Linear independence for Cℓ(1) by using C2ℓ(1)

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