On Twisted Group Algebras of Barnes-Wall Lattices and their Extraspecial Automorphisms

Abstract

Let $m$ be an integer $m\geq 1$ and write $BW_m$ for the Barnes-Wall lattice in $\mathbb{R}^{2^m}$. $BW_m$ possesses a rich symmetry group, a subgroup of which is the extraspecial $2$-group of + type, denoted $E(m)$. One of the main results presented in this dissertation is a theorem that demonstrates for $m\geq 2$, that the action of $E(m)$ on $BW_m$ extends to an action on the associated twisted group algebra $\mathbb{C}_\epsilon[BW_m]$. Additionally, this dissertation establishes that a certain projective representations of the automorphism group of $BW_3/2BW_3, BW_5/2BW_5$, and $BW_7/2BW_7$ can, in fact, be realized as an ordinary representation.