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Results tagged with coxeter-groups
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user 4231
A Coxeter group is a group defined by a presentation by involutions $r_i$ with relators $(r_ir_j)^{m_{ij}}=1$ for certain family $(m_{ij})$ of integers greater than 1.
2
votes
Coordinates of the Weyl vector of $E_8$ (and the 135 classes of $W(E_8)/W(D_8)$)
Concerning the first question in the header (and some of your preparatory remarks), it's useful to keep in mind the Planche VII for $E_8$ at the end of Chapters 4-6 of Bourbaki's treatise Groupes et a …
- 52.9k
5
votes
Accepted
Recursive formula for inverse Kazhdan-Lusztig polynomials
The case of an affine Weyl group is apparently the only one which has been looked at closely. But it may be hard to answer your specific question. As far as I know, there are two relevant papers, …
- 52.9k
3
votes
Accepted
Reduced decomposition for Weyl group elements which support a Bessel function
I'd be extremely surprised if such tables or database existed, mainly because the number of possible reduced decompositions for a Weyl group element tenda to grow very large as the rank increases. …
- 52.9k
7
votes
1
answer
470
views
Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE...
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 …
- 52.9k
2
votes
Bounding weight multiplicities by number of certain Coxeter elements
Though I can't answer the question directly, it may be helpful to clarify some of the issues here. First, it's usually best to focus on the case when $W$ (or the root system of the Lie algebra) is …
3
votes
Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?
Maybe I can provide a belated kind of answer to my own question, which I came across when looking for something else in the older literature. Vinay Deodhar published a paper in 1990 here (just before …
- 52.9k
4
votes
Fundamental invariants for root subsystems
[EDIT] Maybe it's useful after all this time to give a more complete and uniform answer to both of the questions asked, by referring to Theorem 3.4(i) in Springer's 1974 paper on regular elements of f …
- 52.9k
2
votes
Are the orders of the generators of a group and the product of pairs of thereof enough for t...
As Geoff points out, the answer is likely to be no, though it takes some work to apply the known finite group theory here. It should be emphasized in the question that the group considered is finite …
- 52.9k
2
votes
On the vertices of a Coxeter complex
There are different viewpoints in the literature about what constitutes an "affine Coxeter group" and its "Coxeter complex", so it would be helpful to specify what source you are following here. Co …
- 52.9k
12
votes
0
answers
416
views
Reference for class of involutions containing longest element of finite Coxeter group?
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 …
- 52.9k
1
vote
Centralizers of reflections in special subgroups of Coxeter groups
I've tended to lose track of the basic question here, but it may help to focus just on the situation when $W$ is either a finite Coxeter group or an affine Weyl group (assuming in either case that the …
- 52.9k
4
votes
Accepted
The action of the center on the extended Dynkin diagram
The question is perhaps best answered not in the context of Lie theory but in the related setting of affine Weyl groups, where an irreducible root system in Bourbaki's sense leads to an extended Dynki …
- 52.9k
8
votes
Accepted
How to think about the simple reflection $s_0$ in the affine Weyl group?
You are asking several questions here, so it may be useful to separate out what is going on first in the setting of affine reflection groups. This is independent of the application to algebraic grou …
- 52.9k
6
votes
Embed one Coxeter System into another
It's not clear exactly what your first sentence is asking for (there are a variety of surveys and books). For instance, are you only interested in finite Coxeter groups?
The embeddings of the non …
- 52.9k
13
votes
Accepted
Coxeter groups - Parabolic subgroups
The history is definitely somewhat convoluted. Note first that the term "Coxeter group" itself was introduced by Bourbaki in their 1968 volume containing chapters 4-6 of Groupes et algebres de Lie. …
- 52.9k