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We prove stability of the Chari-Loktev bases with respect to the inclusionsof local Weyl modules of the current algebra $\mathfrak{sl}_{r+1}[t]$. This isconjectured in \cite{RRV2} and the $r=1$ case is proved in \cite{RRV1}. LocalWeyl modules being known to be Demazure submodules in the level onerepresentations of the affine Lie algebra $\widehat{\mathfrak{sl}_{r+1}}$, weobtain, by passage to the direct limit, bases for the level one representationsthemselves.

Stability of the Chari-Loktev bases for local Weyl modules of $\mathfrak{sl}_{r+1}[t]$