# Defect 2 spin blocks of symmetric groups and canonical basis coefficients

## TL;DR

This paper develops a combinatorial formula for decomposition numbers of spin representations of symmetric groups in odd characteristic, focusing on defect 2 blocks, and advances understanding of canonical basis coefficients in type A^{(2)}_{2n}.

## Contribution

It introduces a new combinatorial formula for $q$-decomposition numbers in defect 2 blocks, extending Richards's approach to spin representations and canonical bases in type A^{(2)}_{2n}.

## Key findings

- Derived a formula for $q$-decomposition numbers in defect 2 blocks.
- Proved general results on $q$-decomposition numbers.
- Made first substantial progress on canonical bases in type A^{(2)}_{2n}.

## Abstract

This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect $2$, analogous to Richards's formula for defect $2$ blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding "$q$-decomposition numbers", i.e.\ the canonical basis coefficients in the level-$1$ $q$-deformed Fock space of type $A^{(2)}_{2n}$; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic $2n+1$. Along the way, we prove some general results on $q$-decomposition numbers. This paper represents the first substantial progress on canonical bases in type $A^{(2)}_{2n}$.