The level-one partition Fock space and node residues

The partition basis and the declared operators. By [F1] the map $\lambda \mapsto \mathrm{\mid }\lambda ⟩$ is a bijection from partitions to normal semi-infinite words and the stable prefix action gives well-defined linear $F_i,E_i,K_i,K_d$; the weights, residues and size grading are read off literally from $i_k={\lambda }_k-k+1$, so the Definition introduces no new structure beyond the wedge construction.

The regular/restricted equivalence. For each $r\ge 1$, the number of columns of $\lambda$ of height exactly $r$ is ${\lambda }_r-{\lambda }_{r+1}$: such a column has index $c$ with ${\lambda }_{r+1}<c\le {\lambda }_r$. These column heights are the parts of ${\lambda }^{\mathrm{\prime }}$. Thus $\lambda$ is $e$-restricted exactly when every part of ${\lambda }^{\mathrm{\prime }}$ has multiplicity less than $e$, equivalently when ${\lambda }^{\mathrm{\prime }}$ is $e$-regular. Column height itself is unrestricted; for example $(1^e)$ has one column of height $e$ and its only nonzero adjacent row difference is $1<e$.

The boundary example. For $e\ge 2$ the partition $(1^e)$ has parts of multiplicity exactly $e$, so it is not $e$-regular, while its adjacent differences are $0$ except for the final difference $1<e$, so it is $e$-restricted; this shows the two conditions are genuinely different and that the full Fock space contains non-regular labels. All partitions are vertices of the full Fock module; the regular ones will be the labels of the empty component.

Conclusion. The level-one partition Fock space is the charge-$0$ Fock space with weight ${\mathrm{\Lambda }}_0-{\sum }_i{m}_i{\alpha }_i$ and residues $c-r<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>e$, its operators are the stable finite-prefix node operators, and the $e$-regular and $e$-restricted conventions are conjugate to each other with the column-height dictionary.