Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted
BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN $p^{\prime }$ INERTIAL QUOTIENT
Let $k$ be an algebraically closed field of characteristic $p$, and let ${\mathcal{O}}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ be a finite group and $B$ a block of ${\mathcal{O}} G$ with normal abelian defect group and abelian $p^{\prime}$ inertial quotient $L$. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For ${\mathcal{O}}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantized version of the group algebra of the semidirect product $P\rtimes L$.
more » « less- NSF-PAR ID:
- 10121732
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- The Quarterly Journal of Mathematics
- ISSN:
- 0033-5606
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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