Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

verfasst von: Matthew Fayers, Alexander Kleshchev, Lucia Morotti
Abstract: We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block (Formula presented.) is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group (Formula presented.). We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for (Formula presented.) to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.
Organisationseinheit(en): Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Externe Organisation(en): Queen Mary University of London
University of Oregon
Typ: Artikel
Journal: Journal of the London Mathematical Society
Band: 109
Anzahl der Seiten: 49
ISSN: 0024-6107
Publikationsdatum: 31.01.2024
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Allgemeine Mathematik
Elektronische Version(en): https://doi.org/10.1112/jlms.12852 (Zugang: Offen)

BibTeX

@article{243101b2b188458db7ce54c4e009245f,
title = "Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups",
abstract = "We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block (Formula presented.) is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group (Formula presented.). We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for (Formula presented.) to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.",
author = "Matthew Fayers and Alexander Kleshchev and Lucia Morotti",
note = "Funding Information: The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme {\textquoteleft}Groups, representations and applications: new perspectives{\textquoteright} where work on this paper was undertaken. The first author was supported by EPSRC grant EP/W005751/1 (for the purpose of open access, the first author has applied a creative commons attribution (CC BY) licence to any author accepted manuscript version arising). The second author was supported by the NSF grant DMS-2101791, Charles Simonyi Endowment at the Institute for Advanced Study and the Simons Foundation. While finishing writing the paper and working on the revised version, the third author was working at the Mathematisches Institut of the Heinrich-Heine-Universit{\"a}t D{\"u}sseldorf as well as the Department of Mathematics of the University of York. While working at the University of York the third author was supported by the Royal Society grant URF∖R∖221047. ",
year = "2024",
month = jan,
day = "31",
doi = "10.1112/jlms.12852",
language = "English",
volume = "109",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "John Wiley and Sons Ltd",
number = "2",
}