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

In this note we prove linear independence of the combinatorial spanning set for standard Cℓ(1)-module L(kΛ0) by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky's type subspace W(kΛ0) of C2ℓ(1)-module L(kΛ0). It should be noted that the proof of linear independence for the basis of W(kΛ0) is obtained by using simple currents and intertwining operators in the vertex operator algebra L(kΛ0).