A KLR Grading of the Brauer Algebras

We construct a naturally $\mathbb Z$-graded algebra $\mathscr G_n(\delta)$ over $R$ with KLR-like relations and give an explicit isomorphism between $\mathscr G_n(\delta)$ and $\mathscr B_n(\delta)$, 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 $\mathscr G_n(\delta)$.