Specht modules for quiver Hecke algebras of type $C$

We construct and investigate Specht modules ${\mathcal{S}}^{\lambda }$ for cyclotomic quiver Hecke algebras in type $C_{\mathrm{\ell }}^{(1)}$ and $C_{\mathrm{\infty }}$, which are labelled by multipartitions $\lambda$. It is shown that in type $C_{\mathrm{\infty }}$, the Specht module ${\mathcal{S}}^{\lambda }$ has a homogeneous basis indexed by standard tableaux of shape $\lambda$, which yields a graded character formula and good properties with the exact functors $E_i^{\mathrm{\Lambda }}$ and $F_i^{\mathrm{\Lambda }}$. For type $C_{\mathrm{\ell }}^{(1)}$, we propose a conjecture.