Quiver mutation types¶
AUTHORS:
- Gregg Musiker (2012, initial version)
- Christian Stump (2012, initial version)
- Hugh Thomas (2012, initial version)
-
sage.combinat.cluster_algebra_quiver.quiver_mutation_type.
QuiverMutationType
(*args)¶ - Quiver mutation types can be seen as a slight generalization of
- generalized Cartan types.
Background on generalized Cartan types can be found at
For the compendium on the cluster algebra and quiver package in Sage see [MS2011]
A \(B\)-matrix is a skew-symmetrizable \(( n \times n )\)-matrix \(M\). I.e., there exists an invertible diagonal matrix \(D\) such that \(DM\) is skew-symmetric. \(M\) can be encoded as a quiver by having a directed edge from vertex \(i\) to vertex \(j\) with label \((a,b)\) if \(a = M_{i,j} > 0\) and \(b = M_{j,i} < 0\). We consider quivers up to mutation equivalence.
To a quiver mutation type we can associate a generalized Cartan type by sending \(M\) to the generalized Cartan matrix \(C(M)\) obtained by replacing all positive entries by their negatives and adding \(2\)‘s on the main diagonal.
QuiverMutationType
constructs a quiver mutation type object. For more detail on the possible different types, please see the compendium.INPUT:
The input consists either of a quiver mutation type, or of a
letter
(a string), arank
(one integer or a list/tuple of integers), and an optionaltwist
(an integer or a list of integers). There are several different naming conventions for quiver mutation types.Finite type –
letter
is a Dynkin type (A-G), andrank
is the rank.Affine type – there is more than one convention for naming affine types.
- Kac’s notation:
letter
is a Dynkin type,rank
is the rank of the associated finite Dynkin diagram, andtwist
is the twist, which could be 1, 2, or 3. In the special case of affine type A, there is more than one quiver mutation type associated to the Cartan type. In this case only,rank
is a pair of integers (i,j), giving the number of edges pointing clockwise and the number of edges pointing counter-clockwise. The total number of vertices is given by i+j in this case. - Naive notation:
letter
is one of ‘BB’, ‘BC’, ‘BD’, ‘CC’, ‘CD’. The name specifies the two ends of the diagram, which are joined by a path. The total number of vertices is given byrank +1
(to match the indexing people expect because these are affine types). In general,rank
must be large enough for the picture to make sense, but we acceptletter
isBC
andrank=1
. - Macdonald notation: for the dual of an untwisted affine type (such as [‘C’, 6,1]), we accept a twist of -1 (i.e., [‘C’,6,-1]).
- Kac’s notation:
Elliptic type –
letter
is a Dynkin type,rank
is the rank of the finite Dynkin diagram, andtwist
is a tuple of two integers. We follow Saito’s notation.Other shapes:
- Rank 2:
letter
is ‘R2’, andrank
is a pair of integers specifying the label on the unique edge. - Triangle:
letter
isTR
, andrank
is the number of vertices along a side. - T: This defines a quiver shaped like a T.
letter
is ‘T’, and therank
is a triple, whose entries specify the number of vertices along each path from the branch point (counting the branch point). - 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. - Exceptional mutation finite quivers: The two exceptional
mutation finite quivers, found by Derksen-Owen, have
letter
as ‘X’ andrank
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’, andrank
is the total number of vertices. Note that ‘AE’ is of a slightly different form and requiresrank
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. - Infinite type E: It is also possible to obtain infinite-type E
quivers by specifying
letter
as ‘E’ andrank
as the number of vertices.
- Rank 2:
REFERENCES:
- A good reference for finite and affine Dynkin diagrams, including Kac’s notation, is the Wikipedia article Dynkin_diagram.
- A good reference for the skew-symmetrizable elliptic diagrams is “Cluster algebras of finite mutation type via unfolding” by A. Felikson, M. Shapiro, and P. Tumarkin, [FST2012].
EXAMPLES:
Finite types:
sage: QuiverMutationType('A',1) ['A', 1] sage: QuiverMutationType('A',5) ['A', 5] sage: QuiverMutationType('B',2) ['B', 2] sage: QuiverMutationType('B',5) ['B', 5] sage: QuiverMutationType('C',2) ['B', 2] sage: QuiverMutationType('C',5) ['C', 5] sage: QuiverMutationType('D',2) [ ['A', 1], ['A', 1] ] sage: QuiverMutationType('D',3) ['A', 3] sage: QuiverMutationType('D',4) ['D', 4] sage: QuiverMutationType('E',6) ['E', 6] sage: QuiverMutationType('G',2) ['G', 2] sage: QuiverMutationType('A',(1,0),1) ['A', 1] sage: QuiverMutationType('A',(2,0),1) [ ['A', 1], ['A', 1] ] sage: QuiverMutationType('A',(7,0),1) ['D', 7]
Affine types:
sage: QuiverMutationType('A',(1,1),1) ['A', [1, 1], 1] sage: QuiverMutationType('A',(2,4),1) ['A', [2, 4], 1] sage: QuiverMutationType('BB',2,1) ['BB', 2, 1] sage: QuiverMutationType('BB',4,1) ['BB', 4, 1] sage: QuiverMutationType('CC',2,1) ['CC', 2, 1] sage: QuiverMutationType('CC',4,1) ['CC', 4, 1] sage: QuiverMutationType('BC',1,1) ['BC', 1, 1] sage: QuiverMutationType('BC',5,1) ['BC', 5, 1] sage: QuiverMutationType('BD',3,1) ['BD', 3, 1] sage: QuiverMutationType('BD',5,1) ['BD', 5, 1] sage: QuiverMutationType('CD',3,1) ['CD', 3, 1] sage: QuiverMutationType('CD',5,1) ['CD', 5, 1] sage: QuiverMutationType('D',4,1) ['D', 4, 1] sage: QuiverMutationType('D',6,1) ['D', 6, 1] sage: QuiverMutationType('E',6,1) ['E', 6, 1] sage: QuiverMutationType('E',7,1) ['E', 7, 1] sage: QuiverMutationType('E',8,1) ['E', 8, 1] sage: QuiverMutationType('F',4,1) ['F', 4, 1] sage: QuiverMutationType('F',4,-1) ['F', 4, -1] sage: QuiverMutationType('G',2,1) ['G', 2, 1] sage: QuiverMutationType('G',2,-1) ['G', 2, -1] sage: QuiverMutationType('A',3,2) == QuiverMutationType('D',3,2) True
Affine types using Kac’s Notation:
sage: QuiverMutationType('A',1,1) ['A', [1, 1], 1] sage: QuiverMutationType('B',5,1) ['BD', 5, 1] sage: QuiverMutationType('C',5,1) ['CC', 5, 1] sage: QuiverMutationType('A',2,2) ['BC', 1, 1] sage: QuiverMutationType('A',7,2) ['CD', 4, 1] sage: QuiverMutationType('A',8,2) ['BC', 4, 1] sage: QuiverMutationType('D',6,2) ['BB', 5, 1] sage: QuiverMutationType('E',6,2) ['F', 4, -1] sage: QuiverMutationType('D',4,3) ['G', 2, -1]
Elliptic types:
sage: QuiverMutationType('E',6,[1,1]) ['E', 6, [1, 1]] sage: QuiverMutationType('F',4,[2,1]) ['F', 4, [1, 2]] sage: QuiverMutationType('G',2,[3,3]) ['G', 2, [3, 3]]
Mutation finite types:
rank 2 cases:
sage: QuiverMutationType('R2',(1,1)) ['A', 2] sage: QuiverMutationType('R2',(1,2)) ['B', 2] sage: QuiverMutationType('R2',(1,3)) ['G', 2] sage: QuiverMutationType('R2',(1,4)) ['BC', 1, 1] sage: QuiverMutationType('R2',(1,5)) ['R2', [1, 5]] sage: QuiverMutationType('R2',(2,2)) ['A', [1, 1], 1] sage: QuiverMutationType('R2',(3,5)) ['R2', [3, 5]]
Exceptional Derksen-Owen quivers:
sage: QuiverMutationType('X',6) ['X', 6]
(Mainly) mutation infinite types:
Infinite type E:
sage: QuiverMutationType('E',9) ['E', 8, 1] sage: QuiverMutationType('E',10) ['E', 10] sage: QuiverMutationType('E',12) ['E', 12] sage: QuiverMutationType('AE',(2,3)) ['AE', [2, 3]] sage: QuiverMutationType('BE',5) ['BE', 5] sage: QuiverMutationType('CE',5) ['CE', 5] sage: QuiverMutationType('DE',6) ['DE', 6]
Grassmannian types:
sage: QuiverMutationType('GR',(2,4)) ['A', 1] sage: QuiverMutationType('GR',(2,6)) ['A', 3] sage: QuiverMutationType('GR',(3,6)) ['D', 4] sage: QuiverMutationType('GR',(3,7)) ['E', 6] sage: QuiverMutationType('GR',(3,8)) ['E', 8] sage: QuiverMutationType('GR',(3,10)) ['GR', [3, 10]]
Triangular types:
sage: QuiverMutationType('TR',2) ['A', 3] sage: QuiverMutationType('TR',3) ['D', 6] sage: QuiverMutationType('TR',4) ['E', 8, [1, 1]] sage: QuiverMutationType('TR',5) ['TR', 5]
T types:
sage: QuiverMutationType('T',(1,1,1)) ['A', 1] sage: QuiverMutationType('T',(1,1,4)) ['A', 4] sage: QuiverMutationType('T',(1,4,4)) ['A', 7] sage: QuiverMutationType('T',(2,2,2)) ['D', 4] sage: QuiverMutationType('T',(2,2,4)) ['D', 6] sage: QuiverMutationType('T',(2,3,3)) ['E', 6] sage: QuiverMutationType('T',(2,3,4)) ['E', 7] sage: QuiverMutationType('T',(2,3,5)) ['E', 8] sage: QuiverMutationType('T',(2,3,6)) ['E', 8, 1] sage: QuiverMutationType('T',(2,3,7)) ['E', 10] sage: QuiverMutationType('T',(3,3,3)) ['E', 6, 1] sage: QuiverMutationType('T',(3,3,4)) ['T', [3, 3, 4]]
Reducible types:
sage: QuiverMutationType(['A',3],['B',4]) [ ['A', 3], ['B', 4] ]
-
class
sage.combinat.cluster_algebra_quiver.quiver_mutation_type.
QuiverMutationTypeFactory
¶ Bases:
sage.structure.sage_object.SageObject
-
samples
(finite=None, affine=None, elliptic=None, mutation_finite=None)¶ Return a sample of the available quiver mutations types.
INPUT:
finite
affine
elliptic
mutation_finite
All four input keywords default values are
None
. If set toTrue
orFalse
, only these samples are returned.EXAMPLES:
sage: QuiverMutationType.samples() [['A', 1], ['A', 5], ['B', 2], ['B', 5], ['C', 3], ['C', 5], [ ['A', 1], ['A', 1] ], ['D', 5], ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2], ['A', [1, 1], 1], ['A', [4, 5], 1], ['D', 4, 1], ['BB', 5, 1], ['E', 6, [1, 1]], ['E', 7, [1, 1]], ['R2', [1, 5]], ['R2', [3, 5]], ['E', 10], ['BE', 5], ['GR', [3, 10]], ['T', [3, 3, 4]]] sage: QuiverMutationType.samples(finite=True) [['A', 1], ['A', 5], ['B', 2], ['B', 5], ['C', 3], ['C', 5], [ ['A', 1], ['A', 1] ], ['D', 5], ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2]] sage: QuiverMutationType.samples(affine=True) [['A', [1, 1], 1], ['A', [4, 5], 1], ['D', 4, 1], ['BB', 5, 1]] sage: QuiverMutationType.samples(elliptic=True) [['E', 6, [1, 1]], ['E', 7, [1, 1]]] sage: QuiverMutationType.samples(mutation_finite=False) [['R2', [1, 5]], ['R2', [3, 5]], ['E', 10], ['BE', 5], ['GR', [3, 10]], ['T', [3, 3, 4]]]
-
-
sage.combinat.cluster_algebra_quiver.quiver_mutation_type.
QuiverMutationType_Irreducible
¶ The mutation type for a cluster algebra or a quiver. Should not be called directly, but through QuiverMutationType.
-
sage.combinat.cluster_algebra_quiver.quiver_mutation_type.
QuiverMutationType_Reducible
¶ The mutation type for a cluster algebra or a quiver. Should not be called directly, but through QuiverMutationType. Inherits from QuiverMutationType_abstract.
-
sage.combinat.cluster_algebra_quiver.quiver_mutation_type.
QuiverMutationType_abstract
¶ EXAMPLES:
sage: mut_type1 = QuiverMutationType('A',5) sage: mut_type2 = QuiverMutationType('A',5) sage: mut_type3 = QuiverMutationType('A',6) sage: mut_type1 == mut_type2 True sage: mut_type1 == mut_type3 False
-
sage.combinat.cluster_algebra_quiver.quiver_mutation_type.
save_quiver_data
(n, up_to=True, types='ClassicalExceptional', verbose=True)¶ Save mutation classes of certain quivers of ranks up to and equal to
n
or equal ton
toDOT_SAGE/cluster_algebra_quiver/mutation_classes_n.dig6
.This data will then be used to determine quiver mutation types.
INPUT:
n
: the rank (or the upper limit on the rank) of the mutation classes that are being saved.up_to
– (default:True
) ifTrue
, saves data for ranks smaller than or equal ton
. IfFalse
, saves data for rank exactlyn
.types
– (default:’ClassicalExceptional’) if all, saves data for both exceptional mutation-finite quivers and for classical quiver. The input ‘Exceptional’ or ‘Classical’ is also allowed to save only part of this data.