\ \ \ \ \pi_{n}(D_{I,J}) = 0.$$ If $D'_{I,J}$ is another vertex operator satisfying (\[uniquedij\]), $D_{I,J} - D'_{I,J}$ lies in $(\cV_n)_{(2n)} \cap \cI_{n}$, and since there are no relations in $\cH(n)^{O(n)}$ of degree less than $2n+2$, we have $D_{I,J} - D'_{I,J}=0$. There is a distinguished element $D_0 = D_{(0,\dots,n),(0,\dots,n)}$ which is the unique element of $\cI_n$ of minimal weight $n^2+3n+2$. It is annihilated by all operators $J^{2m}(k)$ for $k>2m+1$, which lower the weight by $k-2m-1$.

Given a homogeneous polynomial $p\in gr(\cV_{n})\cong \mathbb{C}[Q_{a,b}|~0\leq a\leq b]$ of degree $k$ in the variables $Q_{a,b}$, a [*normal ordering*]{} of $p$ will be a choice of normally ordered polynomial $P\in (\cV_{n})_{(2k)}$, obtained by replacing $Q_{a,b}$ by $\Omega_{a,b}$, and replacing ordinary products with iterated Wick products of the form (\[iteratedwick\]). Of course $P$ is not unique, but for any choice of $P$ we have $\phi_{2k}(P) = p$, where $\phi_{2k}: (\cV_{n})_{(2k)} \ra (\cV_{n})_{(2k)} /(\cV_{n})_{(2k-1)} \subset gr(\cV_{n})$ is the usual projection. For the rest of this section, $D^{2k}$, $E^{2k}$, $F^{2k}$, etc., will denote elements of $(\cV_{n})_{(2k)}$ which are homogeneous, normally ordered polynomials of degree $k$ in the vertex operators $\Omega_{a,b}$.

Let $D_{I,J}^{2n+2}\in (\cV_{n})_{(2n+2)}$ be some normal ordering of $d_{I,J}$. Then $$\pi_{n}(D_{I,J}^{2n+2}) \in (\cH(n)^{O(n)})_{(2n)},$$ and $\phi_{2n}(\pi_n(D_{I,J}^{2n+2})) \in gr(\mathcal{H}(n)^{O(n)})$ can 