One-to-one correspondence between Fock descendants and polynomial solutions

Let ${\cal F}_n$ be the level $n$ subspace of the Fock module ${\cal F}$, and let ${\cal P}_n$ be the space of order $n$ polynomials $p(\xi_1,\ldots,\xi_n|\eta_1,\ldots,\eta_n)$ whose associated symmetric polynomials $P^{[p]}_{N,k}$ satisfy the kinematic pole condition for every $N$ and $k$. Fock-descendant correspondence. There is a one-to-one correspondence between ${\cal F}_n$ and ${\cal P}_n$, under which each $p\in{\cal P}_n$ defines a level $n$ chiral Fock descendant of $V_a(x)$ through its form factors. The preceding character theorem establishes equality of dimensions, but the asserted correspondence also identifies the polynomial data with descendants.