Root systems & Clifford algebras: from symmetries of viruses to E8 & ADE correspondences

In this talk I present a new take on polyhedral symmetries. I begin by describing that many viruses have icosahedrally symmetric surface structures. I briefly review recent work (with Reidun Twarock and Celine Boehm) to try and extend this symmetry principle also to the interior of viruses and carbon onions via suitable notions of affine extensions of non-crystallographic Coxeter groups. I have argued that in such reflection group settings (a vector space with an inner product) Clifford algebras are very natural objects to consider and in fact provide a very simple reflection formula. Applying this framework to root systems has led to the construction of the exceptional root system E8 from the icosahedron and a proof that each 3D root system induces a corresponding 4D root system. In particular, the Trinity of irreducible 3D root systems (A3,B3,H3) gives rise to the Trinity of exceptional 4D root systems (D4,F4,H4). These exceptional root systems can thus be viewed as intrinsically three-dimensional phenomena. The countably infinite family A1×I2(n) gives rise to I2(n)×I2(n). Arnold had found a very cumbersome and indirect connection between (A3,B3,H3) and (D4,F4,H4) essentially via exponents in the Coxeter plane. This in fact extends to my full correspondence between 3D and 4D root systems, establishing an ADE correspondence related to the McKay correspondence. Furthermore, one can fully factorise the Coxeter element in the Clifford algebra with the exponents and complex structures of the eigenplanes arising purely from the geometry, without the need to complexify the real vector space