In general these groups have firstly been investigated by Coxeter, H.

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Coxeter showed, that these groups are finite as long as the number of strands is less than 6 and infinite else-wise. He showed that each series of finite cubic braid group factors must be an epimorphic image of one of his two series, as long as the groups with less than 5 strands are the full cubic braid groups, whereas the group on 5 strands is not.

This class implements all the groups considered by Coxeter and Assion as finitely presented groups together with the classical realizations given by the authors. It also contains the conversion maps between the two ways of realization. In addition the user can construct other realizations and maps to matrix groups with help of the Burau representation.

Construct cubic braid groups as instance of CubicBraidGroup which have been investigated by J. Assion using the notation S m. For more information type CubicBraidGroup? The number of strands. This argument is passed to the corresponding argument of the classcall of CubicBraidGroup.

Assion using the notation U m. Bases: sage. This class models elements of cubic factor groups of the braid group. It is the element class of the CubicBraidGroup. For more information see the documentation of the parent CubicBraidGroup.

Return the canonical braid preimage of self as Object of the class Braid.

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The preimage of self as instance of Braid. This method uses the same method belonging to Braidbut reduces the indeterminate to a primitive sixth resp. The Burau matrix of the cubic braid coset with entries in the domain given by the options.

This class implements factor groups of the Artin braid group mapping their generators to elements of order 3 see the module header for more information on these groups. Coxeter, see explanation below of enum type CubicBraidGroup. AssionS or CubicBraidGroup.

AssionU the additional relations due to Assion are added:. But anyway, the instances for CubicBraidGroup. Coxeter, CubicBraidGroup. AssionS and CubicBraidGroup. AssionU are different, since they have different classical realizations implemented. Creates an isomorphic image of self as a classical group according to the construction given by Coxeter resp. This boolean does effect the cases of Assion groups when they are realized as projective groups, only.Bases: sage.

Bases: object. Returns all the elements that cover self in Bruhat order. This method returns all neighbors of w under the Coxeter-Knuth relations oriented from left to right. It was defined in [BHZ]. This partial order is not a lattice, as there is no unique maximal element. It can be succintly defined as follows. Return the Bruhat poset of self.

If instead one wants the elements to be plain elements of the Coxeter group, one can use the facade option:. Poset for more on posets and facade posets. Use the symmetric group in the examples for nicer outputand print the edges for a stronger test. See arXiv Delta sequences are certain 2-colored minimal factorizations of c into reflections.

Return the inversion sequence corresponding to the word in indices of simple generators of self. Return True since self is a real reflection group. If Truethen return instead a reduced decomposition of the longest element. Return the permutahedron of self. If function is too slow, switching the base ring to RDF will almost certainly speed things up. The partial order is given by simultaneous inclusion of inversion sets and subgroups attached to every element. The precise description used here can be found in [STW].

Return the longest element of self. Returns the left resp. This is the transitive closure of the union of left and right order.

coxeter group sage

Note that this is not a lattice:. Finite Complex Reflection Groups. Finite Crystals. Navigation index modules next previous Sage 9. CategoryWithAxiom The category of finite Coxeter groups. Finite Category of finite coxeter groups sage: FiniteCoxeterGroups. See also Poset for more on posets and facade posets. Todo Use the symmetric group in the examples for nicer outputand print the edges for a stronger test. See also permutahedron. Note The result is expressed in the root basis coordinates.

Note If function is too slow, switching the base ring to RDF will almost certainly speed things up. UserWarning: This polyhedron data is numerically complicated; cdd could not convert between the inexact V and H representation without loss of data. The resulting object might show inconsistencies.

coxeter group sage

Quick search. Created using Sphinx 3.Hi, I'm trying to implement a certain Coxeter group in Sage, for the purpose of building the corresponding Iwahori-Hecke algebra over it using sage. In the documentation of the constructor of this class, it says that it takes a "Coxeter group" as an argument. However, Coxeter groups as generated by sage.

CoxeterGroup aren't actually instances of a common superclass. Afaict, the methods of IwahoriHeckeAlgebra simply implicitly assume the existence of certain instance methods like W. The equality with IHA4 fails maybe for reasons of internal representation. But one can also work in IHA4.

How it is constructed, or at least mathematically defined. Okay, here are the minimal steps that you need to take for a complete working example, see here :. The requirements on the semantics of these methods are implicit, but can be gathered quite easily from the source file of sage. Subclassing from UniqueRepresentation is necessaryat least if you plan on building the corresponding Iwahori-Hecke algebra over it.

The reason for this is that if you don't subclass from UniqueRepresentationthe object returned by. For this reason, also the following command will fail. I don't exactly know how subclassing from UniqueRepresentation fixes that problem, but it seems to have to do with the magic that Sage uses when dynamically creating classes.

Please start posting anonymously - your entry will be published after you log in or create a new account. Asked: How can I list all the elements of the affine Coxeter group of type A having a specific length.

Automorphism group of Coxeter groups. Path between representatives and normal forms in Coxeter groups. First time here? Check out the FAQ! Hi there! Please sign in help. What is a 'CoxeterGroup'? Your Answer. Add Answer. Question Tools Follow. Copyright Sage, Some rights reserved under creative commons license.

Powered by Askbot version 0. Please note: Askbot requires javascript to work properly, please enable javascript in your browser, here is how.Bases: sage. It is assumed that morphisms in this category preserve the distinguished choice of simple reflections.

In particular, subobjects in this category are parabolic subgroups. In this sense, this category might be better named Coxeter Systems. In the long run we might want to have two distinct categories, one for Coxeter groups with morphisms being just group morphisms and one for Coxeter systems:. Bases: object. Return the list of covers of self in absolute order. Return whether self is smaller than other in the absolute order.

The absolute order is defined analogously to the weak order but using general reflections rather than just simple reflections. This partial order can be used to define noncrossing partitions associated with this Coxeter group.

Return the absolute length of self. The absolute length is the length of the shortest expression of the element as a product of reflections. For permutations in the symmetric groups, the absolute length is the size minus the number of its disjoint cycles. Returns the Demazure or 0-Hecke product of self with another Coxeter group element. See CoxeterGroups. The implementation uses the equivalent condition that any reduced word for other contains a reduced word for self as subword. Algebraic Combin.

Returns all elements that self covers in strong Bruhat order. Returns all elements that cover self in strong Bruhat order. Return the matrix of self in the canonical faithful representation.

Return the set of reflections t such that self t covers self. Return the c -sorting word of self. If positive is Truethen returns the non-descents instead. Caveat: the return type may change to some other iterable tuple, … in the future. Please use keyword arguments also, as the order of the arguments may change as well.

Returns the first left resp.

coxeter group sage

See descents for a description of the options. Return whether self has full support.In mathematicsa Coxeter groupnamed after H. Coxeteris an abstract group that admits a formal description in terms of reflections or kaleidoscopic mirrors. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups ; the symmetry groups of regular polyhedra are an example.

However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced Coxeter as abstractions of reflection groups, and finite Coxeter groups were classified in Coxeter Coxeter groups find applications in many areas of mathematics.

Examples of finite Coxeter groups include the symmetry groups of regular polytopesand the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic planeand the Weyl groups of infinite-dimensional Kac—Moody algebras.

Standard references include Humphreys and Davis Formally, a Coxeter group can be defined as a group with the presentation. The Coxeter matrix can be conveniently encoded by a Coxeter diagramas per the following rules. In particular, two generators commute if and only if they are not connected by an edge.

Furthermore, if a Coxeter graph has two or more connected componentsthe associated group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a direct product of Coxeter groups. The indefinite type is sometimes further subdivided, e.

However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. Coxeter groups are deeply connected with reflection groups. Simply put, Coxeter groups are abstract groups given via a presentationwhile reflection groups are concrete groups given as subgroups of linear groups or various generalizations. The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as a linear representation of a Coxeter group.

For finite reflection groups, this yields an exact correspondence: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space. For infinite Coxeter groups, however, a Coxeter group may not admit a representation as a reflection group.

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Historically, Coxeter proved that every reflection group is a Coxeter group i. The finite Coxeter groups were classified in Coxeterin terms of Coxeter—Dynkin diagrams ; they are all represented by reflection groups of finite-dimensional Euclidean spaces.

Coxeter group

The product of finitely many Coxeter groups in this list is again a Coxeter group, and all finite Coxeter groups arise in this way. Many, but not all of these, are Weyl groups, and every Weyl group can be realized as a Coxeter group. This can be proven by comparing the restrictions on undirected Dynkin diagrams with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6.

Also note that every finitely generated Coxeter group is an automatic group. Note further that the directed Dynkin diagrams B n and C n give rise to the same Weyl group hence Coxeter groupbecause they differ as directed graphs, but agree as undirected graphs — direction matters for root systems but not for the Weyl group; this corresponds to the hypercube and cross-polytope being different regular polytopes but having the same symmetry group. Some properties of the finite irreducible Coxeter groups are given in the following table.

The order of reducible groups can be computed by the product of their irreducible subgroup orders. All symmetry groups of regular polytopes are finite Coxeter groups. Note that dual polytopes have the same symmetry group. There are three series of regular polytopes in all dimensions. The symmetry group of the n - cube and its dual, the n - cross-polytopeis B nand is known as the hyperoctahedral group. The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups.

In two dimensions, the dihedral groupswhich are the symmetry groups of regular polygonsform the series I 2 p. In three dimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedronis H 3known as the full icosahedral group.This implements a general Coxeter group as a matrix group by using the reflection representation.

Bases: sage. UniqueRepresentationsage. For more on creating Coxeter groups, see CoxeterGroup. We can create the matrix representation over different base rings and with different index sets. Using the well-known conversion between Coxeter matrices and Coxeter graphs, we can input a Coxeter graph. Following the standard convention, edges with no label i.

We can also create Coxeter groups from Cartan types using the implementation keyword:. Return the matrix of self in the canonical faithful representation, which is self as a matrix. Return the first left resp. See descents for a description of the options.

Return whether i is a right descent of self. Return the bilinear form associated to self. Return the canonical faithful representation of selfwhich is self. Return the Coxeter matrix of self.

coxeter group sage

Return the fundamental weight with index i. Return the fundamental weights for self. Return whether self is commutative. Return True if this group is finite. Return the order of self.

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These are roots in the Coxeter sense, that all have the same norm. They are given by their coefficients in the base of simple roots, also taken to have all the same norm. The positive roots are listed first, then the negative roots in the same order. The order is the one given by roots. Binary Dihedral Groups. Linear Groups.

Navigation index modules next previous Sage 9. TypeError: unable to convert sqrt 2 to a rational. See also reflections. Quick search.There is something for everyone, from politics to finance, economics to technology.

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