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OP-10216Status unassessedNumber theory

Vinberg's Algorithm · Question 1.4

Arithmetic reflection groups and crystallographic packings · AIMPL workshop record

MSC 11 · Number theory

Editorial mapping of AIMPL source category: Number theory. http://aimpl.org/reflectiongpv/ Not an author-supplied MSC code.

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Problem statement

Suppose Γ\Gamma is non-reflective. After running Vinberg's algorithm to find part of the Coxeter polytope PP, we can look for symmetries in the Coxeter diagram and deduce from these some O⁡(q)\operatorname{O}(q) symmetries of the polytope. Can we use these symmetries to find new roots and possibly the infinite-order symmetry?

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Background

Section context

In this section, Γ<O⁡(n,1)\Gamma < \operatorname{O}(n,1) is taken to be an arithmetic lattice. We'll write Γ=R⋊Aut⁡(P)\Gamma = R \rtimes \operatorname{Aut}(P) where RR is generated by reflections and Aut⁡(P)\operatorname{Aut}(P) is the group of symmetries of the Coxeter polytope PP corresponding to RR.

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References
  1. [1]Arithmetic reflection groups and crystallographic packings · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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