Variants of Coxeter quandles associated with Pin groups

Abstract

We study two families of quandles arising from Coxeter quandles. One is the quandle defined by Andruskiewitsch-Graña, which is the set of roots with binary operation defined by using the negatives of reflections. We observe that this is realized as a conjugation quandle in a Pin group. The other, which we call a rotational $D_n$ quandle, is the set of some right angle rotations in the Coxeter group of type $D_n$ with binary operation given by conjugation. We determine their inner automorphism groups, and observe that they are quite similar.