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

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- by Matthew Fayers PDF
- Represent. Theory
**26**(2022), 134-178 Request permission

## 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 $A^{(2)}_{2n}$.

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## Additional Information

**Matthew Fayers**- Affiliation: Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
- MR Author ID: 680587
- Email: m.fayers@qmul.ac.uk
- Received by editor(s): August 7, 2021
- Received by editor(s) in revised form: September 15, 2021
- Published electronically: March 17, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 134-178 - MSC (2020): Primary 20C30, 20C25, 17B37, 05E10
- DOI: https://doi.org/10.1090/ert/600
- MathSciNet review: 4396039