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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
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
5
votes
Accepted
Eigenvalues for elements of (infinite) Coxeter groups
The eigenvalues of elements of infinite order are certainly not trivial to study,
and as far as I know little has been determined about them.
Keep in mind that arbitrary infinite Coxeter groups are qu …
- 52.9k
3
votes
Does the Poincare series of a Coxeter group always describe a "flag variety"?
Coxeter groups form a very large class of groups defined by generators and relations, whose Poincare series are unlikely to have a common geometric interpretation. However, the Poincare series of an …
- 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
5
votes
0
answers
746
views
Applications (and source) of Bourbaki exercise on root systems with two root lengths?
In Chapters 4-6 of Bourbaki's Groupes et algebres de Lie, Exercise 20 for Section
VI.1 concerns irreducible (reduced) root systems with roots of two lengths: in other
words, systems of types $B_\ell, …
- 52.9k
35
votes
3
answers
2k
views
Is there any need to study Coxeter systems (W,S) with S infinite?
In their treatise Groupes et algebres de Lie, Bourbaki (no doubt heavily influenced by Tits) devoted Chapter IV (1968) to the general theory of what they dubbed "Coxeter systems" $(W,S)$ along with "T …
- 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
2
votes
Accepted
weights and exceptional root systems
EDIT: Reading the question more carefully, I think the difference between the highest weight and an arbitrary Weyl group conjugate will almost never be a single root or muiltiple of a root. (What's …
- 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
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
Spectrum of adjacency matrix of simple Lie algebra.
The formulation is somewhat out of focus, starting with the notation $a_n(x)$ for characteristic polynomial (what is $n$?). The roots indicated do occur in Coxeter's formulation, but not as the eig …
- 52.9k
2
votes
Kazhdan-Luzstig Polynomials and Lower Intervals in the Bruhat Order
The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of c …
- 52.9k
7
votes
Accepted
weyl group representations
Let me add a useful reference book, probably no longer in print but found in many libraries: R.W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley-Interscience, 1985. …
- 52.9k
12
votes
Accepted
Is Soergel's proof of Kazhdan-Lusztig positivity for Weyl groups independent of other proofs?
My understanding is that Soergel's approach applies just to finite Weyl groups and not directly to other finite Coxeter groups (or more generally), since what he can actually prove depends on some of …
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10
votes
Accepted
Representations of finite Coxeter groups
There are many relevant papers, but the most convenient book to consult is:
MR1778802 (2002k:20017) 20C15 (20C08 20F55),
Geck, Meinolf (F-LYON-GD); Pfeiffer,G¨otz (IRL-GLWY)
Characters of finite Coxet …
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