Search Results

Keeping in mind that a generating set of invariant polynomials having the required degrees is not unique, various computations have been recorded in the literature. Those I was aware of before 1990 …

Probably the earliest intrinsic definition of Weyl group occurs in section 1.2 of the groundbreaking paper "Representations of Reductive Groups Over Finite Fields" by Deligne and Lusztig (Ann. of Math …

EDIT: This is a belated attempt (motivated by a question from Yongjun Xu) to answer the question more precisely than I did at first, with more emphasis on careful choice of a Coxeter element when the …

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” reflectio …

As Koen S points out, the longest element of an irreducible Weyl group is treated in an earlier question (in fact, it comes up in several questions). The question asked here presupposes a standard li …

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl group …

The role of the longest element in $W$ emerges only gradually in the Chevalley structure theory. This is developed similarly but in slightly different styles in the three books with the same title Li …

You are asking a number of related questions here, most of which require more reading of Lusztig's papers. See the reference list in my conference paper here, for example. But note first that the n …

It's important to emphasize that the simple group here is typically isomorphic to the rotation subgroup of $W$, which has index 2 and doesn't contain the reflections. So you need to look at the prec …

Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ r …

While looking at a preprint I've just bumped into a question about the longest element $w_0$ of a Weyl group $W$ (say irreducible of a Lie type $A$ - $G$ and of rank $n>1$, to simplify). Suppose t …

It would help to recall Dynkin's definition of "regular semisimple subalgebra" of a complex semisimple Lie algebra (say $\mathfrak{g}$), which has not become standard in later literature. This just …

I'm not sure whether there is an efficient way to answer your question (or a written reference), but it's possible to analyze the situation case-by-case. It's probably best to start with an irreducib …

Rather than overload the question with side remarks, I'll add some background here in community-wiki format to indicate what can be gotten from Springer's relatively elementary treatment of regular el …

First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include …