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Involutions on the the Barnes-Wall lattices and their fixed point sublattices, I

Robert L. Griess Jr
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Abstract

We study the sublattices of the rank $2^d$ Barnes-Wall lattices $\bw d$ which occur as fixed points of involutions. They have ranks $2^{d-1}$ (for dirty involutions) or $2^{d-1}\pm 2^{k-1}$ (for clean involutions), where $k$, the defect, is an integer at most $\frac d 2$. We discuss the involutions on $\bw d$ and determine the isometry groups of the fixed point sublattices for all involutions of defect 1. Transitivity results for the Bolt-Room-Wall group on isometry types of sublattices extend those in \cite{bwy}. Along the way, we classify the orbits of $AGL(d,2)$ on the Reed-Muller codes $RM(2,d)$ and describe {\it cubi sequences} for short codewords, which give them as Boolean sums of codimension 2 affine subspaces.

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Robert L. Griess Jr. 2005-11-03. Involutions on the the Barnes-Wall lattices and their fixed point sublattices, I. https://arxiv.org/abs/math/0511084

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