Alphabeta MathGarside Structure, Normal Forms, and the Center
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1 · Prerequisites
Abstract Algebra
Category Theory
Coxeter Groups
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Positive braid monoid
Lemma
Positive artin relations preserve homogeneous length
Definition
Artin right complements and word reversing
Lemma
Artin right complements satisfy the cube condition
Lemma
Artin positive word reversing is complete
Lemma
The positive braid monoid is left and right cancellative
Definition
Left and right divisibility for positive braids
Lemma
Artin atoms have explicit left and right lcms and complements
Definition
The Garside half twist and simple positive braids
Lemma
Conjugation by the half twist reverses Artin generators
Lemma
Each Artin atom is a left and right divisor of the half twist
Lemma
Every positive braid divides a power of the half twist on both sides
Theorem
Positive braids have left and right gcds and lcms
Theorem
The group of fractions of the positive braid monoid is the Artin braid group
Theorem
Left and right divisibility extend to lattice orders on the braid group
Lemma
Reduced adjacent-transposition words have well-defined positive lifts
Lemma
Simple positive braids are indexed by permutations
Lemma
Delta is the lcm of the artin atoms and has the same left and right divisors
Theorem
Left garside normal form is unique
Corollary
The braid group word problem is decidable by garside normal form
Theorem
Braid groups are torsion free by the garside lattice
Lemma
A central positive braid is a power of delta squared for n greater than two
Theorem
The center of b n is generated by the full twist for n greater than two
Proposition
The center of b two is all of b two
Sources
- Juan Gonzalez-Meneses, Basic results on braid groups, section 4, printed pp. 26-28
- Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 5.1, author manuscript pp. 61-62
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Propositions 2.32-2.33, printed pp. 47-48
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Definitions 4.1-4.2, Example 4.4 and Lemma 4.6, printed pp. 63-65
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Definition 4.21 and Lemma 4.32, printed pp. 68, 73-74
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Definition 4.14 and Example 4.20, printed pp. 66-67
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Example 4.11 and Lemma 4.55, printed pp. 65, 80-81
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Lemma 4.6, Proposition 4.16, Definition 4.48, Lemma 4.55, Proposition 4.51, printed pp. 63-68, 78-83
- Patrick Dehornoy et al., Foundations of Garside Theory, Appendix, Lemmas II.4.60-II.4.63, printed pp. 657-662, and Chapter II, Corollaries 4.45 and 4.47, printed pp. 79-80
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed pp. 26-27 (cancellativity step)
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Proposition 4.44 and Corollary 4.45, printed p. 78
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed pp. 26-27 (prefix order and suffix order)
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed pp. 26-27 (lcm of two atoms)
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Example 4.20, printed pp. 66-67
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter I, Reference Structure 2 and formula (1.6), printed pp. 5-7
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed pp. 27-28
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed pp. 27-28 (sigma_i Delta = Delta sigma_{n-i} and its consequences)
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter IX, Lemma 1.22 and Section 1.3, printed pp. 438-440
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed p. 28 (a ≼ Δ^m and Δ^m ≽ a for some m)
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter IX, Section 1.3, printed pp. 439-440
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed pp. 26-27
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter II, Section 4, printed pp. 63-83
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed p. 28 (Ore's condition and the embedding of B_n^+)
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter IX, Lemma 1.22, printed p. 438
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed p. 28 (extension of the order to B_n)
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4.1, printed pp. 29-30
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter IX, Proposition 1.10 (exchange) and Corollary 1.11(ii) (Matsumoto), printed pp. 434-435
- M. Macauley, Math 4120 lecture notes: Generating sets for S_n
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4, printed pp. 26-29
- J. Birman and T. Brendle, Braids: A Survey, Section 5.1
- Patrick Dehornoy et al., Foundations of Garside Theory, Chapter IX, printed pp. 433-438
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4.1, printed pp. 28-30
- J. Gonzalez-Meneses, Basic results on braid groups, Sections 4 and 4.1, printed pp. 26-30
- J. Gonzalez-Meneses, Basic results on braid groups, Proposition 4.1, printed p. 30
- J. Gonzalez-Meneses, Basic results on braid groups, Theorem 4.2, printed pp. 30-31
- J. Birman and T. Brendle, Braids: A Survey, Section 5.2
- J. Gonzalez-Meneses, Basic results on braid groups, Section 4.3, printed p. 31