Linear independence for $C_{\mathrm{\ell }}^{(1)}$ by using $C_{2\mathrm{\ell }}^{(1)}$

In this note we prove linear independence of the combinatorial spanning set for standard $C_{\mathrm{\ell }}^{(1)}$-module $L(k{\mathrm{\Lambda }}_0)$ by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky’s type subspace $W(k{\mathrm{\Lambda }}_0)$ of $C_{2\mathrm{\ell }}^{(1)}$-module $L(k{\mathrm{\Lambda }}_0)$. It should be noted that the proof of linear independence for the basis of $W(k{\mathrm{\Lambda }}_0)$ is obtained by using simple currents and intertwining operators in the vertex operator algebra $L(k{\mathrm{\Lambda }}_0)$.