Colourings of exceptional uniform polytopes of types $\mathrm{E}_6$ and $\mathrm{E}_7$

N. Vavilov   ; V. Migrin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXIV, Tome 517 (2022), pp. 36-54

Résumé

We compute the cycle indices of the Weyl group $W(\mathrm{E}_6)$ in its action on the vertices of the Schläli polytope $(\mathrm{E}_6, \varphi_1)$ and of the Weyl group $W(\mathrm{E}_7)$ in its action on the vertices of the Hesse polytope $(\mathrm{E}_7, \varphi_7)$. This is done purely by hand using the following visual aids – weight diagrams of the corresponding representations to encode the action of the Weyl groups on the polytopes, and the enhanced Dynkin diagrams of the corresponding root systems to encode the conjugacy classes of the Weyl groups themselves, in the style of Carter and Stekolshchik.

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