Linear Independence for $A_1^{(1)}$ by Using $C_2^{(1)}$

Abstract

In the previous paper, the authors proved linear independence of the combinatorial spanning set for standard $C_{\ell }^{(1)}$-module $L(k\hat{I}{›}_0)$ by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky's type subspace $W(k\hat{I}{›}_0)$ of $C_{2\ell }^{(1)}$-module $L(k\hat{I}{›}_0)$. In this note we extend this argument for $C_1^{(1)}\cong A_1^{(1)}$ to all standard $A_1^{(1)}$-modules $L(\hat{I}›)$. In the proof we use a coefficient of an intertwining operator of the type $\left(\genfrac{}{}{0ex}{}{L(\hat{I}{›}_2)}{L(\hat{I}{›}_1)L(\hat{I}{›}_1)}\right)$ for standard $C_2^{(1)}$-modules.