Random endomorphisms of spherical reflection groups
Abstract
The goal of this paper is to understand the set End(W) of endomorphisms of an irreducible spherical reflection group W. We do this in two ways: numerically, by deriving an explicit formula for |End(W)|; and probabilistically, by exploring the question what does a random endomorphism of W look like? For example, we show that as n∞ the probability that a random endomorphism of Wn is an automorphism tends to 12 if Wn=C2n or Dn, to 14 if Wn=C2n+1, and to 1 if Wn=An.
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