Leading terms of relations for standard modules of affine Lie algebras $C_n\text{sp}(1)$

In this paper we give a combinatorial parametrization of leading terms of defining relations for level $k$ standard modules for affine Lie algebra of type $C_n\text{sp}(1)$. Using this parametrization we conjecture colored Rogers-Ramanujan type combinatorial identities for $n\ge 2$ and $k\ge 2$; the identity in the case $n=k=1$ is equivalent to one of Capparelli’s identities.