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

TL;DR

This paper compares the canonical bases of level-1 quantized Fock spaces in affine types A^{(1)} and A^{(2)}, providing explicit formulas and methods to derive bases in type A^{(2)} from type A^{(1)}.

Contribution

It introduces a method to derive the canonical basis in type A^{(2)}_{2n} from the canonical basis in type A^{(1)}_n, including explicit formulas for extremal weight spaces.

Findings

Derived explicit formulas for canonical bases in extremal weight spaces.

Established a method to relate bases between affine types A^{(1)} and A^{(2)}.

Set the stage for future work on decomposition numbers of RoCK blocks.

Abstract

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.