TY - JOUR

T1 - Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

AU - Fayers, Matthew

AU - Kleshchev, Alexander

AU - Morotti, Lucia

N1 - Funding Information: The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme ‘Groups, representations and applications: new perspectives’ 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-Universität Düsseldorf 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.

PY - 2024/1/31

Y1 - 2024/1/31

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85183846153&partnerID=8YFLogxK

U2 - 10.1112/jlms.12852

DO - 10.1112/jlms.12852

M3 - Article

AN - SCOPUS:85183846153

VL - 109

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 2

M1 - e12852

ER -