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Added primitive period 1 quivers and Gale-Robinson quivers
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Gregg Musiker authored and Frédéric Chapoton committed Jan 7, 2014
1 parent 037277a commit 741ecfb
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5 changes: 5 additions & 0 deletions src/sage/combinat/cluster_algebra_quiver/quiver.py
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Original file line number Diff line number Diff line change
Expand Up @@ -210,6 +210,11 @@ def __init__( self, data, frozen=None ):
quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], tuple(data[1]) )._digraph )
quiv._mutation_type = mutation_type
self.__init__( quiv )
# Primitive Period 1 quivers are always products of affine A type quivers but a different element of the quiver equivalence class
elif data[0] == 'P1' and type( data[1]) is list:
quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], tuple(data[1]) )._digraph )
quiv.mutation_type()
self.__init__( quiv )
else:
self.__init__( mutation_type.standard_quiver() )
elif len(data) == 3 and type( data[0] ) is str:
Expand Down
139 changes: 138 additions & 1 deletion src/sage/combinat/cluster_algebra_quiver/quiver_mutation_type.py
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Original file line number Diff line number Diff line change
Expand Up @@ -322,6 +322,10 @@ def _samples(self):
* Grassmannian: This defines the cluster algebra (without coefficients) corresponding to the cluster algebra with coefficients which is the co-ordinate ring of a Grassmannian. ``letter`` is 'GR'. ``rank`` is a pair of integers (`k`, `n`) with 'k' < 'n' specifying the Grassmannian of `k`-planes in `n`-space. This defines a quiver given by a (k-1) x (n-k-1) grid where each square is cyclically oriented.
* Primitive Period 1: As in Fordy-Marsh (http://arxiv.org/abs/0904.0200), these are the primitive quivers whose mutations simply cyclically permute them. ``letter`` is 'P1' and ``rank`` is a pair of integers, (N, k) specifying the number of vertices and the source of arrow whose target is vertex 0. We require 0 < k <= N/2.
* Gale Robinson: As in Example 8.7 of Fordy Marsh (http://arxiv.org/abs/0904.0200). ``letter`` is 'GaRo' and ``rank`` is a triple of integers given by n, the number of vertices, r, and s, satisfying 0 < r <= s <= n/2.
* Exceptional mutation finite quivers: The two exceptional mutation finite quivers, found by Derksen-Owen, have ``letter`` as 'X' and ``rank`` 6 or 7, equal to the number of vertices.
* AE, BE, CE, DE: Quivers are built of one end which looks like type (affine A), B, C, or D, and the other end which looks like type E (i.e., it consists of two antennae, one of length one, and one of length two). ``letter`` is 'AE', 'BE', 'CE', or 'DE', and ``rank`` is the total number of vertices. Note that 'AE' is of a slightly different form and requires ``rank`` to be a pair of integers (i,j) just as in the case of affine type A. See Exercise 4.3 in Kac's book Infinite Dimensional Lie Algebras for more details.
Expand Down Expand Up @@ -559,6 +563,34 @@ def _samples(self):
sage: QuiverMutationType('T',(3,3,4))
['T', [3, 3, 4]]
Primitive Period 1 types::
sage: QuiverMutationType('P1', (4,2))
['P1', [4, 2]]
sage: QuiverMutationType('P1', (4,1))
['P1', [4, 1]]
sage: QuiverMutationType('P1', (5,2))
['P1', [5, 2]]
sage: QuiverMutationType('P1', (5,1))
['P1', [5, 1]]
sage: QuiverMutationType('P1', (6,3))
['P1', [6, 3]]
sage: QuiverMutationType('P1', (6,2))
['P1', [6, 2]]
sage: QuiverMutationType('P1', (6,1))
['P1', [6, 1]]
Gale-Robinson types::
sage: QuiverMutationType('GaRo', (4,1,2))
['GaRo', [4, 1, 2]]
sage: QuiverMutationType('GaRo', (5,1,2))
['GaRo', [5, 1, 2]]
sage: QuiverMutationType('GaRo', (6,1,2))
['GaRo', [6, 1, 2]]
sage: QuiverMutationType('GaRo', (6,2,3))
['GaRo', [6, 2, 3]]
Reducible types::
sage: QuiverMutationType(['A',3],['B',4])
Expand Down Expand Up @@ -1458,6 +1490,111 @@ def __init__(self, letter, rank, twist=None):
else:
_mutation_type_error( data )

# type GaRo (Gale-Robinson)
elif letter == 'GaRo':
if twist == None and type(rank) is list and len(rank) == 3 and all( rank[i] in ZZ for i in [0,1,2] ) and rank[0] > 0 and rank[1] > 0 and rank[2] > rank[1] and rank[2] <= rank[0]/2:
N, r, s = rank
self._rank = N
self._digraph.allow_multiple_edges(True)
zero_again = False
if s < N/2:
# First create primitive period 1 quiver of type ['P1', N, s]
for i in range(N):
if zero_again:
self._digraph.add_edge(i, (i+s) % N, 1 )
else:
if (i+s) %N == 0:
zero_again = True
self._digraph.add_edge(i, 0, 1)
else:
self._digraph.add_edge(i+s, i, 1)
# Secondly, subtract from it (i.e. add the reverse of) a primitive period 1 quiver of type ['P1', N, r]
zero_again = False
for i in range(N):
if zero_again:
self._digraph.add_edge( (i+r) % N, i, 1 )
else:
if (i+r) %N == 0:
zero_again = True
self._digraph.add_edge(0, i, 1)
else:
self._digraph.add_edge(i, i+r, 1)
# Thirdly, subtract from it a primitive period 1 quiver of type ['P1', N, s-r] on vertices (r+1, ..., N-r)
zero_again = False
for i in range(N-2*r):
if zero_again:
self._digraph.add_edge(((i+(s-r)) % (N-2*r) ) + r, i + r, 1 )
else:
if (i+(s-r)) % (N-2*r) == 0:
zero_again = True
self._digraph.add_edge(0+r, i+r, 1)
else:
self._digraph.add_edge(i + r, i+(s-r) + r, 1)
elif s == N/2:
# This case is similar to the above except we have to double-up the case for s = N/2
#
# First create primitive period 1 quiver of type ['P1', N, N/2]
for i in range(N):
if zero_again:
self._digraph.add_edge(i, (i+s) % N, 1 )
else:
if (i+s) %N == 0:
zero_again = True
self._digraph.add_edge(i, 0, 1)
else:
self._digraph.add_edge(i+s, i, 1)
# Secondly, subtract from it (i.e. add the reverse of) two copies of a primitive period 1 quiver of type ['P1', N, r]
zero_again = False
for i in range(N):
if zero_again:
self._digraph.add_edge( (i+r) % N, i, 1 )
else:
if (i+r) %N == 0:
zero_again = True
self._digraph.add_edge(0, i, 1)
else:
self._digraph.add_edge(i, i+r, 1)
# Thirdly, subtract from it two copies of a primitive period 1 quiver of type ['P1', N, N/2-r] on vertices (r+1, ..., N-r)
zero_again = False
for i in range(N-2*r):
if zero_again:
self._digraph.add_edge(((i+(s-r)) % (N-2*r) ) + r, i + r, 1 )
else:
if (i+(s-r)) % (N-2*r) == 0:
zero_again = True
self._digraph.add_edge(0+r, i+r, 1)
else:
self._digraph.add_edge(i + r, i+(s-r) + r, 1)
#self._digraph = GR_graph(N,r,s)

# Lastly, we get rid of the two-cycles that were created
while any ( (b,a) in self._digraph.edges(labels=False) for (a,b) in self._digraph.edges(labels=False) ):
for i in range(N):
for j in range(N):
if ((i,j) in self._digraph.edges(labels=False)) and ((j,i) in self._digraph.edges(labels=False)):
self._digraph.delete_edges( [(i,j, 1), (j,i, 1)] )
else:
_mutation_type_error( data )

# type P1 (primitive period 1)
elif letter == 'P1':
if twist == None and type(rank) is list and len(rank) ==2 and all( rank[i] in ZZ and rank[i] > 0 for i in [0,1]) and rank[1] <= rank[0]/2:
N, k = rank
self._rank = N
zero_again = False
for i in range(N):
if zero_again:
self._digraph.add_edge(i, (i+k) % N )
else:
if (i+k) %N == 0:
zero_again = True
self._digraph.add_edge(i, 0)
else:
self._digraph.add_edge(i+k, i)
else:
_mutation_type_error( data )


# type R2 (rank 2 finite mutation types)
elif letter == 'R2':
if twist == None and type(rank) is list and len(rank) == 2 and all( rank[i] in ZZ and rank[i] > 0 for i in [0,1] ):
Expand Down Expand Up @@ -2278,4 +2415,4 @@ def _edge_list_to_matrix( edges, n, m ):
M[v2,v1] = b
if v2 < n:
M[v1,v2] = a
return M
return M

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