  Let $V=\C^n$ be endowed with an orthogonal form and $G=\Or(V)$ be the
corresponding orthogonal group. Brauer showed in 1937 that there is a
surjective homomorphism $\nu:B_r(n)\to\End_G(V^{\otimes r})$, where $B_r(n)$ is
the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$
has remained elusive. In this paper we show that, in analogy with the case of
$\GL(V)$, for $r\geq n+1$, $\nu$ has kernel which is generated by a single
idempotent element $E$, and we give a simple explicit formula for $E$. Using
the theory of cellular algebras, we show how $E$ may be used to determine the
multiplicities of the irreducible representations of $\Or(V)$ in $V^{\ot r}$.
We also show how our results extend to the case where $\C$ is replaced by an
appropriate field of positive characteristic, and comment on quantum analogues
of our results.
