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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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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


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:
  • 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:
  • MathSciNet review: 4396039