CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every classical braid group is linear

Statement

Assume AC. For every $n \geq 1$ the representation $ρ_{L K B}$ embeds $B_{n}$ into ${G L}_{(\binom{n}{2})} (Λ)$ , hence also into ${G L}_{(\binom{n}{2})} (Q (q, t))$ ; therefore every classical braid group is linear.

Facts & Assumptions

Given: the braid group $B_{n}$ , the ring $Λ = Z [q^{\pm 1}, t^{\pm 1}]$ and its fraction field $K = Q (q, t)$ .

[F1]

The Lawrence-Krammer-Bigelow representation is faithful: $ρ_{L K B} : B_{n} \to {G L}_{(\binom{n}{2})} (Λ)$ has trivial kernel.

[F2]

The integral LKB module is free of rank n choose two: the target is the full matrix group of the free $Λ$ -module of rank $(\binom{n}{2})$ , and extension of scalars to $K$ presents it as ${G L}_{(\binom{n}{2})} (K)$ .

Proof

1.1F1F2

By [F1] the homomorphism $ρ_{L K B}$ is injective, so $B_{n}$ is isomorphic to a subgroup of the matrix group ${G L}_{(\binom{n}{2})} (Λ)$ ; this already exhibits a faithful finite-dimensional representation of $B_{n}$ over the commutative ring $Λ$ .

2.1F1F2algebra∎

Extending scalars along the inclusion $Λ ↪ K$ gives a group homomorphism ${G L}_{(\binom{n}{2})} (Λ) \to {G L}_{(\binom{n}{2})} (K)$ which is injective, because its entries are the entries of the matrix and the inclusion $Λ ↪ K$ is injective; the composite with $ρ_{L K B}$ is therefore an injective homomorphism from $B_{n}$ to ${G L}_{(\binom{n}{2})} (K)$ . By [F2] the size $(\binom{n}{2})$ is finite for every $n$ , so $B_{n}$ is a linear group. For $n = 1$ the group $B_{1}$ is trivial and therefore linear as well.

Depends on

Used by

A linear representation need not be faithful Counterexample

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471-486 (standard reference, not scraped)
Bigelow, The Lawrence-Krammer representation, arXiv:math/0204057v1 (standard reference, not scraped)
Krammer, Braid groups are linear, Ann. of Math. 155 (2002) 131-156 (standard reference, not scraped)