A KLR Grading of the Brauer Algebras

We construct a naturally $\mathbb{Z}$-graded algebra ${\mathcal{G}}_n(\delta )$ over $R$ with KLR-like relations and give an explicit isomorphism between ${\mathcal{G}}_n(\delta )$ and ${\mathcal{B}}_n(\delta )$, the Brauer algebras over $R$, when $R$ is a field of characteristic 0. This isomorphism allows us to exhibit a non-trivial $\mathbb{Z}$-grading on the Brauer algebras over a field of characteristic 0. As a byproduct of the proof, we also construct an explicit homogeneous cellular basis for ${\mathcal{G}}_n(\delta )$.