Polyhedral computation with symmetry
The programs lrs and cdd allow to compute the dual description of a polytope. Unfortunately, in many cases, we cannot apply them for memory and time reasons. Starting from the Adjacency Decomposition method of Christof & Reinelt I progressively build an extremely good algorithm for computing dual description of polytopes with symmetry.
- Perfect forms are prominent object in the geometry and number theory of quadratic form. We computed all perfect forms in dimension 8 and found 10916 of them, a technical feat deemed impossible before. The main challenge was the computation of all facets of a polyhedral cone in dimension 36 with 120 extreme rays. It has about 25,075,566,937,584 facets which fall into 83,092 orbits.
- The Leech lattice is the most remarkable lattice of geometry of numbers. We determine the facets of its contact polytope. There are 1,197,362,269,604,214,277,200 facets in 232 orbits.
L D. Bremner, M. Dutour Sikirić, A. Schuermann, Polyhedral representation conversion up to symmetries, CRM proceedings, volume 48.
L M. Dutour Sikirić, A. Schuermann, F. Vallentin, Classification of eight dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21--32.
L M. Dutour Sikirić, A. Schuermann, F. Vallentin, The contact polytope of the Leech lattice, submitted.