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Forthcoming research seminars

Maths HEP Lunchtime Seminars: From E_8 to viruses - affine extensions of (non-crystallographic) Coxeter groups

Motivated by recent results in mathematical virology, we present novel
asymmetric Z[tau]-integer-valued affine extensions of the
non-crystallographic Coxeter groups H_2, H_3 and H_4 derived in a
Kac-Moody-type formalism. In particular, we show that the affine
reflection planes which extend the Coxeter group H_3 generate (twist)
translations along 2-, 3- and 5-fold axes of icosahedral symmetry, and we
classify these translations in terms of the Fibonacci recursion relation
applied to different start values. We thus provide an explanation of
previous results concerning affine extensions of icosahedral symmetry in a
Coxeter group context, and extend this analysis to the case of the
non-crystallographic Coxeter groups H_2 and H_4. These results will
enable new applications of group theory in physics (quasicrystals), biology
(viruses) and chemistry (fullerenes).

We furthermore show how such affine extensions of non-crystallographic
Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and
projections of the more familiar affine extensions of the root systems
E_8, D_6 and A_4. This broader and more conventional context of
crystallographic lattices (such as E_8) suggests potential for
applications in high energy physics, integrable systems and modular form
theory. By inverting the projection, we make the case for admitting
different number fields in the Cartan matrix, which could open up enticing
possibilities in hyperbolic geometry and rational conformal field theory.

Contact tomas.andrade@durham.ac.uk or jonathan.pearson@durham.ac.uk or mukund.rangamani@durham.ac.uk or b.s.withers@durham.ac.uk for more information