def __classcall_private__(cls, cartan_type, r, s):
"""
Normalize the input arguments to ensure unique representation.
EXAMPLES::
sage: KRT1 = KirillovReshetikhinTableaux(CartanType(['A',3,1]), 2, 3)
sage: KRT2 = KirillovReshetikhinTableaux(['A',3,1], 2, 3)
sage: KRT1 is KRT2
True
"""
ct = CartanType(cartan_type)
assert ct.is_affine()
if ct.is_untwisted_affine():
if ct.letter == 'D':
if r == ct.n or r == ct.n - 1:
return KRTableauxSpin(ct, r, s)
return KRTableauxTypeVertical(ct, r, s)
if ct.letter == 'B':
if r == ct.n:
return KRTableauxBn(ct, r, s)
return KRTypeVertical(ct, r, s)
if ct.letter == 'A' or (ct.letter == 'C' and r == ct.n):
return KRTableauxRectangle(ct, r, s)
else:
if ct.dual().letter == 'B':
return KRTableauxTypeVertical(ct, r, s)
raise NotImplementedError
def __classcall_private__(cls, cartan_type, r, s):
"""
Normalize the input arguments to ensure unique representation.
EXAMPLES::
sage: KRT1 = KirillovReshetikhinTableaux(CartanType(['A',3,1]), 2, 3)
sage: KRT2 = KirillovReshetikhinTableaux(['A',3,1], 2, 3)
sage: KRT1 is KRT2
True
"""
ct = CartanType(cartan_type)
assert ct.is_affine()
if ct.is_untwisted_affine():
if ct.letter == 'D':
if r == ct.n or r == ct.n - 1:
return KRTableauxSpin(ct, r, s)
return KRTableauxTypeVertical(ct, r, s)
if ct.letter == 'B':
if r == ct.n:
return KRTableauxBn(ct, r, s)
return KRTypeVertical(ct, r, s)
if ct.letter == 'A' or (ct.letter == 'C' and r == ct.n):
return KRTableauxRectangle(ct, r, s)
else:
if ct.dual().letter == 'B':
return KRTableauxTypeVertical(ct, r, s)
raise NotImplementedError
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)
def find_cartan_type_from_matrix(CM):
"""
Find a Cartan type by direct comparison of matrices 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: M = matrix([[2,-1,-1], [-1,2,-1], [-1,-1,2]])
sage: find_cartan_type_from_matrix(M)
['A', 2, 1]
sage: M = matrix([[2,-1,0], [-1,2,-2], [0,-1,2]])
sage: find_cartan_type_from_matrix(M)
['C', 3]
sage: M = matrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]])
sage: find_cartan_type_from_matrix(M)
"""
n = CM.ncols()
# Build the list to test based upon rank
if n == 1:
return CartanType(['A', 1])
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 += [['G',2,1], ['D',4,3]]
test += [['C',n], ['BC',n-1,2], ['C',n-1,1]]
if n >= 4:
if n == 4:
test += [['F',4], ['G',2,1], ['D',4,3]]
test += [['D',n], ['B',n-1,1]]
if n == 5:
test += [['F',4,1], ['D',n-1,1]]
elif 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
for x in test:
ct = CartanType(x)
if ct.cartan_matrix() == CM:
return ct
if ct == ct.dual():
continue
ct = ct.dual()
if ct.cartan_matrix() == CM:
return ct
return None
def find_cartan_type_from_matrix(CM):
"""
Find a Cartan type by direct comparison of matrices 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: M = matrix([[2,-1,-1], [-1,2,-1], [-1,-1,2]])
sage: find_cartan_type_from_matrix(M)
['A', 2, 1]
sage: M = matrix([[2,-1,0], [-1,2,-2], [0,-1,2]])
sage: find_cartan_type_from_matrix(M)
['C', 3]
sage: M = matrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]])
sage: find_cartan_type_from_matrix(M)
"""
n = CM.ncols()
# Build the list to test based upon rank
if n == 1:
return CartanType(['A', 1])
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 += [['G',2,1], ['D',4,3]]
test += [['C',n], ['BC',n-1,2], ['C',n-1,1]]
if n >= 4:
if n == 4:
test += [['F',4], ['G',2,1], ['D',4,3]]
test += [['D',n], ['B',n-1,1]]
if n == 5:
test += [['F',4,1], ['D',n-1,1]]
elif 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
for x in test:
ct = CartanType(x)
if ct.cartan_matrix() == CM:
return ct
if ct == ct.dual():
continue
ct = ct.dual()
if ct.cartan_matrix() == CM:
return ct
return None
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)
