Artin Groups#
Artin groups are implemented as a particular case of finitely presented groups. For finite-type Artin groups, there is a specific left normal form using the Garside structure associated to the lift the long element of the corresponding Coxeter group.
AUTHORS:
Travis Scrimshaw (2018-02-05): Initial version
- class sage.groups.artin.ArtinGroup(coxeter_matrix, names)#
Bases:
FinitelyPresentedGroupAn Artin group.
Fix an index set \(I\). Let \(M = (m_{ij})_{i,j \in I}\) be a
Coxeter matrix. An Artin group is a group \(A_M\) that has a presentation given by generators \(\{ s_i \mid i \in I \}\) and relations\[\underbrace{s_i s_j s_i \cdots}_{m_{ij}} = \underbrace{s_j s_i s_j \cdots}_{\text{$m_{ji}$ factors}}\]for all \(i,j \in I\) with the usual convention that \(m_{ij} = \infty\) implies no relation between \(s_i\) and \(s_j\). There is a natural corresponding Coxeter group \(W_M\) by imposing the additional relations \(s_i^2 = 1\) for all \(i \in I\). Furthermore, there is a natural section of \(W_M\) by sending a reduced word \(s_{i_1} \cdots s_{i_{\ell}} \mapsto s_{i_1} \cdots s_{i_{\ell}}\).
Artin groups \(A_M\) are classified based on the Coxeter data:
\(A_M\) is of finite type or spherical if \(W_M\) is finite;
\(A_M\) is of affine type if \(W_M\) is of affine type;
\(A_M\) is of large type if \(m_{ij} \geq 4\) for all \(i,j \in I\);
\(A_M\) is of extra-large type if \(m_{ij} \geq 5\) for all \(i,j \in I\);
\(A_M\) is right-angled if \(m_{ij} \in \{2,\infty\}\) for all \(i,j \in I\).
Artin groups are conjectured to have many nice properties:
Artin groups are torsion free.
Finite type Artin groups have \(Z(A_M) = \ZZ\) and infinite type Artin groups have trivial center.
Artin groups have solvable word problems.
\(H_{W_M} / W_M\) is a \(K(A_M, 1)\)-space, where \(H_W\) is the hyperplane complement of the Coxeter group \(W\) acting on \(\CC^n\).
These conjectures are known when the Artin group is finite type and a number of other cases. See, e.g., [GP2012] and references therein.
INPUT:
coxeter_data– data defining a Coxeter matrixnames– string or list/tuple/iterable of strings (default:'s'); the generator names or name prefix
EXAMPLES:
sage: A.<a,b,c> = ArtinGroup(['B',3]); A # needs sage.rings.number_field Artin group of type ['B', 3] sage: ArtinGroup(['B',3]) # needs sage.rings.number_field Artin group of type ['B', 3]
The input must always include the Coxeter data, but the
namescan either be a string representing the prefix of the names or the explicit names of the generators. Otherwise the default prefix of's'is used:sage: ArtinGroup(['B',2]).generators() # needs sage.rings.number_field (s1, s2) sage: ArtinGroup(['B',2], 'g').generators() # needs sage.rings.number_field (g1, g2) sage: ArtinGroup(['B',2], 'x,y').generators() # needs sage.rings.number_field (x, y)
REFERENCES:
See also
- Element#
alias of
ArtinGroupElement
- an_element()#
Return an element of
self.EXAMPLES:
sage: A = ArtinGroup(['B',2]) # needs sage.rings.number_field sage: A.an_element() # needs sage.rings.number_field s1
- as_permutation_group()#
Return an isomorphic permutation group.
This raises a
ValueErrorerror since Artin groups are infinite and have no corresponding permutation group.EXAMPLES:
sage: Gamma = graphs.CycleGraph(5) sage: G = RightAngledArtinGroup(Gamma) sage: G.as_permutation_group() Traceback (most recent call last): ... ValueError: the group is infinite sage: A = ArtinGroup(['D',4], 'g') # needs sage.rings.number_field sage: A.as_permutation_group() # needs sage.rings.number_field Traceback (most recent call last): ... ValueError: the group is infinite
- cardinality()#
Return the number of elements of
self.OUTPUT:
Infinity.
EXAMPLES:
sage: Gamma = graphs.CycleGraph(5) sage: G = RightAngledArtinGroup(Gamma) sage: G.cardinality() +Infinity sage: A = ArtinGroup(['A',1]) # needs sage.rings.number_field sage: A.cardinality() # needs sage.rings.number_field +Infinity
- coxeter_group()#
Return the Coxeter group of
self.EXAMPLES:
sage: A = ArtinGroup(['D',4]) # needs sage.rings.number_field sage: A.coxeter_group() # needs sage.rings.number_field Finite Coxeter group over Integer Ring with Coxeter matrix: [1 3 2 2] [3 1 3 3] [2 3 1 2] [2 3 2 1]
- coxeter_matrix()#
Return the Coxeter matrix of
self.EXAMPLES:
sage: A = ArtinGroup(['B',3]) # needs sage.rings.number_field sage: A.coxeter_matrix() # needs sage.rings.number_field [1 3 2] [3 1 4] [2 4 1]
- coxeter_type()#
Return the Coxeter type of
self.EXAMPLES:
sage: A = ArtinGroup(['D',4]) # needs sage.rings.number_field sage: A.coxeter_type() # needs sage.rings.number_field Coxeter type of ['D', 4]
- index_set()#
Return the index set of
self.OUTPUT:
A tuple.
EXAMPLES:
sage: A = ArtinGroup(['E',7]) # needs sage.rings.number_field sage: A.index_set() # needs sage.rings.number_field (1, 2, 3, 4, 5, 6, 7)
- order()#
Return the number of elements of
self.OUTPUT:
Infinity.
EXAMPLES:
sage: Gamma = graphs.CycleGraph(5) sage: G = RightAngledArtinGroup(Gamma) sage: G.cardinality() +Infinity sage: A = ArtinGroup(['A',1]) # needs sage.rings.number_field sage: A.cardinality() # needs sage.rings.number_field +Infinity
- some_elements()#
Return a list of some elements of
self.EXAMPLES:
sage: A = ArtinGroup(['B',3]) # needs sage.rings.number_field sage: A.some_elements() # needs sage.rings.number_field [s1, s1*s2*s3, (s1*s2*s3)^3]
- class sage.groups.artin.ArtinGroupElement(parent, x, check=True)#
Bases:
FinitelyPresentedGroupElementAn element of an Artin group.
It is a particular case of element of a finitely presented group.
EXAMPLES:
sage: # needs sage.rings.number_field sage: A.<s1,s2,s3> = ArtinGroup(['B',3]) sage: A Artin group of type ['B', 3] sage: s1 * s2 / s3 / s2 s1*s2*s3^-1*s2^-1 sage: A((1, 2, -3, -2)) s1*s2*s3^-1*s2^-1
- coxeter_group_element(W=None)#
Return the corresponding Coxeter group element under the natural projection.
INPUT:
W– (default:self.parent().coxeter_group()) the image Coxeter group
OUTPUT:
An element of the Coxeter group
W.EXAMPLES:
sage: # needs sage.rings.number_field sage: B.<s1,s2,s3> = ArtinGroup(['B',3]) sage: b = s1 * s2 / s3 / s2 sage: b1 = b.coxeter_group_element(); b1 [ 1 -1 0] [ 2 -1 0] [ a -a 1] sage: b.coxeter_group_element().reduced_word() [1, 2, 3, 2] sage: A.<s1,s2,s3> = ArtinGroup(['A',3]) sage: c = s1 * s2 *s3 sage: c1 = c.coxeter_group_element(); c1 [4, 1, 2, 3] sage: c1.reduced_word() [3, 2, 1] sage: c.coxeter_group_element(W=SymmetricGroup(4)) (1,4,3,2) sage: A.<s1,s2,s3> = BraidGroup(4) sage: c = s1 * s2 * s3^-1 sage: c0 = c.coxeter_group_element(); c0 [4, 1, 2, 3] sage: c1 = c.coxeter_group_element(W=SymmetricGroup(4)); c1 (1,4,3,2)
From an element of the Coxeter group it is possible to recover the image by the standard section to the Artin group:
sage: # needs sage.rings.number_field sage: B(b1) s1*s2*s3*s2 sage: A(c0) s1*s2*s3 sage: A(c0) == A(c1) True
- exponent_sum()#
Return the exponent sum of
self.OUTPUT:
Integer.
EXAMPLES:
sage: # needs sage.rings.number_field sage: A = ArtinGroup(['E',6]) sage: b = A([1, 4, -3, 2]) sage: b.exponent_sum() 2 sage: b = A([]) sage: b.exponent_sum() 0 sage: B = BraidGroup(5) sage: b = B([1, 4, -3, 2]) sage: b.exponent_sum() 2 sage: b = B([]) sage: b.exponent_sum() 0
- class sage.groups.artin.FiniteTypeArtinGroup(coxeter_matrix, names)#
Bases:
ArtinGroupA finite-type Artin group.
An Artin group is finite-type or spherical if the corresponding Coxeter group is finite. Finite type Artin groups are known to be torsion free, have a Garside structure given by \(\Delta\) (see
delta()) and have a center generated by \(\Delta\).See also
EXAMPLES:
sage: ArtinGroup(['E',7]) # needs sage.rings.number_field Artin group of type ['E', 7]
Since the word problem for finite-type Artin groups is solvable, their Cayley graph can be locally obtained as follows (see github issue #16059):
sage: def ball(group, radius): ....: ret = set() ....: ret.add(group.one()) ....: for length in range(1, radius): ....: for w in Words(alphabet=group.gens(), length=length): ....: ret.add(prod(w)) ....: return ret sage: A = ArtinGroup(['B',3]) # needs sage.rings.number_field sage: GA = A.cayley_graph(elements=ball(A, 4), generators=A.gens()); GA # needs sage.rings.number_field Digraph on 32 vertices
Since the Artin group has nontrivial relations, this graph contains less vertices than the one associated to the free group (which is a tree):
sage: F = FreeGroup(3) sage: GF = F.cayley_graph(elements=ball(F, 4), generators=F.gens()); GF # needs sage.combinat Digraph on 40 vertices
- Element#
alias of
FiniteTypeArtinGroupElement
- delta()#
Return the \(\Delta\) element of
self.EXAMPLES:
sage: A = ArtinGroup(['B',3]) # needs sage.rings.number_field sage: A.delta() # needs sage.rings.number_field s3*(s2*s3*s1)^2*s2*s1 sage: A = ArtinGroup(['G',2]) # needs sage.rings.number_field sage: A.delta() # needs sage.rings.number_field (s2*s1)^3 sage: B = BraidGroup(5) sage: B.delta() s0*s1*s2*s3*s0*s1*s2*s0*s1*s0
- class sage.groups.artin.FiniteTypeArtinGroupElement(parent, x, check=True)#
Bases:
ArtinGroupElementAn element of a finite-type Artin group.
- left_normal_form()#
Return the left normal form of
self.OUTPUT:
A tuple of simple generators in the left normal form. The first element is a power of \(\Delta\), and the rest are elements of the natural section lift from the corresponding Coxeter group.
EXAMPLES:
sage: # needs sage.rings.number_field sage: A = ArtinGroup(['B',3]) sage: A([1]).left_normal_form() (1, s1) sage: A([-1]).left_normal_form() (s1^-1*(s2^-1*s1^-1*s3^-1)^2*s2^-1*s3^-1, s3*(s2*s3*s1)^2*s2) sage: A([1, 2, 2, 1, 2]).left_normal_form() (1, s1*s2*s1, s2*s1) sage: A([3, 3, -2]).left_normal_form() (s1^-1*(s2^-1*s1^-1*s3^-1)^2*s2^-1*s3^-1, s3*s1*s2*s3*s2*s1, s3, s3*s2*s3) sage: A([1, 2, 3, -1, 2, -3]).left_normal_form() (s1^-1*(s2^-1*s1^-1*s3^-1)^2*s2^-1*s3^-1, (s3*s1*s2)^2*s1, s1*s2*s3*s2) sage: A([1,2,1,3,2,1,3,2,3,3,2,3,1,2,3,1,2,3,1,2]).left_normal_form() ((s3*(s2*s3*s1)^2*s2*s1)^2, s3*s2) sage: B = BraidGroup(4) sage: b = B([1, 2, 3, -1, 2, -3]) sage: b.left_normal_form() (s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1, s0*s1*s2*s1*s0, s0*s2*s1) sage: c = B([1]) sage: c.left_normal_form() (1, s0)