2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle

2019/12/06 by Mckernon, Elliot
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1912.03222

Abstract

We consider a block B of a finite group with defect group D ≅ (C_2^m)ⁿ and inertial quotient 𝔼 containing a Singer cycle (an element of order 2ⁿ-1). This implies 𝔼 = E \rtimes F, where E ≅ C_2ⁿ-1, F ≤ C_n, and E acts transitively on the elements in D of order 2, and freely on D \backslash \1\. We classify the basic Morita equivalence classes of B over a complete discrete valuation ring O: when m=1, B is basic Morita equivalent to the principal block of one of SL₂(2ⁿ) \rtimes F, D \rtimes 𝔼, or J₁ (where J₁ occurs only when n=3). When m>1, B is basic Morita equivalent to D \rtimes 𝔼.

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