Endotrivial modules for the quaternion group and iterated Jokers in chromatic homotopy theory

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• Andrew Baker (University of Glasgow)
• Friday 31 May 2024, 09:30-10:30
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TRH - Topology, representation theory and higher structures

The Joker is a famous, very singular example of an endotrivial module over the 8-dimension subHopf algebra of the mod 2 Steenrod algebra generated by Sq^{1 and Sq}2. It is known that this can be realised as the cohomology of two distinct Spanier-Whitehead dual spectra. Similarly, the double and iterated double are also realisable, but then the process stops. In the chromatic world, the double versions give rise to objects whose Morava K-theory at height 2 involves endotrivial modules over the quaternion group of order 8 which lives inside the corresponding Morava stabilizer group. This gives a somewhat surprising connection between endotriviality in two different contexts. I will explain how all this works and discuss some possible generalisations to higher chromatic heights.

This talk is part of the Isaac Newton Institute Seminar Series series.

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