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Implementation of the general Burau representation for Artin groups.
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‎src/doc/en/reference/references/index.rst‎

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@@ -568,6 +568,10 @@ REFERENCES:
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Springer Verlag 2006; pre-print available at
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http://eprint.iacr.org/2005/200
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.. [BQ2024] Asilata Bapat and Hoel Queffelec.
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*Some remarks about the faithfulness of the Burau representation
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of Artin-Tits groups*. Preprint, 2024. :arxiv:`2409.00144`.
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.. [BBS1982] \L. Blum, M. Blum, and M. Shub. Comparison of Two
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Pseudo-Random Number Generators. *Advances in Cryptology:
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Proceedings of Crypto '82*, pp.61--78, 1982.

‎src/sage/groups/artin.py‎

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AUTHORS:
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- Travis Scrimshaw (2018-02-05): Initial version
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- Travis Scrimshaw (2025-01-30): Allowed general construction; implemented
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general Burau representation
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.. TODO::
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Implement affine type Artin groups with their well-known embeddings into
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the classical braid group.
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"""
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# ****************************************************************************
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# Copyright (C) 2018 Travis Scrimshaw <tcscrims at gmail.com>
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# Copyright (C) 2018-2025 Travis Scrimshaw <tcscrims at gmail.com>
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#
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# This program is free software: you can redistribute it and/or modify
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# it under the terms of the GNU General Public License as published by
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from sage.groups.free_group import FreeGroup
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from sage.groups.finitely_presented import FinitelyPresentedGroup, FinitelyPresentedGroupElement
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from sage.misc.cachefunc import cached_method
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from sage.misc.lazy_attribute import lazy_attribute
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from sage.rings.infinity import Infinity
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from sage.structure.richcmp import richcmp, rich_to_bool
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from sage.structure.unique_representation import UniqueRepresentation
@@ -156,6 +164,132 @@ def coxeter_group_element(self, W=None):
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In = W.index_set()
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return W.prod(s[In[abs(i)-1]] for i in self.Tietze())
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def burau_matrix(self, var='t', reduced=False):
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r"""
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Return the Burau matrix of the Artin group element.
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Following [BQ2024]_, the (generalized) Burau representation of an
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Artin group is defined by deforming the reflection representation
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of the corresponding Coxeter group. However, we substitute
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`q \mapsto -t` from [BQ2024]_ to match the unitary
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INPUT:
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- ``var`` -- string (default: ``'t'``); the name of the
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variable in the entries of the matrix
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OUTPUT:
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The Burau matrix of the Artin group element over the Laurent
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polynomial ring in the variable ``var``.
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EXAMPLES::
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sage: A.<s1,s2,s3> = ArtinGroup(['B',3])
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sage: B1 = s1.burau_matrix()
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sage: B2 = s2.burau_matrix()
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sage: B3 = s3.burau_matrix()
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sage: [B1, B2, B3]
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[
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[-t^2 t 0] [ 1 0 0] [ 1 0 0]
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[ 0 1 0] [ t -t^2 a*t] [ 0 1 0]
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[ 0 0 1], [ 0 0 1], [ 0 a*t -t^2]
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]
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sage: B1 * B2 * B1 == B2 * B1 * B2
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True
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sage: B2 * B3 * B2 * B3 == B3 * B2 * B3 * B2
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True
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sage: B1 * B3 == B3 * B1
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True
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sage: (~s1).burau_matrix() * B1 == 1
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True
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We verify the example in Theorem 4.1 of [BQ2024]_::
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sage: A.<s1,s2,s3,s4> = ArtinGroup(['A', 3, 1])
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sage: [g.burau_matrix() for g in A.gens()]
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[
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[-t^2 t 0 t] [ 1 0 0 0] [ 1 0 0 0]
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[ 0 1 0 0] [ t -t^2 t 0] [ 0 1 0 0]
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[ 0 0 1 0] [ 0 0 1 0] [ 0 t -t^2 t]
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[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1],
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<BLANKLINE>
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[ 1 0 0 0]
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[ 0 1 0 0]
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[ 0 0 1 0]
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[ t 0 t -t^2]
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]
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sage: a = s3^2 * s4 * s3 * s2 *s1 * ~s3 * s4 * s3 * s2 * s1^-2 * s4
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sage: b = s1^2 * ~s2 * s4 * s1 * ~s3 * s2 * ~s4 * s3 * s1 * s4 * s1 * ~s2 * s4^-2 * s3
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sage: alpha = a * s3 * ~a
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sage: beta = b * s2 * ~b
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sage: elm = alpha * beta * ~alpha * ~beta
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sage: print(elm.Tietze())
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(3, 3, 4, 3, 2, 1, -3, 4, 3, 2, -1, -1, 4, 3, -4, 1, 1, -2, -3,
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-4, 3, -1, -2, -3, -4, -3, -3, 1, 1, -2, 4, 1, -3, 2, -4, 3, 1,
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4, 1, -2, -4, -4, 3, 2, -3, 4, 4, 2, -1, -4, -1, -3, 4, -2, 3,
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-1, -4, 2, -1, -1, 3, 3, 4, 3, 2, 1, -3, 4, 3, 2, -1, -1, 4,
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-3, -4, 1, 1, -2, -3, -4, 3, -1, -2, -3, -4, -3, -3, 1, 1, -2,
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4, 1, -3, 2, -4, 3, 1, 4, 1, -2, -4, -4, 3, -2, -3, 4, 4, 2,
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-1, -4, -1, -3, 4, -2, 3, -1, -4, 2, -1, -1)
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sage: elm.burau_matrix()
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[1 0 0 0]
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[0 1 0 0]
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[0 0 1 0]
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[0 0 0 1]
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Next, we show ``elm`` is not the identity by using the embedding of
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the affine braid group `\widetilde{B}_n \to B_{n+1}`::
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sage: B.<t1,t2,t3,t4> = BraidGroup(5)
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sage: D = t1 * t2 * t3 * t4^2
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sage: t0 = D * t3 * ~D
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sage: t0*t1*t0 == t1*t0*t1
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True
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sage: t0*t2 == t2*t0
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True
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sage: t0*t3*t0 == t3*t0*t3
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True
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sage: T = [t0, t1, t2, t3]
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sage: emb = B.prod(T[i-1] if i > 0 else ~T[-i-1] for i in elm.Tietze())
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sage: emb.is_one()
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False
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Since the Burau representation does not respect the group embedding,
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the corresponding `B_5` element's Burau matrix is not the identity
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(Bigelow gave an example of the representation not being faithful for
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`B_5`, but it is still open for `B_4`)::
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sage: emb.burau_matrix() != 1
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True
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We also verify the result using the elements in [BQ2024]_ Remark 4.2::
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sage: ap = s3 * s1 * s2 * s1 * ~s3 * s4 * s2 * s3 * s2 * ~s3 * s1^-2 * s4
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sage: bp = s1 * ~s4 * s1^2 * s3^-2 * ~s2 * s4 * s1 * ~s3 * s2 * ~s4 * s3 * s1 * s4 * s1 * ~s2 * s4^-2 * s3
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sage: alpha = ap * s3 * ~ap
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sage: beta = bp * s2 * ~bp
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sage: elm = alpha * beta * ~alpha * ~beta
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sage: elm.burau_matrix()
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[1 0 0 0]
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[0 1 0 0]
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[0 0 1 0]
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[0 0 0 1]
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REFERENCES:
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- [BQ2024]_
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"""
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gens, invs = self.parent()._burau_generators
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MS = gens[0].parent()
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ret = MS.prod(gens[i-1] if i > 0 else invs[-i-1] for i in self.Tietze())
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if var == 't':
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return ret
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from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
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poly_ring = LaurentPolynomialRing(ret.base_ring().base_ring(), var)
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return ret.change_ring(poly_ring)
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class FiniteTypeArtinGroupElement(ArtinGroupElement):
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"""
@@ -453,7 +587,7 @@ def __classcall_private__(cls, coxeter_data, names=None):
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from sage.groups.raag import RightAngledArtinGroup
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return RightAngledArtinGroup(coxeter_data.coxeter_graph(), names)
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if not coxeter_data.is_finite():
456-
raise NotImplementedError
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return super().__classcall__(cls, coxeter_data, names)
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if coxeter_data.coxeter_type().cartan_type().type() == 'A':
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from sage.groups.braid import BraidGroup
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return BraidGroup(coxeter_data.rank()+1, names)
@@ -476,15 +610,17 @@ def __init__(self, coxeter_matrix, names):
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rels = []
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# Generate the relations based on the Coxeter graph
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I = coxeter_matrix.index_set()
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gens = free_group.gens()
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for ii, i in enumerate(I):
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for j in I[ii + 1:]:
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for jj, j in enumerate(I[ii + 1:], start=ii+1):
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m = coxeter_matrix[i, j]
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if m == Infinity: # no relation
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continue
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elt = [i, j] * m
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for ind in range(m, 2*m):
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elt[ind] = -elt[ind]
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rels.append(free_group(elt))
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elt = (gens[ii] * gens[jj]) ** (m // 2)
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if m % 2 == 1:
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elt = elt * gens[ii] * ~gens[jj]
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elt = elt * (~gens[ii] * ~gens[jj]) ** (m // 2)
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rels.append(elt)
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FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
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def _repr_(self):
@@ -688,6 +824,153 @@ def _standard_lift(self, w):
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"""
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return self(self._standard_lift_Tietze(w))
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@lazy_attribute
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def _burau_generators(self):
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"""
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The Burau matrices for the generators of ``self`` and their inverses.
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EXAMPLES::
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sage: A = ArtinGroup(['G',2])
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sage: A._burau_generators
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[[
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[-t^2 a*t] [ 1 0]
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[ 0 1], [ a*t -t^2]
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],
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[
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[ -t^-2 a*t^-1] [ 1 0]
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[ 0 1], [a*t^-1 -t^-2]
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]]
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sage: A = ArtinGroup(['H',3])
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sage: A._burau_generators
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[[
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[-t^2 t 0] [ 1 0 0]
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[ 0 1 0] [ t -t^2 (1/2*a + 1/2)*t]
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[ 0 0 1], [ 0 0 1],
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<BLANKLINE>
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[ 1 0 0]
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[ 0 1 0]
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[ 0 (1/2*a + 1/2)*t -t^2]
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],
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[
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[-t^-2 t^-1 0]
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[ 0 1 0]
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[ 0 0 1],
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<BLANKLINE>
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[ 1 0 0]
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[ t^-1 -t^-2 (1/2*a + 1/2)*t^-1]
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[ 0 0 1],
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<BLANKLINE>
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[ 1 0 0]
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[ 0 1 0]
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[ 0 (1/2*a + 1/2)*t^-1 -t^-2]
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]]
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sage: CM = matrix([[1,4,7], [4,1,10], [7,10,1]])
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sage: A = ArtinGroup(CM)
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sage: gens = A._burau_generators[0]; gens
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[
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[ -t^2 (E(8) - E(8)^3)*t (-E(7)^3 - E(7)^4)*t]
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[ 0 1 0]
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[ 0 0 1],
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<BLANKLINE>
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[ 1 0 0]
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[ (E(8) - E(8)^3)*t -t^2 (E(20) - E(20)^9)*t]
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[ 0 0 1],
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<BLANKLINE>
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[ 1 0 0]
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[ 0 1 0]
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[(-E(7)^3 - E(7)^4)*t (E(20) - E(20)^9)*t -t^2]
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]
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sage: all(gens[i] * A._burau_generators[1][i] == 1 for i in range(3))
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True
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sage: B1, B2, B3 = gens
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sage: (B1 * B2)^2 == (B2 * B1)^2
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True
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sage: (B2 * B3)^5 == (B3 * B2)^5
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True
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sage: B1 * B3 * B1 * B3 * B1 * B3 * B1 == B3 * B1 * B3 * B1 * B3 * B1 * B3
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True
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"""
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data = self.coxeter_type()
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coxeter_matrix = self.coxeter_matrix()
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n = coxeter_matrix.rank()
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# Determine the base field
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if data.is_simply_laced():
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from sage.rings.integer_ring import ZZ
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base_ring = ZZ
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elif data.is_finite():
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from sage.rings.number_field.number_field import QuadraticField
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letter = data.cartan_type().type()
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if letter in ['B', 'C', 'F']:
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base_ring = QuadraticField(2)
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elif letter == 'G':
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base_ring = QuadraticField(3)
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elif letter == 'H':
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base_ring = QuadraticField(5)
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else:
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from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField
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base_ring = UniversalCyclotomicField()
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else:
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from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField
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base_ring = UniversalCyclotomicField()
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# Construct the matrices
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from sage.matrix.args import SparseEntry
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from sage.matrix.matrix_space import MatrixSpace
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from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
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import sage.rings.abc
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poly_ring = LaurentPolynomialRing(base_ring, 't')
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q = -poly_ring.gen()
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MS = MatrixSpace(poly_ring, n, sparse=True)
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one = MS.one()
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# FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty
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if isinstance(base_ring, sage.rings.abc.UniversalCyclotomicField):
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E = base_ring.gen
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def val(x):
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if x == -1:
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return 2 * q
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elif x == 1:
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return 1 + q**2
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else:
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return q * (E(2*x) + ~E(2*x))
940+
elif isinstance(base_ring, sage.rings.abc.NumberField_quadratic):
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from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField
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E = UniversalCyclotomicField().gen
943+
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def val(x):
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if x == -1:
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return 2 * q
947+
elif x == 1:
948+
return 1 + q**2
949+
else:
950+
return q * base_ring((E(2 * x) + ~E(2 * x)).to_cyclotomic_field())
951+
else:
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def val(x):
953+
if x == -1:
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return 2 * q
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elif x == 1:
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return 1 + q**2
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elif x == 2:
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return 0
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elif x == 3:
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return q
961+
else:
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from sage.functions.trig import cos
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from sage.symbolic.constants import pi
964+
return q * base_ring(2 * cos(pi / x))
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index_set = data.index_set()
966+
gens = [one - MS([SparseEntry(i, j, val(coxeter_matrix[index_set[i], index_set[j]]))
967+
for j in range(n)])
968+
for i in range(n)]
969+
invs = [one - q**-2 * MS([SparseEntry(i, j, val(coxeter_matrix[index_set[i], index_set[j]]))
970+
for j in range(n)])
971+
for i in range(n)]
972+
return [gens, invs]
973+
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Element = ArtinGroupElement
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