Lawrence–Krammer representation

Consider the braid group ${\displaystyle B_{n}}$ to be the mapping class group of a disc with n marked points, ${\displaystyle P_{n}}$. The Lawrence–Krammer representation is defined as the action of ${\displaystyle B_{n}}$ on the homology of a certain covering space of the configuration space ${\displaystyle C_{2}P_{n}}$. Specifically, the first integral homology group of ${\displaystyle C_{2}P_{n}}$ is isomorphic to ${\displaystyle \mathbb {Z} ^{n+1}}$, and the subgroup of ${\displaystyle H_{1}(C_{2}P_{n},\mathbb {Z} )}$ invariant under the action of ${\displaystyle B_{n}}$ is primitive, free abelian, and of rank 2. Generators for this invariant subgroup are denoted by ${\displaystyle q,t}$.

The covering space of ${\displaystyle C_{2}P_{n}}$ corresponding to the kernel of the projection map

${\displaystyle \pi _{1}(C_{2}P_{n})\to \mathbb {Z} ^{2}\langle q,t\rangle }$

is called the Lawrence–Krammer cover and is denoted ${\displaystyle {\overline {C_{2}P_{n}}}}$. Diffeomorphisms of${\displaystyle P_{n}}$ act on ${\displaystyle P_{n}}$, thus also on ${\displaystyle C_{2}P_{n}}$, moreover they lift uniquely to diffeomorphisms of ${\displaystyle {\overline {C_{2}P_{n}}}}$ which restrict to the identity on the co-dimension two boundary stratum (where both points are on the boundary circle). The action of ${\displaystyle B_{n}}$ on

${\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} ),}$

thought of as a

${\displaystyle \mathbb {Z} \langle t^{\pm },q^{\pm }\rangle }$-module,

is the Lawrence–Krammer representation. The group ${\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} )}$ is known to be a free ${\displaystyle \mathbb {Z} \langle t^{\pm },q^{\pm }\rangle }$-module, of rank ${\displaystyle n(n-1)/2}$.

Using Bigelow's conventions for the Lawrence–Krammer representation, generators for the group ${\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} )}$ are denoted ${\displaystyle v_{j,k}}$ for ${\displaystyle 1\leq j<k\leq n}$. Letting ${\displaystyle \sigma _{i}}$ denote the standard Artin generators of the braid group, we obtain the expression:

${\displaystyle \sigma _{i}\cdot v_{j,k}=\left\{{\begin{array}{lr}v_{j,k}&i\notin \{j-1,j,k-1,k\},\\qv_{i,k}+(q^{2}-q)v_{i,j}+(1-q)v_{j,k}&i=j-1\\v_{j+1,k}&i=j\neq k-1,\\qv_{j,i}+(1-q)v_{j,k}-(q^{2}-q)tv_{i,k}&i=k-1\neq j,\\v_{j,k+1}&i=k,\\-tq^{2}v_{j,k}&i=j=k-1.\end{array}}\right.}$

Stephen Bigelow and Daan Krammer have given independent proofs that the Lawrence–Krammer representation is faithful.

The Lawrence–Krammer representation preserves a non-degenerate sesquilinear form which is known to be negative-definite Hermitian provided ${\displaystyle q,t}$ are specialized to suitable unit complex numbers (q near 1 and t near i). Thus the braid group is a subgroup of the unitary group of square matrices of size ${\displaystyle n(n-1)/2}$. Recently it has been shown that the image of the Lawrence–Krammer representation is a dense subgroup of the unitary group in this case.

The sesquilinear form has the explicit description:

${\displaystyle \langle v_{i,j},v_{k,l}\rangle =-(1-t)(1+qt)(q-1)^{2}t^{-2}q^{-3}\left\{{\begin{array}{lr}-q^{2}t^{2}(q-1)&i=k<j<l{\text{ or }}i<k<j=l\\-(q-1)&k=i<l<j{\text{ or }}k<i<j=l\\t(q-1)&i<j=k<l\\q^{2}t(q-1)&k<l=i<j\\-t(q-1)^{2}(1+qt)&i<k<j<l\\(q-1)^{2}(1+qt)&k<i<l<j\\(1-qt)(1+q^{2}t)&k=i,j=l\\0&{\text{otherwise}}\\\end{array}}\right.}$

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