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Lifting of elements of Weyl groups. (English) Zbl 1395.20024
Let $$G$$ be a reductive algebraic group, let $$T\leq G$$ be a Cartan subgroup, let $$N$$ be the nomalizer of $$T$$ in $$G$$ and let $$W=N/T$$ be its corresponding Weyl group. In this paper, the authors discuss the problem of determining the orders of the elements of $$N$$ that are lifts of elements of $$W$$. Whilst it is straightforward to see that an element of $$W$$ of order $$d$$ will lift to an element of order either $$d$$ or $$2d$$, it is harder to determine precisely which holds and it is this question that is addressed here.
The first main theorem of this paper is concerned with the case in which elements lift to elements of the same order, the case of the underlying field having characteristic 2 being very different from the rest. The other two main results of this paper are more technical and focus specifically on the case in which the characteristic of the underlying field is not 2.

MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20G99 Linear algebraic groups and related topics
Keywords:
algebraic groups; Weyl groups; Tits group
Full Text:
References:
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