Dec 2014

The rhombic dodecahedron

SU(4) lattice and Weyl group with all chambers
This is a Mathematica plot of the Euler dual of the truncated cube, a rhombic dodecahedron. It is the fundamental domain for the maximal torus of SU(4). It can be viewed as the union of a cube with 6 square based pyramids, one on each face, with dihedral angle such that the adjacent pyramid faces are coplanar. The dual has 8 triangular faces (one across each vertex of the cube) and 6 square faces - one centred on each face of the cube - yielding 14 vertices of the Weyl object above. The truncated cube has 12 vertices, one for each edge of the cube yielding 12 faces on our Weyl object and they therefore both have 26 edges. The rotation group of the truncated cube preserves the triangular faces so it is isomorphic to the rotation group of the cube, i.e., S_ 4. The black lines denote the intersection of the reflection planes for the Weyl group of SU(4) with the maximal torus.

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The image at the top left is a Mathematica plot of the spectrum of the Dirac operator for the universal cover of SL(2,R). The spectrum is localised over the unit disc in the complex plane which parameterises the Principal Series representations for the group, and the single point of contact between the upper branches and the lower (left and right and right as drawn) illustrates the spectral gap that occurs except at one representation on the boundary of the disc. This is the "limit of discrete series" representation, and corresponds to the fact that the Dirac element is generator in K-theory. The full calculation is given in our paper "The local spectrum of the Dirac operator for the universal cover of SL2(ℝ)" which can be found on ArXiv at http://arxiv.org/abs/1406.0365.