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\title{A Note on the Unitarity Property of the Gassner Invariant}
\author{Dror Bar-Natan}
\institute{University of Toronto}
\runningtitle{Unitarity Of Gassner}
\runningauthor{Dror Bar-Natan}
\date{\today; first edition: June 29, 2014.}
\begin{abstract}\begin{center}\parbox[t]{130mm}{We give a 3-page description of the Gassner invariant (or representation) of braids (or pure braids), along with a description and a proof of its unitarity property.
\mbox{}
%keywords
{\bf Keywords:} Braids, Unitarity, Gassner, Burau.}\end{center}
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\avtogl{Dror Bar-Natan}{Unitarity Of Gassner}
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The%
\blfootnote{The full \TeX\ sources are at \url{http://drorbn.net/AcademicPensieve/2014-06/UnitarityOfGassner/}. Updated less often: \arXiv{1406.7632}.}%
\blfootnote{This work was partially supported by NSERC grant RGPIN 262178.}%
\blfootnote{{\em 2010 Mathematics Subject Classification.} 57M25.}%
unitarity of the Gassner representation~\cite{Gassner:OnBraidGroups} of the pure braid group was discussed by many authors (e.g.~\cite{Long:LinReps, Abdulrahim:Faithfulness, KirkLivingstonWang:Gassner}) and from several points of view, yet without exposing how utterly simple the formulas turn out to be\footnote{Partially this is because the formulas are simplest when extended a ``Gassner invariant'' defined on the full braid group, but then it is not a representation and it is not unitary. Yet it has an easy ``unitarity property''; see below.}. When the present author needed quick and easy formulas, he couldn't find them. This note is written in order to rectify this situation (but with no discussion of theory). I was heavily influenced by a similar discussion of the unitarity of the Burau representation in~\cite[Section~3.1.2]{KasselTuraev:BraidGroups}.
Let $n$ be a natural number. The braid group $B_n$ on $n$ strands is the group with generators $\sigma_i$, for $1\leq i\leq n-1$, and with relations $\sigma_i\sigma_j=\sigma_j\sigma_i$ when $|i-j|>1$ and $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ when $1\leq i\leq n-2$. A standard way to depict braids, namely elements of $B_n$, is as follows:
\[ \raisebox{6mm}{$b_0=\sigma_1\sigma_3^{-1}\sigma_2$:}\qquad \includegraphics[width=1in]{B.pdf} \]
Braids are made of strands that are indexed $1$ through $n$ at the bottom. The generator $\sigma_i$ denotes a positive crossing between the strand at position $\#i$ as counted just below the horizontal level of that crossing, and the strand just to its right. Note that with the strands indexed at the bottom, the two strands participating in a crossing corresponding to $\sigma_i$ may have arbitrary indices, depending on the permutation induced by the braids below the level of that crossing.
Let $t$ be a formal variable and let $U_i(t)=U_{n;i}(t)$ denote the $n\times n$ identity matrix with its $2\times 2$ block at rows $i$ and $i+1$ and columns $i$ and $i+1$ replaced by $\begin{pmatrix} 1-t & 1 \\ t & 0 \end{pmatrix}$, as in the following example:
\[
U_{5;3}(t) = \begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1-t & 1 & 0 \\
0 & 0 & t & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}.
\]
Let $U^{-1}_i(t)$ be the inverse of $U_i(t)$; it is the $n\times n$ identity matrix with the block at $\{i,i+1\}\times\{i,i+1\}$ replaced by $\begin{pmatrix} 0 & \bar{t} \\ 1 & 1-\bar{t} \end{pmatrix}$, where $\bar{t}$ denotes $t^{-1}$.
Let $b$ be a braid $b=\prod_{\alpha=1}^k \sigma_{i_\alpha}^{s_\alpha}$, where the $s_\alpha$ are signs and where products are taken from left to right. Let $j_\alpha$ be the index of the ``over'' strand at crossing $\#\alpha$ in $b$. The Gassner invariant $\Gamma(b)$ of $b$ is given by
\[ \Gamma(b)\coloneqq \prod_{\alpha=1}^k U_{i_\alpha}^{s_\alpha}(t_{j_\alpha}). \]
It is a Laurent polynomial in $n$ formal variables $t_1,\ldots,t_n$, with coefficients in $\bbZ$.
\begin{center}
\null\qquad\includegraphics[width=1.8in]{R3.pdf}
\newline
{\footnotesize Figure. The braids $\sigma_1\sigma_2\sigma_1$ and $\sigma_2\sigma_1\sigma_2$.}
\end{center}
As an example, for the braids in the Figure, $\Gamma(\sigma_1\sigma_2\sigma_1) = U_1(t_1)U_2(t_1)U_1(t_2)$ and $\Gamma(\sigma_2\sigma_1\sigma_2) = U_2(t_2)U_1(t_1)U_2(t_1)$. The equality of these two matrix products constitutes the bulk of the proof of the well-definedness of $\Gamma$, and the rest is even easier. The verification of this equality is a routine exercise in $3\times 3$ matrix multiplication. Impatient readers may find it in the {\sl Mathematica} notebook that accompanies this note,~\cite{Notebook}.
A second example is the braid $b_0$ of the first figure. Here and in~\cite{Notebook},
\begin{multline*}
\Gamma(b_0)=U_1(t_1)U_3^{-1}(t_4)U_2(t_1) \\
= \begin{pmatrix}
1-t_1 & 1-t_1 & 1 & 0 \\
t_1 & 0 & 0 & 0 \\
0 & 0 & 0 & \bar{t_4} \\
0 & t_1 & 0 & 1-\bar{t_4}
\end{pmatrix}
\end{multline*}
Given a permutation $\tau=[\tau 1,\ldots,\tau n]$ of $1,\ldots,n$, let $\Omega(\tau)$ be the triangular $n\times n$ matrix
\[ \Omega(\tau)\coloneqq
\begin{pmatrix}
(1-t_{\tau 1})^{-1} & 0 & \cdots & 0 \\
1 & (1-t_{\tau 2})^{-1} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \ldots & (1-t_{\tau n})^{-1}
\end{pmatrix}
\]
(diagonal entries $(1-t_{\tau i})^{-1}$, $1$'s below the diagonal, $0$'s above). Let $\iota$ denote the identity permutation $[1,2,\ldots,n]$.
\noindent{\bf Theorem.} Let $b$ be a braid that induces a strand permutation $\tau=[\tau 1,\ldots,\tau n]$ (meaning, the strand indices that appear at the top of $b$ are $\tau 1,\tau 2,\ldots,\tau n$). Let $\gamma=\Gamma(b)$ be the Gassner invariant of $b$. Then $\gamma$ satisfies the ``unitarity property''
\begin{equation} \label{eq:unitarity}
\Omega(\tau)\gamma^{-1}=\bar{\gamma}^T\Omega(\iota),
\end{equation}
\[
\text{or equivalently,}\qquad
\gamma^{-1}=\Omega(\tau)^{-1}\bar{\gamma}^T\Omega(\iota),
\]
where $\bar{\gamma}$ is $\gamma$ subject to the substitution $\forall i\, t_i\to \bar{t_i}\coloneqq t_i^{-1}$, and $\bar{\gamma}^T$ is the transpose matrix of $\bar{\gamma}$.
\noindent{\it Proof.} A direct and simple-minded computation proves Equation~\eqref{eq:unitarity} for $b=\sigma_i$ and for $b=\sigma_i^{-1}$, namely for $\gamma=U_i(t_i)$ and for $\gamma=U_i^{-1}(t_{i+1})$ (impatient readers see~\cite{Notebook}), and then, clearly, using the second form of Equation~\eqref{eq:unitarity}, the statement generalizes to products with all the intermediate $\Omega(\tau)^{-1}\Omega(\tau)$ pairs cancelling out nicely. \qed
If the Gassner invariant $\Gamma$ is restricted to pure braids, namely to braids that induce the identity permutation, it becomes multiplicative and then it can be called ``the Gassner representation'' (in general $\Gamma$ can be recast as a homomorphism into $M_{n\times n}(\bbZ[t_i,\bar{t_i}])\rtimes S_n$, where $S_n$ acts on matrices by permuting the variables $t_i$ appearing in their entries).
For pure braids $\Omega(\tau)=\Omega(\iota)\eqqcolon\Omega$ and hence by conjugating (in the $t_i\to1/t_i$ sense) and transposing Equation~\eqref{eq:unitarity} and replacing $\gamma$ by $\gamma^{-1}$, we find that the theorem also holds if $\Omega$ is replaced by $\bar{\Omega}^T$. Hence, extending the coefficients to $\bbC$, the theorem also holds if $\Omega$ is replaced by $\Psi\coloneqq i\Omega-i\bar{\Omega}^T$, which is formally Hermitian ($\bar{\Psi}^T=\Psi$).
If the $t_i$'s are specialized to complex numbers of unit norm then inversion is the same as complex conjugation. If also the $t_i$'s are sufficiently close to $1$ and have positive imaginary parts, then $\Psi$ is dominated by its main diagonal entries, which are real, positive, and large, and hence $\Psi$ is positive definite and genuinely Hermitian. Thus in that case, the Gassner representation is unitary in the standard sense of the word, relative to the inner product on $\bbC^n$ defined by $\Psi$.
We remark is that the Gassner representation easily extends to a representation of pure v/w-braids. See e.g.~\cite[Sections 2.1.2 and 2.2]{WKO1}, where the generators $\sigma_{ij}$ are described (they are {\em not} generators of the ordinary pure braid group). Simply set $\Gamma(\sigma_{ij})^{\pm 1}=U_{ij}^{\pm 1}$ where $U_{ij}$ is the $n\times n$ identity matrix with its $2\times 2$ block at rows $i$ and $j$ and columns $i$ and $j$ replaced by $\begin{pmatrix} 1 & 1-t_i \\ 0 & t_i \end{pmatrix}$. Yet on v/w-braids $\Gamma$ does not satisfy the unitarity property of this note and I'd be very surprised if it is at all unitary.
We also remark that there is an alternative form $\Gamma'$ for the Gassner representation of pure v/w-braids, defined by $\Gamma'(\sigma_{ij})^{\pm 1}=V_{ij}^{\pm 1}$ where $V_{ij}$ is the $n\times n$ identity matrix with its $2\times 2$ block at rows $i$ and $j$ and columns $i$ and $j$ replaced by $\begin{pmatrix} 1 & 1-t_j \\ 0 & t_i \end{pmatrix}$. Clearly, $U_{ij}$ and $V_{ij}$ are conjugate; $V_{ij}=D^{-1}U_{ij}D$, with $D$ the diagonal matrix whose $(i,i)$ entry is $1-t_i$ for every $i$. Hence on ordinary pure braids and for appropriate values of the $t_i$'s (as above), $\Gamma'$ is also unitary, relative to the Hermitian inner product defined by the matrix
\[ \Psi' \coloneqq \bar{D}^T\Psi D=i\bar{D}^T(\Omega-\bar{\Omega}^T)D \]
whose printed form is better avoided (yet it appears at the end of~\cite{Notebook}).
\begin{thebibliography}{}
\bibitem[\protect\citeauthoryear{Abdulrahim}{1997}]{Abdulrahim:Faithfulness} M.~N.~Abdulrahim,
{\em A Faithfulness Criterion for the Gassner Representation of the Pure Braid Group,}
Proceedings of the American Mathematical Society {\bf 125-5} (1997) 1249--1257.
\bibitem[\protect\citeauthoryear{Bar-Natan}{2014}]{Notebook} D.~Bar-Natan,
{\tt UnitarityOfGassnerDemo.nb},
a {\sl Mathematica} noteboook at \url{http://drorbn.net/AcademicPensieve/2014-06/UnitarityOfGassner/}.
\bibitem[\protect\citeauthoryear{Bar-Natan and Dancso}{2014}]{WKO1} D.~Bar-Natan and Z.~Dancso,
{\em Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial,}
\url{http://drorbn.net/AcademicPensieve/Projects/WKO1/} and \arXiv{1405.1956}.
\bibitem[\protect\citeauthoryear{Gassner}{1959}]{Gassner:OnBraidGroups} B.~J.~Gassner,
{\em On Braid Groups,}
Ph.D. thesis, New York Univeristy, 1959.
\bibitem[\protect\citeauthoryear{Kassel and Turaev}{2008}]{KasselTuraev:BraidGroups} C.~Kassel and V.~Turaev,
{\em Braid Groups,}
Springer GTM {\bf 247}, 2008.
\bibitem[\protect\citeauthoryear{Kirk et al.}{2001}]{KirkLivingstonWang:Gassner} P.~Kirk, C.~Livingston, and Z.~Wang,
{\em The Gassner Representation for String Links,}
Communications in Contemporary Mathematics {\bf 3-1} (2001) 87--136, \arXiv{math/9806035}.
\bibitem[\protect\citeauthoryear{Long}{1989}]{Long:LinReps} D.~D.~Long,
{\em On the Linear Representation of Braid Groups,}
Transactions of the American Mathematical Society {\bf 311-2} (1989) 535--560.
\end{thebibliography}
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\newpage
Dror Bar-Natan,
Department of Mathematics,
University of Toronto,
Toronto Ontario M5S 2E4,
Canada,
\email{drorbn@math.toronto.edu},
\url{http://www.math.toronto.edu/~drorbn}.
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