English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We show that the projectivized complex reflection group $\Gamma$ of theunique $(1+i)$-modular Hermitian $\mathbb{Z}[i]$-module of signature $(9,1)$ isa new arithmetic reflection group in $PU(9,1)$. We find $32$ complexreflections of order four generating $\Gamma$. The mirrors of these $32$reflections form the vertices of a sort of Coxeter-Dynkin diagram $D$ for$\Gamma$ that encode Coxeter-type generators and relations for $\Gamma$. Thevertices of $D$ can be indexed by sixteen points and sixteen affine hyperplanesin $\mathbb{F}_2^4$. The edges of $D$ are determined by the finite geometry ofthese points and hyperplanes. The group of automorphisms of the diagram $D$ is$2^4 \colon (2^3 \colon L_3(2)) \colon 2$. This group transitively permutes the$32$ mirrors of generating reflections and fixes an unique point $\tau$ in$\mathbb{C} H^9$. These $32$ mirrors are precisely the mirrors closest to$\tau$. These results are strikingly similar to the results satisfied by thecomplex hyperbolic reflection group at the center of Allcock's monstrousproposal.

A new complex reflection group in $PU(9,1)$ and the Barnes-Wall lattice