

sage: ct.symmetrizer() 

Finite family {0: 2, 1: 2, 2: 2, 3: 2, 4: 1} 

sage: ct.cartan_matrix().symmetrizer() 

Finite family {0: 2, 1: 2, 2: 2, 3: 2, 4: 1} 

""" 

sym = self.is_symmetrizable(True) 

if not sym: 

raise ValueError("the Cartan matrix is not symmetrizable") 

iset = self.index_set() 

# The result from is_symmetrizable needs to be scaled 

# to integer coefficients 

from sage.arith.all import LCM 

from sage.rings.all import QQ 

scalar = LCM([QQ(x).denominator() for x in sym]) 

return Family( {iset[i]: ZZ(val*scalar) for i, val in enumerate(sym)} ) 

 

@cached_method 

def symmetrized_matrix(self): 

""" 

Return the symmetrized matrix of ``self`` if symmetrizable. 

 

EXAMPLES:: 

 

sage: cm = CartanMatrix(['B',4,1]) 

sage: cm.symmetrized_matrix() 

[ 4 0 -2 0 0] 

[ 0 4 -2 0 0] 

[-2 -2 4 -2 0] 

[ 0 0 -2 4 -2] 

[ 0 0 0 -2 2] 

""" 

M = matrix.diagonal(list(self.symmetrizer())) * self 

M.set_immutable() 

return M 

 

########################################################################## 

# Cartan type methods 

 

def index_set(self): 

""" 

Return the index set of ``self``. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['A',1,1]) 

sage: C.index_set() 

(0, 1) 

sage: C = CartanMatrix(['E',6]) 

sage: C.index_set() 

(1, 2, 3, 4, 5, 6) 

""" 

return self._index_set 

 

def cartan_type(self): 

""" 

Return the Cartan type of ``self`` or ``self`` if unknown. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['A',4,1]) 

sage: C.cartan_type() 

['A', 4, 1] 

 

If the Cartan type is unknown:: 

 

sage: C = CartanMatrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]]) 

sage: C.cartan_type() 

[ 2 -1 -2] 

[-1 2 -1] 

[-2 -1 2] 

""" 

if self._cartan_type is None: 

return self 

return self._cartan_type 

 

def subtype(self, index_set): 

""" 

Return a subtype of ``self`` given by ``index_set``. 

 

A subtype can be considered the Dynkin diagram induced from 

the Dynkin diagram of ``self`` by ``index_set``. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['F',4]) 

sage: S = C.subtype([1,2,3]) 

sage: S 

[ 2 -1 0] 

[-1 2 -1] 

[ 0 -2 2] 

sage: S.index_set() 

(1, 2, 3) 

""" 

ind = self.index_set() 

I = [ind.index(i) for i in index_set] 

return CartanMatrix(self.matrix_from_rows_and_columns(I, I), index_set) 

 

def rank(self): 

r""" 

Return the rank of ``self``. 

 

EXAMPLES:: 

 

sage: CartanMatrix(['C',3]).rank() 

3 

sage: CartanMatrix(["A2","B2","F4"]).rank() 

8 

""" 

return self.ncols() 

 

def relabel(self, relabelling): 

""" 

Return the relabelled Cartan matrix. 

 

EXAMPLES:: 

 

sage: CM = CartanMatrix(['C',3]) 

sage: R = CM.relabel({1:0, 2:4, 3:1}); R 

[ 2 0 -1] 

[ 0 2 -1] 

[-1 -2 2] 

sage: R.index_set() 

(0, 1, 4) 

sage: CM 

[ 2 -1 0] 

[-1 2 -2] 

[ 0 -1 2] 

""" 

return self.dynkin_diagram().relabel(relabelling, inplace=False).cartan_matrix() 

 

@cached_method 

def dynkin_diagram(self): 

""" 

Return the Dynkin diagram corresponding to ``self``. 

 

EXAMPLES:: 

 

sage: C = CartanMatrix(['A',2]) 

sage: C.dynkin_diagram() 

O---O 

1 2 

A2 

sage: C = CartanMatrix(['F',4,1]) 

sage: C.dynkin_diagram() 

O---O---O=>=O---O 

0 1 2 3 4 

F4~ 

sage: C = CartanMatrix([[2,-4],[-4,2]]) 

sage: C.dynkin_diagram() 

Dynkin diagram of rank 2 

""" 

from sage.combinat.root_system.dynkin_diagram import DynkinDiagram 

if self._cartan_type is not None: 

return DynkinDiagram(self._cartan_type) 

return DynkinDiagram(self) 

 

def cartan_matrix(self): 

r""" 

Return the Cartan matrix of ``self``. 

 

EXAMPLES:: 

 

sage: CartanMatrix(['C',3]).cartan_matrix() 

[ 2 -1 0] 

[-1 2 -2] 

[ 0 -1 2] 

""" 

return self 

 

def dual(self): 

r""" 

Return the dual Cartan matrix of ``self``, which is obtained by taking 

the transpose. 

 

EXAMPLES:: 

 

sage: ct = CartanType(['C',3]) 

sage: M = CartanMatrix(ct); M 

[ 2 -1 0] 

[-1 2 -2] 

[ 0 -1 2] 

sage: M.dual() 

[ 2 -1 0] 

[-1 2 -1] 

[ 0 -2 2] 

sage: M.dual() == CartanMatrix(ct.dual()) 

True 

sage: M.dual().cartan_type() == ct.dual() 

True 

 

An example with arbitrary Cartan matrices:: 

 

sage: cm = CartanMatrix([[2,-5], [-2, 2]]); cm 

[ 2 -5] 

[-2 2] 

sage: cm.dual() 

[ 2 -2] 

[-5 2] 

sage: cm.dual() == CartanMatrix(cm.transpose()) 

True 

sage: cm.dual().dual() == cm 

True 

""" 

if self._cartan_type is not None: 

return CartanMatrix(self._cartan_type.dual()) 

return CartanMatrix(self.transpose()) 

 

def is_simply_laced(self): 

""" 

Implements :meth:`CartanType_abstract.is_simply_laced()`. 

 

A Cartan matrix is simply-laced if all non diagonal entries are `0` 

or `-1`. 

 

EXAMPLES:: 

 

sage: cm = CartanMatrix([[2, -1, -1, -1], [-1, 2, -1, -1], [-1, -1, 2, -1], [-1, -1, -1, 2]]) 

sage: cm.is_simply_laced() 

True 

""" 

for i in range(self.nrows()): 

for j in range(i+1, self.ncols()): 

if self[i, j] < -1 or self[j, i] < -1: 

return False 

return True 

 

def is_crystallographic(self): 

""" 

Implements :meth:`CartanType_abstract.is_crystallographic`. 

 

A Cartan matrix is crystallographic if it is symmetrizable. 

 

EXAMPLES:: 

 

sage: CartanMatrix(['F',4]).is_crystallographic() 

True 

""" 

return self.is_symmetrizable() 

 

def column_with_indices(self, j): 

""" 

Return the `j^{th}` column `(a_{i,j})_i` of ``self`` as a container 

(or iterator) of tuples `(i, a_{i,j})` 

 

EXAMPLES:: 

 

sage: M = CartanMatrix(['B',4]) 

sage: [ (i,a) for (i,a) in M.column_with_indices(3) ] 

[(3, 2), (2, -1), (4, -2)] 

""" 

return self.dynkin_diagram().column(j) 

 

def row_with_indices(self, i): 

""" 

Return the `i^{th}` row `(a_{i,j})_j` of ``self`` as a container 

(or iterator) of tuples `(j, a_{i,j})` 

 

EXAMPLES:: 

 

sage: M = CartanMatrix(['C',4]) 

sage: [ (i,a) for (i,a) in M.row_with_indices(3) ] 

[(3, 2), (2, -1), (4, -2)] 

""" 

return self.dynkin_diagram().row(i) 

 

@cached_method 

def is_finite(self): 

""" 

Return ``True`` if ``self`` is a finite type or ``False`` otherwise. 

 

A generalized Cartan matrix is finite if the determinant of all its 

principal submatrices (see :meth:`principal_submatrices`) is positive. 

Such matrices have a positive definite symmetrized matrix. Note that a 

finite matrix may consist of multiple blocks of Cartan matrices each 

having finite Cartan type. 

 

EXAMPLES:: 

 

sage: M = CartanMatrix(['C',4]) 

sage: M.is_finite() 

True 

sage: M = CartanMatrix(['D',4,1]) 

sage: M.is_finite() 

False 

sage: M = CartanMatrix([[2, -4], [-3, 2]]) 

sage: M.is_finite() 

False 

""" 

if self._cartan_type is None: 

if not self.is_symmetrizable(): 

return False 

return self.symmetrized_matrix().is_positive_definite() 

return self._cartan_type.is_finite() 

 

@cached_method 

def is_affine(self): 

""" 

Return ``True`` if ``self`` is an affine type or ``False`` otherwise. 

 

A generalized Cartan matrix is affine if all of its indecomposable 

blocks are either finite (see :meth:`is_finite`) or have zero 

determinant with all proper principal minors positive. 

 

EXAMPLES:: 

 

sage: M = CartanMatrix(['C',4]) 

sage: M.is_affine() 

False 

sage: M = CartanMatrix(['D',4,1]) 

sage: M.is_affine() 

True 

sage: M = CartanMatrix([[2, -4], [-3, 2]]) 

sage: M.is_affine() 

False 

""" 

if self._cartan_type is None: 

if self.det() != 0: 

return False 

for b in self.indecomposable_blocks(): 

if b.det() < 0 or not all( 

a.det() > 0 for a in b.principal_submatrices(proper=True)): 

return False 

return True 

return self._cartan_type.is_affine() 

 

@cached_method 

def is_hyperbolic(self, compact=False): 

""" 

Return if ``True`` if ``self`` is a (compact) hyperbolic type 

or ``False`` otherwise. 

 

An indecomposable generalized Cartan matrix is hyperbolic if it has 

negative determinant and if any proper connected subdiagram of its 

Dynkin diagram is of finite or affine type. It is compact hyperbolic 

if any proper connected subdiagram has finite type. 

 

INPUT: 

 

- ``compact`` -- if ``True``, check if matrix is compact hyperbolic 

 

EXAMPLES:: 

 

sage: M = CartanMatrix([[2,-2,0],[-2,2,-1],[0,-1,2]]) 

sage: M.is_hyperbolic() 

True 

sage: M.is_hyperbolic(compact=True) 

False 

sage: M = CartanMatrix([[2,-3],[-3,2]]) 

sage: M.is_hyperbolic() 

True 

sage: M = CartanMatrix(['C',4]) 

sage: M.is_hyperbolic() 

False 

""" 

if not self.is_indefinite() or not self.is_indecomposable(): 

return False 

 

D = self.dynkin_diagram() 

verts = tuple(D.vertex_iterator()) 

for v in verts: 

l = set(verts)-set((v,)) 

subg = D.subgraph(vertices=l) 

if compact and not subg.is_finite(): 

return False 

elif not subg.is_finite() and not subg.is_affine(): 

return False 

return True 

 

@cached_method 

def is_lorentzian(self): 

""" 

Return ``True`` if ``self`` is a Lorentzian type or ``False`` otherwise. 

 

A generalized Cartan matrix is Lorentzian if it has negative determinant 

and exactly one negative eigenvalue. 

 

EXAMPLES:: 

 

sage: M = CartanMatrix([[2,-3],[-3,2]]) 

sage: M.is_lorentzian() 

True 

sage: M = CartanMatrix([[2,-1],[-1,2]]) 

sage: M.is_lorentzian() 

False 

""" 

if self.det() >= 0: 

return False 

return sum(1 for x in self.eigenvalues() if x < 0) == 1 

 

@cached_method 

def is_indefinite(self): 

""" 

Return if ``self`` is an indefinite type or ``False`` otherwise. 

 

EXAMPLES:: 

 

sage: M = CartanMatrix([[2,-3],[-3,2]]) 

sage: M.is_indefinite() 

True 

sage: M = CartanMatrix("A2") 

sage: M.is_indefinite() 

False 

""" 

return not self.is_finite() and not self.is_affine() 

 

@cached_method 

def is_indecomposable(self): 

""" 

Return if ``self`` is an indecomposable matrix or ``False`` otherwise. 

 

EXAMPLES:: 

 

sage: M = CartanMatrix(['A',5]) 

sage: M.is_indecomposable() 

True 

sage: M = CartanMatrix([[2,-1,0],[-1,2,0],[0,0,2]]) 

sage: M.is_indecomposable() 

False 

""" 

comp_num = self.dynkin_diagram().connected_components_number() 

# consider the empty matrix to be indecomposable 

return comp_num <= 1 

 

def principal_submatrices(self, proper=False): 

""" 

Return a list of all principal submatrices of ``self``. 

 

INPUT: 

 

- ``proper`` -- if ``True``, return only proper submatrices 

 

EXAMPLES:: 

 

sage: M = CartanMatrix(['A',2]) 

sage: M.principal_submatrices() 

[ 

[ 2 -1] 

[], [2], [2], [-1 2] 

] 

sage: M.principal_submatrices(proper=True) 

[[], [2], [2]] 

 

""" 

iset = list(range(self.ncols())) 

ret = [] 

for l in powerset(iset): 

if not proper or (proper and l != iset): 

ret.append(self.matrix_from_rows_and_columns(l,l)) 

return ret 

 

@cached_method 

def indecomposable_blocks(self): 

""" 

Return a tuple of all indecomposable blocks of ``self``. 

 

EXAMPLES:: 

 

sage: M = CartanMatrix(['A',2]) 

sage: M.indecomposable_blocks() 

( 

[ 2 -1] 

[-1 2] 

) 

sage: M = CartanMatrix([['A',2,1],['A',3,1]]) 

sage: M.indecomposable_blocks() 

( 

[ 2 -1 0 -1] 

[-1 2 -1 0] [ 2 -1 -1] 

[ 0 -1 2 -1] [-1 2 -1] 

[-1 0 -1 2], [-1 -1 2] 

) 

""" 

subgraphs = self.dynkin_diagram().connected_components_subgraphs() 

return tuple(CartanMatrix(subg._matrix_().rows()) for subg in subgraphs) 

 

def is_generalized_cartan_matrix(M): 

""" 

Return ``True`` if ``M`` is a generalized Cartan matrix. For a definition 

of a generalized Cartan matrix, see :class:`CartanMatrix`. 

 

EXAMPLES:: 

 

sage: from sage.combinat.root_system.cartan_matrix import is_generalized_cartan_matrix 

sage: M = matrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]]) 

sage: is_generalized_cartan_matrix(M) 

True 

sage: M = matrix([[2,-1,-2], [-1,2,-1], [0,-1,2]]) 

sage: is_generalized_cartan_matrix(M) 

False 

sage: M = matrix([[1,-1,-2], [-1,2,-1], [-2,-1,2]]) 

sage: is_generalized_cartan_matrix(M) 

False 

 

A non-symmetrizable example:: 

 

sage: M = matrix([[2,-1,-2], [-1,2,-1], [-1,-1,2]]) 

sage: is_generalized_cartan_matrix(M) 

True 

""" 

if not is_Matrix(M): 

return False 

if not M.is_square(): 

return False 

n = M.ncols() 

for i in range(n): 

if M[i,i] != 2: 

return False 

for j in range(i+1, n): 

if M[i,j] > 0 or M[j,i] > 0: 

return False 

elif M[i,j] == 0 and M[j,i] != 0: 

return False 

elif M[j,i] == 0 and M[i,j] != 0: 

return False 

return True 

 

def find_cartan_type_from_matrix(CM): 

r""" 

Find a Cartan type by direct comparison of Dynkin diagrams given from 

the generalized Cartan matrix ``CM`` and return ``None`` if not found. 

 

INPUT: 

 

- ``CM`` -- a generalized Cartan matrix 

 

EXAMPLES:: 

 

sage: from sage.combinat.root_system.cartan_matrix import find_cartan_type_from_matrix 

sage: CM = CartanMatrix([[2,-1,-1], [-1,2,-1], [-1,-1,2]]) 

sage: find_cartan_type_from_matrix(CM) 

['A', 2, 1] 

sage: CM = CartanMatrix([[2,-1,0], [-1,2,-2], [0,-1,2]]) 

sage: find_cartan_type_from_matrix(CM) 

['C', 3] relabelled by {1: 0, 2: 1, 3: 2} 

sage: CM = CartanMatrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]]) 

sage: find_cartan_type_from_matrix(CM) 

""" 

types = [] 

for S in CM.dynkin_diagram().connected_components_subgraphs(): 

S = DiGraph(S) # We need a simple digraph here 

n = S.num_verts() 

# Build the list to test based upon rank 

if n == 1: 

types.append(CartanType(['A', 1])) 

continue 

 

test = [['A', n]] 

if n >= 2: 

if n == 2: 

test += [['G',2], ['A',2,2]] 

test += [['B',n], ['A',n-1,1]] 

if n >= 3: 

if n == 3: 

test.append(['G',2,1]) 

test += [['C',n], ['BC',n-1,2], ['C',n-1,1]] 

if n >= 4: 

if n == 4: 

test.append(['F',4]) 

test += [['D',n], ['B',n-1,1]] 

if n >= 5: 

if n == 5: 

test.append(['F',4,1]) 

test.append(['D',n-1,1]) 

if n == 6: 

test.append(['E',6]) 

elif n == 7: 

test += [['E',7], ['E',6,1]] 

elif n == 8: 

test += [['E',8], ['E',7,1]] 

elif n == 9: 

test.append(['E',8,1]) 

 

# Test every possible Cartan type and its dual 

found = False 

for x in test: 

ct = CartanType(x) 

T = DiGraph(ct.dynkin_diagram()) # We need a simple digraph here 

iso, match = T.is_isomorphic(S, certificate=True, edge_labels=True) 

if iso: 

types.append(ct.relabel(match)) 

found = True 

break 

 

if ct == ct.dual(): 

continue # self-dual, so nothing more to test 

 

ct = ct.dual() 

T = DiGraph(ct.dynkin_diagram()) # We need a simple digraph here 

iso, match = T.is_isomorphic(S, certificate=True, edge_labels=True) 

if iso: 

types.append(ct.relabel(match)) 

found = True 

break 

if not found: 

return None 

 

return CartanType(types)