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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Defect $2$ spin blocks of symmetric groups and canonical basis coefficients
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by Matthew Fayers
Represent. Theory 26 (2022), 134-178
DOI: https://doi.org/10.1090/ert/600
Published electronically: March 17, 2022

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

References
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Bibliographic 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