TY - CONF

T1 - Exceptional uniform polytopes of the E_6, E_7 and E_8 symmetry types

AU - Migrin, Victor

AU - Vavilov, Nikolai

PY - 2021/4/18

Y1 - 2021/4/18

N2 - The goal of this project is to review the structure and geometry of the nine Gosset—Elte uniform polytopes in dimensions 6 through 8, of exceptional symmetry types E6, E7 and E8. These polytopes were extensively studied by Coxeter, Conway, Sloane, Moody, Patera, McMullen, and many other remarkable mathematicians. We develop a new easier approach towards their combinatorial and geometric properties. In particular, we propose a new way to describe the faces of these polytopes, and their adjacencies, inscribed subpolytopes, compounds, independent subsets, foldings, and the like. Our main tools — weight diagrams, description of root subsystems and conjugacy classes of the Weyl group — are elementary and standard in the representation theory of algebraic groups. But we believe their specific use in the study of polytopes might be new, and considerably simplifies computations. As an illustration of our methods that seems to be new, we calculate the cycle indices for the actions of the Weyl groups on the faces of these polytopes. With our tools, this can be done by hand in the easier cases, such as the Schläfli and Hesse polytopes for E6 and E7. Nevertheless, the senior polytopes and the case of E8 require the use of computers anyway, even after all possible simplifications. Since our actual new results are mostly of technical and/or computational nature, the talk itself will be mostly expository, explaining the background and the basic ideas of our approach, and presenting much easier proofs of the classical results.

AB - The goal of this project is to review the structure and geometry of the nine Gosset—Elte uniform polytopes in dimensions 6 through 8, of exceptional symmetry types E6, E7 and E8. These polytopes were extensively studied by Coxeter, Conway, Sloane, Moody, Patera, McMullen, and many other remarkable mathematicians. We develop a new easier approach towards their combinatorial and geometric properties. In particular, we propose a new way to describe the faces of these polytopes, and their adjacencies, inscribed subpolytopes, compounds, independent subsets, foldings, and the like. Our main tools — weight diagrams, description of root subsystems and conjugacy classes of the Weyl group — are elementary and standard in the representation theory of algebraic groups. But we believe their specific use in the study of polytopes might be new, and considerably simplifies computations. As an illustration of our methods that seems to be new, we calculate the cycle indices for the actions of the Weyl groups on the faces of these polytopes. With our tools, this can be done by hand in the easier cases, such as the Schläfli and Hesse polytopes for E6 and E7. Nevertheless, the senior polytopes and the case of E8 require the use of computers anyway, even after all possible simplifications. Since our actual new results are mostly of technical and/or computational nature, the talk itself will be mostly expository, explaining the background and the basic ideas of our approach, and presenting much easier proofs of the classical results.

KW - полуправильные многогранники

KW - многогранники Госсета-Эльте

KW - корневые многогранники

KW - весовые диаграммы

KW - исключительные системы корней

M3 - Abstract

Y2 - 19 April 2021 through 24 April 2021

ER -