Two examples are given where combinatorial counting of relations among relations for C_n^(1) standard modules at level 5 and C_3^(1) at higher levels matches the representation-theoretic dimension in a trapezoid of negative root vectors.
Classical freeness of $\widehat{\mathfrak{sl}}_n$ at level $1$ via combinatorics
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abstract
We use a family of Rogers--Ramanujan-type combinatorial identities of Dousse--Konan involving coloured partitions to prove classical freeness of the simple vertex operator algebras based on $\widehat{\mathfrak{sl}}_n$ at level $1$. These identities are used to produce Gr\"obner bases for the relevant arc algebras.
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Two examples of combinatorial relations among relations of $C_{n}\sp{(1)}$-standard modules for higher levels
Two examples are given where combinatorial counting of relations among relations for C_n^(1) standard modules at level 5 and C_3^(1) at higher levels matches the representation-theoretic dimension in a trapezoid of negative root vectors.