Subword complex¶
Fix a Coxeter system \((W,S)\). The subword complex \(\mathcal{SC}(Q,w)\) associated to a word \(Q \in S^*\) and an element \(w \in W\) is the simplicial complex whose ground set is the set of positions in \(Q\) and whose facets are complements of sets of positions defining a reduced expression for \(w\).
A subword complex is a shellable sphere if and only if the Demazure product of \(Q\) equals \(w\), otherwise it is a shellable ball.
The code is optimized to be used with ReflectionGroup, it works as well with CoxeterGroup, but many methods fail for WeylGroup.
EXAMPLES:
sage: W = ReflectionGroup(['A',3]); I = list(W.index_set()) # optional - gap3
sage: Q = I + W.w0.coxeter_sorting_word(I); Q # optional - gap3
[1, 2, 3, 1, 2, 3, 1, 2, 1]
sage: S = SubwordComplex(Q,W.w0) # optional - gap3
sage: for F in S: print("{} {}".format(F, F.root_configuration())) # optional - gap3
(0, 1, 2) [(1, 0, 0), (0, 1, 0), (0, 0, 1)]
(0, 1, 8) [(1, 0, 0), (0, 1, 0), (0, 0, -1)]
(0, 2, 6) [(1, 0, 0), (0, 1, 1), (0, -1, 0)]
(0, 6, 7) [(1, 0, 0), (0, 0, 1), (0, -1, -1)]
(0, 7, 8) [(1, 0, 0), (0, -1, 0), (0, 0, -1)]
(1, 2, 3) [(1, 1, 0), (0, 0, 1), (-1, 0, 0)]
(1, 3, 8) [(1, 1, 0), (-1, 0, 0), (0, 0, -1)]
(2, 3, 4) [(1, 1, 1), (0, 1, 0), (-1, -1, 0)]
(2, 4, 6) [(1, 1, 1), (-1, 0, 0), (0, -1, 0)]
(3, 4, 5) [(0, 1, 0), (0, 0, 1), (-1, -1, -1)]
(3, 5, 8) [(0, 1, 0), (-1, -1, 0), (0, 0, -1)]
(4, 5, 6) [(0, 1, 1), (-1, -1, -1), (0, -1, 0)]
(5, 6, 7) [(-1, 0, 0), (0, 0, 1), (0, -1, -1)]
(5, 7, 8) [(-1, 0, 0), (0, -1, 0), (0, 0, -1)]
Testing that the implementation also works with CoxeterGroup:
sage: W = CoxeterGroup(['A',3]); I = list(W.index_set())
sage: Q = I + W.w0.coxeter_sorting_word(I); Q
[1, 2, 3, 1, 2, 3, 1, 2, 1]
sage: S = SubwordComplex(Q,W.w0); S
Subword complex of type ['A', 3] for Q = (1, 2, 3, 1, 2, 3, 1, 2, 1) and pi = [1, 2, 3, 1, 2, 1]
sage: P = S.increasing_flip_poset(); P; len(P.cover_relations())
Finite poset containing 14 elements
21
The root configuration works:
sage: for F in S: print("{} {}".format(F, F.root_configuration()))
(0, 1, 2) [(1, 0, 0), (0, 1, 0), (0, 0, 1)]
(0, 1, 8) [(1, 0, 0), (0, 1, 0), (0, 0, -1)]
(0, 2, 6) [(1, 0, 0), (0, 1, 1), (0, -1, 0)]
(0, 6, 7) [(1, 0, 0), (0, 0, 1), (0, -1, -1)]
(0, 7, 8) [(1, 0, 0), (0, -1, 0), (0, 0, -1)]
(1, 2, 3) [(1, 1, 0), (0, 0, 1), (-1, 0, 0)]
(1, 3, 8) [(1, 1, 0), (-1, 0, 0), (0, 0, -1)]
(2, 3, 4) [(1, 1, 1), (0, 1, 0), (-1, -1, 0)]
(2, 4, 6) [(1, 1, 1), (-1, 0, 0), (0, -1, 0)]
(3, 4, 5) [(0, 1, 0), (0, 0, 1), (-1, -1, -1)]
(3, 5, 8) [(0, 1, 0), (-1, -1, 0), (0, 0, -1)]
(4, 5, 6) [(0, 1, 1), (-1, -1, -1), (0, -1, 0)]
(5, 6, 7) [(-1, 0, 0), (0, 0, 1), (0, -1, -1)]
(5, 7, 8) [(-1, 0, 0), (0, -1, 0), (0, 0, -1)]
And the weight configuration also works:
sage: W = CoxeterGroup(['A',2])
sage: w = W.from_reduced_word([1,2,1])
sage: SC = SubwordComplex([1,2,1,2,1],w)
sage: F = SC([1,2])
sage: F.extended_weight_configuration()
[(4/3, 2/3), (2/3, 4/3), (-2/3, 2/3), (2/3, 4/3), (-2/3, 2/3)]
sage: F.extended_weight_configuration(coefficients=(1,2))
[(4/3, 2/3), (4/3, 8/3), (-2/3, 2/3), (4/3, 8/3), (-2/3, 2/3)]
One finally can compute the brick polytope, using all functionality on weight configurations, though it does not realize to live in real space:
sage: W = CoxeterGroup(['A',3]); I = list(W.index_set())
sage: Q = I + W.w0.coxeter_sorting_word(I)
sage: S = SubwordComplex(Q,W.w0)
sage: S.brick_polytope()
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 14 vertices
sage: W = CoxeterGroup(['H',3]); I = list(W.index_set())
sage: Q = I + W.w0.coxeter_sorting_word(I)
sage: S = SubwordComplex(Q,W.w0)
sage: S.brick_polytope()
doctest:...: RuntimeWarning: the polytope is build with rational vertices
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 32 vertices
AUTHORS:
Christian Stump: initial version
Vincent Pilaud: greedy flip algorithm, minor improvements, documentation
REFERENCES:
- KnuMil
Knutson and Miller. Subword complexes in Coxeter groups. Adv. Math., 184(1):161-176, 2004.
- PilStu(1,2)
Pilaud and Stump. Brick polytopes of spherical subword complexes and generalized associahedra. Adv. Math. 276:1-61, 2015.
- class sage.combinat.subword_complex.SubwordComplex(Q, w, algorithm='inductive')¶
Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.topology.simplicial_complex.SimplicialComplexFix a Coxeter system \((W,S)\). The subword complex \(\mathcal{SC}(Q,w)\) associated to a word \(Q \in S^*\) and an element \(w \in W\) is the simplicial complex whose ground set is the set of positions in \(Q\) and whose facets are complements of sets of positions defining a reduced expression for \(w\).
A subword complex is a shellable sphere if and only if the Demazure product of \(Q\) equals \(w\), otherwise it is a shellable ball.
Warning
This implementation only works for groups build using
CoxeterGroup, and does not work with groups build usingWeylGroup.EXAMPLES:
As an example, dual associahedra are subword complexes in type \(A_{n-1}\) given by the word \([1, \dots, n, 1, \dots, n, 1, \dots, n-1, \dots, 1, 2, 1]\) and the permutation \(w_0\).
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w); SC # optional - gap3 Subword complex of type ['A', 2] for Q = (1, 2, 1, 2, 1) and pi = [1, 2, 1] sage: SC.facets() # optional - gap3 [(0, 1), (0, 4), (1, 2), (2, 3), (3, 4)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w); SC Subword complex of type ['A', 2] for Q = (1, 2, 1, 2, 1) and pi = [1, 2, 1] sage: SC.facets() [(0, 1), (0, 4), (1, 2), (2, 3), (3, 4)]
REFERENCES: [KnuMil], [PilStu]
- Element¶
alias of
SubwordComplexFacet
- barycenter()¶
Return the barycenter of the brick polytope of
self.See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: SC.barycenter() # optional - gap3 (2/3, 4/3) sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: SC.barycenter() (4/3, 8/3)
- brick_fan()¶
Return the brick fan of
self.It is the normal fan of the brick polytope of
self. It is formed by the cones generated by the weight configurations of the facets ofself.See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.brick_fan() # optional - gap3 Rational polyhedral fan in 2-d lattice N sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.brick_fan() Rational polyhedral fan in 2-d lattice N
- brick_polytope(coefficients=None)¶
Return the brick polytope of
self.This polytope is the convex hull of the brick vectors of
self.INPUT:
coefficients – (optional) a list of coefficients used to scale the fundamental weights
See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: X = SC.brick_polytope(); X # optional - gap3 A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices sage: Y = SC.brick_polytope(coefficients=[1,2]); Y # optional - gap3 A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices sage: X == Y # optional - gap3 False sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: X = SC.brick_polytope(); X A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices sage: W = ReflectionGroup(['H',3]) # optional - gap3 sage: c = W.index_set(); Q = c + tuple(W.w0.coxeter_sorting_word(c)) # optional - gap3 sage: SC = SubwordComplex(Q,W.w0) # optional - gap3 sage: SC.brick_polytope() # optional - gap3 A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 32 vertices
- brick_vectors(coefficients=None)¶
Return the list of all brick vectors of facets of
self.INPUT:
coefficients – (optional) a list of coefficients used to scale the fundamental weights
See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: SC.brick_vectors() # optional - gap3 [(5/3, 7/3), (5/3, 1/3), (2/3, 7/3), (-1/3, 4/3), (-1/3, 1/3)] sage: SC.brick_vectors(coefficients=(1,2)) # optional - gap3 [(7/3, 11/3), (7/3, 2/3), (4/3, 11/3), (-2/3, 5/3), (-2/3, 2/3)] sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: SC.brick_vectors() [(10/3, 14/3), (10/3, 2/3), (4/3, 14/3), (-2/3, 8/3), (-2/3, 2/3)] sage: SC.brick_vectors(coefficients=(1,2)) [(14/3, 22/3), (14/3, 4/3), (8/3, 22/3), (-4/3, 10/3), (-4/3, 4/3)]
- cartan_type()¶
Return the Cartan type of
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.cartan_type() # optional - gap3 ['A', 2] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.cartan_type() ['A', 2]
- cover_relations(label=False)¶
Return the set of cover relations in the associated poset.
INPUT:
label – boolean (default
False) whether or not to label the cover relations by the position of flip
OUTPUT:
a list of pairs of facets
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: sorted(SC.cover_relations()) # optional - gap3 [((0, 1), (0, 4)), ((0, 1), (1, 2)), ((0, 4), (3, 4)), ((1, 2), (2, 3)), ((2, 3), (3, 4))] sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: sorted(SC.cover_relations()) [((0, 1), (0, 4)), ((0, 1), (1, 2)), ((0, 4), (3, 4)), ((1, 2), (2, 3)), ((2, 3), (3, 4))]
- dimension()¶
Return the dimension of
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: SC.dimension() # optional - gap3 1 sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: SC.dimension() 1
- facets()¶
Return all facets of
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.facets() # optional - gap3 [(0, 1), (0, 4), (1, 2), (2, 3), (3, 4)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.facets() [(0, 1), (0, 4), (1, 2), (2, 3), (3, 4)]
- greedy_facet(side='positive')¶
Return the negative (or positive) greedy facet of
self.This is the lexicographically last (or first) facet of
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.greedy_facet(side="positive") # optional - gap3 (0, 1) sage: SC.greedy_facet(side="negative") # optional - gap3 (3, 4) sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.greedy_facet(side="positive") (0, 1) sage: SC.greedy_facet(side="negative") (3, 4)
- group()¶
Return the group associated to
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.group() # optional - gap3 Irreducible real reflection group of rank 2 and type A2 sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.group() Finite Coxeter group over Integer Ring with Coxeter matrix: [1 3] [3 1]
- increasing_flip_graph(label=True)¶
Return the increasing flip graph of the subword complex.
OUTPUT:
a directed graph
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: SC.increasing_flip_graph() # optional - gap3 Digraph on 5 vertices sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: SC.increasing_flip_graph() Digraph on 5 vertices
- increasing_flip_poset()¶
Return the increasing flip poset of the subword complex.
OUTPUT:
a poset
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: SC.increasing_flip_poset() # optional - gap3 Finite poset containing 5 elements sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: SC.increasing_flip_poset() Finite poset containing 5 elements
- interval(I, J)¶
Return the interval [I,J] in the increasing flip graph subword complex.
INPUT:
I, J – two facets
OUTPUT:
a set of facets
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: F = SC([1,2]) # optional - gap3 sage: SC.interval(F, F) # optional - gap3 {(1, 2)} sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: F = SC([1,2]) sage: SC.interval(F, F) {(1, 2)}
- is_ball()¶
Return
Trueif the subword complexselfis a ball.This is the case if and only if it is not a sphere.
EXAMPLES:
sage: W = ReflectionGroup(['A',3]) # optional - gap3 sage: w = W.from_reduced_word([2,3,2]) # optional - gap3 sage: SC = SubwordComplex([3,2,3,2,3], w) # optional - gap3 sage: SC.is_ball() # optional - gap3 False sage: SC = SubwordComplex([3,2,1,3,2,3], w) # optional - gap3 sage: SC.is_ball() # optional - gap3 True sage: W = CoxeterGroup(['A',3]) sage: w = W.from_reduced_word([2,3,2]) sage: SC = SubwordComplex([3,2,3,2,3], w) sage: SC.is_ball() False
- is_double_root_free()¶
Return
Trueifselfis double-root-free.This means that the root configurations of all facets do not contain a root twice.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.is_double_root_free() # optional - gap3 True sage: SC = SubwordComplex([1,1,2,2,1,1], w) # optional - gap3 sage: SC.is_double_root_free() # optional - gap3 True sage: SC = SubwordComplex([1,2,1,2,1,2], w) # optional - gap3 sage: SC.is_double_root_free() # optional - gap3 False sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.is_double_root_free() True
- is_pure()¶
Return
Truesince all subword complexes are pure.EXAMPLES:
sage: W = ReflectionGroup(['A',3]) # optional - gap3 sage: w = W.from_reduced_word([2,3,2]) # optional - gap3 sage: SC = SubwordComplex([3,2,3,2,3], w) # optional - gap3 sage: SC.is_pure() # optional - gap3 True sage: W = CoxeterGroup(['A',3]) sage: w = W.from_reduced_word([2,3,2]) sage: SC = SubwordComplex([3,2,3,2,3], w) sage: SC.is_pure() True
- is_root_independent()¶
Return
Trueifselfis root-independent.This means that the root configuration of any (or equivalently all) facets is linearly independent.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: SC.is_root_independent() # optional - gap3 True sage: SC = SubwordComplex([1,2,1,2,1,2], W.w0) # optional - gap3 sage: SC.is_root_independent() # optional - gap3 False sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: SC.is_root_independent() True
- is_sphere()¶
Return
Trueif the subword complexselfis a sphere.EXAMPLES:
sage: W = ReflectionGroup(['A',3]) # optional - gap3 sage: w = W.from_reduced_word([2,3,2]) # optional - gap3 sage: SC = SubwordComplex([3,2,3,2,3], w) # optional - gap3 sage: SC.is_sphere() # optional - gap3 True sage: SC = SubwordComplex([3,2,1,3,2,3], w) # optional - gap3 sage: SC.is_sphere() # optional - gap3 False sage: W = CoxeterGroup(['A',3]) sage: w = W.from_reduced_word([2,3,2]) sage: SC = SubwordComplex([3,2,3,2,3], w) sage: SC.is_sphere() True
- kappa_preimages()¶
Return a dictionary containing facets of
selfas keys, and list of elements ofself.group()as values.See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: kappa = SC.kappa_preimages() # optional - gap3 sage: for F in SC: print("{} {}".format(F, [w.reduced_word() for w in kappa[F]])) # optional - gap3 (0, 1) [[]] (0, 4) [[2], [2, 1]] (1, 2) [[1]] (2, 3) [[1, 2]] (3, 4) [[1, 2, 1]] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: kappa = SC.kappa_preimages() sage: for F in SC: print("{} {}".format(F, [w.reduced_word() for w in kappa[F]])) (0, 1) [[]] (0, 4) [[2], [2, 1]] (1, 2) [[1]] (2, 3) [[1, 2]] (3, 4) [[1, 2, 1]]
- minkowski_summand(i)¶
Return the \(i\) th Minkowski summand of
self.INPUT:
\(i\) – an integer defining a position in the word \(Q\)
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], W.w0) # optional - gap3 sage: SC.minkowski_summand(1) # optional - gap3 A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex sage: W = CoxeterGroup(['A',2]) sage: SC = SubwordComplex([1,2,1,2,1], W.w0) sage: SC.minkowski_summand(1) A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex
- pi()¶
Return the element in the Coxeter group associated to
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.pi().reduced_word() # optional - gap3 [1, 2, 1] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.pi().reduced_word() [1, 2, 1]
- word()¶
Return the word in the simple generators associated to
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: SC.word() # optional - gap3 (1, 2, 1, 2, 1) sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: SC.word() (1, 2, 1, 2, 1)
- class sage.combinat.subword_complex.SubwordComplexFacet(parent, positions, facet_test=True)¶
Bases:
sage.topology.simplicial_complex.Simplex,sage.structure.element.ElementA facet of a subword complex.
Facets of the subword complex \(\mathcal{SC}(Q,w)\) are complements of sets of positions in \(Q\) defining a reduced expression for \(w\).
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1], w) # optional - gap3 sage: F = SC[0]; F # optional - gap3 (0, 1) sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1], w) sage: F = SC[0]; F (0, 1)
- brick_vector(coefficients=None)¶
Return the brick vector of
self.This is the sum of the weight vectors in the extended weight configuration.
INPUT:
coefficients – (optional) a list of coefficients used to scale the fundamental weights
See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: F.extended_weight_configuration() # optional - gap3 [(2/3, 1/3), (1/3, 2/3), (-1/3, 1/3), (1/3, 2/3), (-1/3, 1/3)] sage: F.brick_vector() # optional - gap3 (2/3, 7/3) sage: F.brick_vector(coefficients=[1,2]) # optional - gap3 (4/3, 11/3) sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]) sage: F.brick_vector() (4/3, 14/3) sage: F.brick_vector(coefficients=[1,2]) (8/3, 22/3)
- extended_root_configuration()¶
Return the extended root configuration of
self.Let \(Q = q_1 \dots q_m \in S^*\) and \(w \in W\). The extended root configuration of a facet \(I\) of \(\mathcal{SC}(Q,w)\) is the sequence \(\mathsf{r}(I, 1), \dots, \mathsf{r}(I, m)\) of roots defined by \(\mathsf{r}(I, k) = \Pi Q_{[k-1] \smallsetminus I} (\alpha_{q_k})\), where \(\Pi Q_{[k-1] \smallsetminus I}\) is the product of the simple reflections \(q_i\) for \(i \in [k-1] \smallsetminus I\) in this order.
The extended root configuration is used to perform flips efficiently.
See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: F.extended_root_configuration() # optional - gap3 [(1, 0), (1, 1), (-1, 0), (1, 1), (0, 1)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F (1, 2) sage: F.extended_root_configuration() [(1, 0), (1, 1), (-1, 0), (1, 1), (0, 1)]
- extended_weight_configuration(coefficients=None)¶
Return the extended weight configuration of
self.Let \(Q = q_1 \dots q_m \in S^*\) and \(w \in W\). The extended weight configuration of a facet \(I\) of \(\mathcal{SC}(Q,w)\) is the sequence \(\mathsf{w}(I, 1), \dots, \mathsf{w}(I, m)\) of weights defined by \(\mathsf{w}(I, k) = \Pi Q_{[k-1] \smallsetminus I} (\omega_{q_k})\), where \(\Pi Q_{[k-1] \smallsetminus I}\) is the product of the simple reflections \(q_i\) for \(i \in [k-1] \smallsetminus I\) in this order.
The extended weight configuration is used to compute the brick vector.
INPUT:
coefficients – (optional) a list of coefficients used to scale the fundamental weights
See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]) # optional - gap3 sage: F.extended_weight_configuration() # optional - gap3 [(2/3, 1/3), (1/3, 2/3), (-1/3, 1/3), (1/3, 2/3), (-1/3, 1/3)] sage: F.extended_weight_configuration(coefficients=(1,2)) # optional - gap3 [(2/3, 1/3), (2/3, 4/3), (-1/3, 1/3), (2/3, 4/3), (-1/3, 1/3)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]) sage: F.extended_weight_configuration() [(4/3, 2/3), (2/3, 4/3), (-2/3, 2/3), (2/3, 4/3), (-2/3, 2/3)] sage: F.extended_weight_configuration(coefficients=(1,2)) [(4/3, 2/3), (4/3, 8/3), (-2/3, 2/3), (4/3, 8/3), (-2/3, 2/3)]
- flip(i, return_position=False)¶
Return the facet obtained after flipping position
iinself.INPUT:
i– position in the word \(Q\) (integer).return_position– boolean (default:False) tells whether the new position should be returned as well.
OUTPUT:
The new subword complex facet.
The new position if
return_positionisTrue.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: F.flip(1) # optional - gap3 (2, 3) sage: F.flip(1, return_position=True) # optional - gap3 ((2, 3), 3) sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F (1, 2) sage: F.flip(1) (2, 3) sage: F.flip(1, return_position=True) ((2, 3), 3)
- is_vertex()¶
Return
Trueifselfis a vertex of the brick polytope ofself.parent.A facet is a vertex of the brick polytope if its root cone is pointed. Note that this property is always satisfied for root-independent subword complexes.
See also
EXAMPLES:
sage: W = ReflectionGroup(['A',1]) # optional - gap3 sage: w = W.from_reduced_word([1]) # optional - gap3 sage: SC = SubwordComplex([1,1,1],w) # optional - gap3 sage: F = SC([0,1]); F.is_vertex() # optional - gap3 True sage: F = SC([0,2]); F.is_vertex() # optional - gap3 False sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1,2,1],w) # optional - gap3 sage: F = SC([0,1,2,3]); F.is_vertex() # optional - gap3 True sage: F = SC([0,1,2,6]); F.is_vertex() # optional - gap3 False sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1,2,1],w) sage: F = SC([0,1,2,3]); F.is_vertex() True sage: F = SC([0,1,2,6]); F.is_vertex() False
- kappa_preimage()¶
Return the fiber of
selfunder the \(\kappa\) map.The \(\kappa\) map sends an element \(w \in W\) to the unique facet of \(I \in \mathcal{SC}(Q,w)\) such that the root configuration of \(I\) is contained in \(w(\Phi^+)\). In other words, \(w\) is in the preimage of
selfunder \(\kappa\) if and only if \(w^{-1}\) sends every root in the root configuration to a positive root.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: F.kappa_preimage() # optional - gap3 [(1,4)(2,3)(5,6)] sage: F = SC([0,4]); F # optional - gap3 (0, 4) sage: F.kappa_preimage() # optional - gap3 [(1,3)(2,5)(4,6), (1,2,6)(3,4,5)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F (1, 2) sage: F.kappa_preimage() [ [-1 1] [ 0 1] ] sage: F = SC([0,4]); F (0, 4) sage: F.kappa_preimage() [ [ 1 0] [-1 1] [ 1 -1], [-1 0] ]
- plot(list_colors=None, labels=[], thickness=3, fontsize=14, shift=(0, 0), compact=False, roots=True, **args)¶
In type \(A\) or \(B\), plot a pseudoline arrangement representing the facet
self.Pseudoline arrangements are graphical representations of facets of types A or B subword complexes.
INPUT:
list_colors– list (default:[]) to change the colors of the pseudolines.labels– list (default:[]) to change the labels of the pseudolines.thickness– integer (default:3) for the thickness of the pseudolines.fontsize– integer (default:14) for the size of the font used for labels.shift– couple of coordinates (default:(0,0)) to change the origin.compact– boolean (default:False) to require a more compact representation.roots– boolean (default:True) to print the extended root configuration.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F.plot() # optional - gap3 Graphics object consisting of 26 graphics primitives sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F.plot() Graphics object consisting of 26 graphics primitives sage: W = ReflectionGroup(['B',3]) # optional - gap3 sage: c = W.from_reduced_word([1,2,3]) # optional - gap3 sage: Q = c.reduced_word()*2 + W.w0.coxeter_sorting_word(c) # optional - gap3 sage: SC = SubwordComplex(Q, W.w0) # optional - gap3 sage: F = SC[15]; F.plot() # optional - gap3 Graphics object consisting of 52 graphics primitives
REFERENCES: [PilStu]
- root_cone()¶
Return the polyhedral cone generated by the root configuration of
self.See also
EXAMPLES:
sage: W = ReflectionGroup(['A',1]) # optional - gap3 sage: w = W.from_reduced_word([1]) # optional - gap3 sage: SC = SubwordComplex([1,1,1],w) # optional - gap3 sage: F = SC([0,2]); F.root_cone() # optional - gap3 1-d cone in 1-d lattice N sage: W = CoxeterGroup(['A',1]) sage: w = W.from_reduced_word([1]) sage: SC = SubwordComplex([1,1,1],w) sage: F = SC([0,2]); F.root_cone() 1-d cone in 1-d lattice N
- root_configuration()¶
Return the root configuration of
self.Let \(Q = q_1 \dots q_m \in S^*\) and \(w \in W\). The root configuration of a facet \(I = [i_1, \dots, i_n]\) of \(\mathcal{SC}(Q,w)\) is the sequence \(\mathsf{r}(I, i_1), \dots, \mathsf{r}(I, i_n)\) of roots defined by \(\mathsf{r}(I, k) = \Pi Q_{[k-1] \smallsetminus I} (\alpha_{q_k})\), where \(\Pi Q_{[k-1] \smallsetminus I}\) is the product of the simple reflections \(q_i\) for \(i \in [k-1] \smallsetminus I\) in this order.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: F.root_configuration() # optional - gap3 [(1, 1), (-1, 0)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F (1, 2) sage: F.root_configuration() # optional - gap3 [(1, 1), (-1, 0)]
- show(*kwds, **args)¶
Show the facet
self.See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F.show() # optional - gap3
- upper_root_configuration()¶
Return the positive roots of the root configuration of
self.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: F.root_configuration() # optional - gap3 [(1, 1), (-1, 0)] sage: F.upper_root_configuration() # optional - gap3 [(1, 0)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F (1, 2) sage: F.upper_root_configuration() [(1, 0)]
- weight_cone()¶
Return the polyhedral cone generated by the weight configuration of
self.See also
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: WC = F.weight_cone(); WC # optional - gap3 2-d cone in 2-d lattice N sage: WC.rays() # optional - gap3 N( 1, 2), N(-1, 1) in 2-d lattice N sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F (1, 2) sage: WC = F.weight_cone(); WC 2-d cone in 2-d lattice N
- weight_configuration()¶
Return the weight configuration of
self.Let \(Q = q_1 \dots q_m \in S^*\) and \(w \in W\). The weight configuration of a facet \(I = [i_1, \dots, i_n]\) of \(\mathcal{SC}(Q,w)\) is the sequence \(\mathsf{w}(I, i_1), \dots, \mathsf{w}(I, i_n)\) of weights defined by \(\mathsf{w}(I, k) = \Pi Q_{[k-1] \smallsetminus I} (\omega_{q_k})\), where \(\Pi Q_{[k-1] \smallsetminus I}\) is the product of the simple reflections \(q_i\) for \(i \in [k-1] \smallsetminus I\) in this order.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: w = W.from_reduced_word([1,2,1]) # optional - gap3 sage: SC = SubwordComplex([1,2,1,2,1],w) # optional - gap3 sage: F = SC([1,2]); F # optional - gap3 (1, 2) sage: F.weight_configuration() # optional - gap3 [(1/3, 2/3), (-1/3, 1/3)] sage: W = CoxeterGroup(['A',2]) sage: w = W.from_reduced_word([1,2,1]) sage: SC = SubwordComplex([1,2,1,2,1],w) sage: F = SC([1,2]); F (1, 2) sage: F.weight_configuration() [(2/3, 4/3), (-2/3, 2/3)]