If we consider the index $2$ subgroup of a Weyl group consisting of the isometries with determinant $1$ (the 'special' Weyl group), is it known that it is generated by rotations around some fixed axes? For example, it is true that the group of rotational isometries of the hypercube is generated by rotations around the principal (coordinate) axes? Thank you.
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2$\begingroup$ You might consider choosing a more informative title in order to make more people see this question. $\endgroup$– Stefan Kohl ♦Commented Dec 6, 2015 at 16:35
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1$\begingroup$ You should also clarify what do you mean by "rotations around some fixed axes" and "principal (coordinate) axes". $\endgroup$– MishaCommented Dec 6, 2015 at 18:04
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If I had a cube in three dimensional space with points consisting of all point whose coordinates have absolute value equal to one then rotations about the coordinate axes that map the cube to itself would leave the faces invariant. However there are rotations about the corners that leave no face fixed.
In four dimensions rotations do not have fixed axes only fixed points and also some rotations have fixed planes. So I think the answer to your question as written is no.
answered Dec 7, 2015 at 3:06