Comparing Fock spaces in types $A^{(1)}$ and $A^{(2)}$

We compare the canonical bases of level-$1$ quantised Fock spaces in affine types $A^{(1)}$ and $A^{(2)}$, showing how to derive the canonical basis in type $A^{(2)}_{2n}$ from the the canonical basis in type $A^{(1)}_n$ in certain weight spaces. In particular, we derive an explicit formula for the canonical basis in extremal weight spaces, which correspond to RoCK blocks of double covers of symmetric groups. In a forthcoming paper with Kleshchev and Morotti we will use this formula to find the decomposition numbers for RoCK blocks of double covers with abelian defect.